diff --git a/books/bookvol10.3.pamphlet b/books/bookvol10.3.pamphlet index 3d20015..b306b68 100644 --- a/books/bookvol10.3.pamphlet +++ b/books/bookvol10.3.pamphlet @@ -21873,6 +21873,15 @@ complex number.) \end{quote} + +A key difficulty is creating test suites which generate DoubleFloats. +We need to be bit-exact in the results. This can be achieved using the +common lisp {\tt integer-decode-float} function. We cover that function +in two different ways, with {\tt machineFraction} which translates the +results into an Axiom {\bf Fraction(Integer)} and {\tt integerDecode} +which just returns a list of the multiple values from the +{\tt integer-decode-float} function. + <>= )set break resume )sys rm -f DoubleFloat.output @@ -21880,7 +21889,8 @@ complex number.) )set message test on )set message auto off )clear all ---S 1 of 10 + +--S 1 of 13 2.71828 --R --R @@ -21888,7 +21898,7 @@ complex number.) --R Type: Float --E 1 ---S 2 of 10 +--S 2 of 13 2.71828@DoubleFloat --R --R @@ -21896,7 +21906,7 @@ complex number.) --R Type: DoubleFloat --E 2 ---S 3 of 10 +--S 3 of 13 2.71828 :: DoubleFloat --R --R @@ -21904,7 +21914,7 @@ complex number.) --R Type: DoubleFloat --E 3 ---S 4 of 10 +--S 4 of 13 eApprox : DoubleFloat := 2.71828 --R --R @@ -21912,13 +21922,13 @@ eApprox : DoubleFloat := 2.71828 --R Type: DoubleFloat --E 4 ---S 5 of 10 +--S 5 of 13 avg : List DoubleFloat -> DoubleFloat --R --R Type: Void --E 5 ---S 6 of 10 +--S 6 of 13 avg l == empty? l => 0 :: DoubleFloat reduce(_+,l) / #l @@ -21926,7 +21936,7 @@ avg l == --R Type: Void --E 6 ---S 7 of 10 +--S 7 of 13 avg [] --R --R Compiling function avg with type List DoubleFloat -> DoubleFloat @@ -21935,7 +21945,7 @@ avg [] --R Type: DoubleFloat --E 7 ---S 8 of 10 +--S 8 of 13 avg [3.4,9.7,-6.8] --R --R @@ -21943,7 +21953,7 @@ avg [3.4,9.7,-6.8] --R Type: DoubleFloat --E 8 ---S 9 of 10 +--S 9 of 13 cos(3.1415926)$DoubleFloat --R --R @@ -21951,15 +21961,43 @@ cos(3.1415926)$DoubleFloat --R Type: DoubleFloat --E 9 ---S 10 of 10 +--S 10 of 13 cos(3.1415926 :: DoubleFloat) --R --R --R (10) - 0.99999999999999856 --R Type: DoubleFloat --E 10 + +--S 11 of 13 +a:DFLOAT := -1.0/3.0 +--R +--R +--R (11) - 0.33333333333333331 +--R Type: DoubleFloat +--E 11 + +--S 12 of 13 +integerDecode a +--R +--R +--R (12) [6004799503160661,- 54,- 1] +--R Type: List Integer +--E 12 + +--S 13 of 13 +machineFraction a +--R +--R +--R 6004799503160661 +--R (13) - ----------------- +--R 18014398509481984 +--R Type: Fraction Integer +--E 13 + )spool )lisp (bye) + @ <>= @@ -22044,6 +22082,33 @@ arguments themselves are already of type DoubleFloat. By far, the most common usage of DoubleFloat is for functions to be graphed. +You can get the exact machine-specific bits of a DoubleFloat in two ways. +The first is to use the integerDecode function to break the DoubleFloat +into components. + + a := -1.0/3.0 + + -0.33333333333333331 + + integerDecode a + + [6004799503160661,- 54,- 1] + +This is the mantissa, exact to 54 bits, the power of 2, and the sign. +Thus it is: + + 6004799503160661 * 2^-54 + +You can get the same information as a fraction with + + machineFraction a + + 6004799503160661 + - ----------------- + 18014398509481984 + +where the denominator 18014398509481984 is 2^54 + See Also: 0 )help Float @@ -22113,68 +22178,70 @@ o )show DoubleFloat \cross{DFLOAT}{gcdPolynomial} \\ \cross{DFLOAT}{hash} & \cross{DFLOAT}{increasePrecision} & -\cross{DFLOAT}{inv} & -\cross{DFLOAT}{latex} \\ +\cross{DFLOAT}{integerDecode} & +\cross{DFLOAT}{inv} \\ +\cross{DFLOAT}{latex} & \cross{DFLOAT}{lcm} & \cross{DFLOAT}{log} & -\cross{DFLOAT}{log10} & -\cross{DFLOAT}{log2} \\ +\cross{DFLOAT}{log10} \\ +\cross{DFLOAT}{log2} & +\cross{DFLOAT}{machineFraction} & \cross{DFLOAT}{mantissa} & -\cross{DFLOAT}{max} & +\cross{DFLOAT}{max} \\ \cross{DFLOAT}{min} & -\cross{DFLOAT}{multiEuclidean} \\ +\cross{DFLOAT}{multiEuclidean} & \cross{DFLOAT}{negative?} & -\cross{DFLOAT}{norm} & +\cross{DFLOAT}{norm} \\ \cross{DFLOAT}{nthRoot} & -\cross{DFLOAT}{OMwrite} \\ +\cross{DFLOAT}{OMwrite} & \cross{DFLOAT}{one?} & -\cross{DFLOAT}{order} & +\cross{DFLOAT}{order} \\ \cross{DFLOAT}{patternMatch} & -\cross{DFLOAT}{pi} \\ +\cross{DFLOAT}{pi} & \cross{DFLOAT}{polygamma} & -\cross{DFLOAT}{positive?} & +\cross{DFLOAT}{positive?} \\ \cross{DFLOAT}{precision} & -\cross{DFLOAT}{prime?} \\ +\cross{DFLOAT}{prime?} & \cross{DFLOAT}{principalIdeal} & -\cross{DFLOAT}{rationalApproximation} & +\cross{DFLOAT}{rationalApproximation} \\ \cross{DFLOAT}{recip} & -\cross{DFLOAT}{retract} \\ +\cross{DFLOAT}{retract} & \cross{DFLOAT}{retractIfCan} & -\cross{DFLOAT}{round} & +\cross{DFLOAT}{round} \\ \cross{DFLOAT}{sample} & -\cross{DFLOAT}{sec} \\ +\cross{DFLOAT}{sec} & \cross{DFLOAT}{sech} & -\cross{DFLOAT}{sign} & +\cross{DFLOAT}{sign} \\ \cross{DFLOAT}{sin} & -\cross{DFLOAT}{sinh} \\ +\cross{DFLOAT}{sinh} & \cross{DFLOAT}{sizeLess?} & -\cross{DFLOAT}{sqrt} & +\cross{DFLOAT}{sqrt} \\ \cross{DFLOAT}{squareFree} & -\cross{DFLOAT}{squareFreePart} \\ +\cross{DFLOAT}{squareFreePart} & \cross{DFLOAT}{subtractIfCan} & -\cross{DFLOAT}{tan} & +\cross{DFLOAT}{tan} \\ \cross{DFLOAT}{tanh} & -\cross{DFLOAT}{truncate} \\ +\cross{DFLOAT}{truncate} & \cross{DFLOAT}{unit?} & -\cross{DFLOAT}{unitCanonical} & +\cross{DFLOAT}{unitCanonical} \\ \cross{DFLOAT}{unitNormal} & -\cross{DFLOAT}{wholePart} \\ +\cross{DFLOAT}{wholePart} & \cross{DFLOAT}{zero?} & -\cross{DFLOAT}{?*?} & +\cross{DFLOAT}{?*?} \\ \cross{DFLOAT}{?**?} & -\cross{DFLOAT}{?+?} \\ +\cross{DFLOAT}{?+?} & \cross{DFLOAT}{?-?} & -\cross{DFLOAT}{-?} & +\cross{DFLOAT}{-?} \\ \cross{DFLOAT}{?/?} & -\cross{DFLOAT}{?$<$?} \\ +\cross{DFLOAT}{?$<$?} & \cross{DFLOAT}{?$<=$?} & -\cross{DFLOAT}{?=?} & +\cross{DFLOAT}{?=?} \\ \cross{DFLOAT}{?$>$?} & -\cross{DFLOAT}{?$>=$?} \\ +\cross{DFLOAT}{?$>=$?} & \cross{DFLOAT}{?\^{}?} & -\cross{DFLOAT}{?quo?} & +\cross{DFLOAT}{?quo?} \\ \cross{DFLOAT}{?rem?} & -\cross{DFLOAT}{?\~{}=?} +\cross{DFLOAT}{?\~{}=?} && \end{tabular} <>= @@ -22235,10 +22302,29 @@ DoubleFloat(): Join(FloatingPointSystem, DifferentialRing, OpenMath, rationalApproximation: (%, NonNegativeInteger) -> Fraction Integer ++ rationalApproximation(f, n) computes a rational approximation ++ r to f with relative error \spad{< 10**(-n)}. - rationalApproximation: (%, NonNegativeInteger, NonNegativeInteger) -> Fraction Integer - ++ rationalApproximation(f, n, b) computes a rational - ++ approximation r to f with relative error \spad{< b**(-n)} - ++ (that is, \spad{|(r-f)/f| < b**(-n)}). + rationalApproximation: (%, NonNegativeInteger, NonNegativeInteger) -> _ + Fraction Integer + ++ rationalApproximation(f, n, b) computes a rational + ++ approximation r to f with relative error \spad{< b**(-n)} + ++ (that is, \spad{|(r-f)/f| < b**(-n)}). + machineFraction : % -> Fraction Integer + ++ machineFraction(x) returns a bit-exact fraction of the machine + ++ floating point number using the common lisp integer-decode-float + ++ function. See Steele, ISBN 0-13-152414-3 p354 + ++ This function can be used to print results which do not depend + ++ on binary-to-decimal conversions + ++ + ++X a:DFLOAT:=-1.0/3.0 + ++X machineFraction a + integerDecode : % -> List Integer + ++ integerDecode(x) returns the multiple values of the common + ++ lisp integer-decode-float function. + ++ See Steele, ISBN 0-13-152414-3 p354. This function can be used + ++ to ensure that the results are bit-exact and do not depend on + ++ the binary-to-decimal conversions. + ++ + ++X a:DFLOAT:=-1.0/3.0 + ++X integerDecode a == add format: String := "~G" @@ -22292,6 +22378,18 @@ DoubleFloat(): Join(FloatingPointSystem, DifferentialRing, OpenMath, -- complex to get the correct behaviour. --makeComplex(x: %):% == COMPLEX(x, 0$%)$Lisp + machineFraction(df:%):Fraction(Integer) == + numer:Integer:=INTEGER_-DECODE_-FLOAT_-NUMERATOR(df)$Lisp + denom:Integer:=INTEGER_-DECODE_-FLOAT_-DENOMINATOR(df)$Lisp + sign:Integer:=INTEGER_-DECODE_-FLOAT_-SIGN(df)$Lisp + sign*numer/denom + + integerDecode(df:%):List(Integer) == + numer:Integer:=INTEGER_-DECODE_-FLOAT_-NUMERATOR(df)$Lisp + exp:Integer:=INTEGER_-DECODE_-FLOAT_-EXPONENT(df)$Lisp + sign:Integer:=INTEGER_-DECODE_-FLOAT_-SIGN(df)$Lisp + [numer,exp,sign] + base() == FLOAT_-RADIX(0$%)$Lisp mantissa x == manexp(x).MANTISSA exponent x == manexp(x).EXPONENT diff --git a/books/bookvol5.pamphlet b/books/bookvol5.pamphlet index 59b112a..404e339 100644 --- a/books/bookvol5.pamphlet +++ b/books/bookvol5.pamphlet @@ -29028,6 +29028,41 @@ See Steele Common Lisp 1990 p308 @ +\defun{integer-decode-float-numerator}{Machine specific float numerator} +This is used in the DoubleFloat integerDecode function +<>= +(defun integer-decode-float-numerator (x) + (integer-decode-float x)) + +@ + +\defun{integer-decode-float-denominator}{Machine specific float denominator} +This is used in the DoubleFloat integerDecode function +<>= +(defun integer-decode-float-denominator (x) + (multiple-value-bind (mantissa exponent sign) (integer-decode-float x) + (expt 2 (abs exponent)))) + +@ + +\defun{integer-decode-float-sign}{Machine specific float sign} +This is used in the DoubleFloat integerDecode function +<>= +(defun integer-decode-float-sign (x) + (multiple-value-bind (mantissa exponent sign) (integer-decode-float x) + sign)) + +@ + +\defun{integer-decode-float-exponent}{Machine specific float bit length} +This is used in the DoubleFloat integerDecode function +<>= +(defun integer-decode-float-exponent (x) + (multiple-value-bind (mantissa exponent sign) (integer-decode-float x) + exponent)) + +@ + \defun{manexp}{Decode floating-point values} This function is used by DoubleFloat to implement the ``mantissa'' and ``exponent'' functions. @@ -29473,6 +29508,10 @@ See Steele Common Lisp 1990 pp305-307 <> <> <> +<> +<> +<> +<> <> <> <> diff --git a/changelog b/changelog index c389367..8949c10 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,7 @@ +20090101 tpd src/axiom-website/patches.html 20090101.01.tpd.patch +20100101 tpd src/input/ei.input rewrite using machineFraction +20100101 tpd books/bookvol5 support dfloat machineFraction, integerDecode +20100101 tpd books/bookvol10.3 add dfloat machineFraction, integerDecode 20091231 tpd src/axiom-website/patches.html 20091231.02.tpd.patch 20091231 tpd src/interp/i-syscmd.lisp remove dead code 20091231 tpd books/bookvol5 add banner and cleanup globals diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 97562f1..34c102f 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -2346,5 +2346,7 @@ books/bookvol5 treeshake i-syscmd
books/bookvol5 treeshake i-syscmd
20091231.01.tpd.patch books/bookvol5 add banner and cleanup globals
+20090101.01.tpd.patch +books/bookvol10.3 add dfloat machineFraction, integerDecode
diff --git a/src/input/ei.input.pamphlet b/src/input/ei.input.pamphlet index 04bed32..941a023 100644 --- a/src/input/ei.input.pamphlet +++ b/src/input/ei.input.pamphlet @@ -40,7 +40,7 @@ with the scaling factor $t=(-20/x)-1$ \begin{chunk}{*} --S 2 of 20 -aChebyshev:=_ +aChebyshev:List Fraction Integer:=[machineFraction (x::DFLOAT) for x in _ [0.191217322586055345391519326510E1,_ -0.420835505286848437550974986680E-01,_ 0.172281962728432678337118157835E-02,_ @@ -81,54 +81,147 @@ aChebyshev:=_ -0.165272509838212659649744302314E-28,_ 0.459223035877302702795636377166E-29,_ -0.129006276721326384737453212670E-29,_ - 0.366271848103200259081177078922E-30] + 0.366271848103200259081177078922E-30]] --R --R --R (3) ---R [1.9121732258 6055345391 51932651, - 0.0420835505 2868484375 5097498668, ---R 0.0017228196 2728432678 3371181578 35, ---R - 0.0000991578 2173444563 6455984232 2973, ---R 0.0000071760 9316802277 5052655906 65592, ---R - 0.6152733145 0951269682 7956791331 E -6, ---R 0.6024857106 5627583129 399970161 E -7, ---R - 0.6573848845 2883048229 5894189637 E -8, ---R 0.7853167541 8323998199 4810079871 E -9, ---R - 0.1013730288 0038789855 4202774257 E -9, ---R 0.1399770413 2267686027 7823488623 E -10, ---R - 0.2051008376 7838189961 8962318711 E -11, ---R 0.3168388726 0024778181 4907985818 E -12, ---R - 0.5132760082 8391806541 5984751899 E -13, ---R 0.8680933040 7665493418 7433687383 E -14, ---R - 0.1527015040 9030849719 8572355351 E -14, ---R 0.2784686251 6493573965 0105251453 E -15, ---R - 0.5249890437 4217669680 8472933696 E -16, ---R 0.1020717991 2485612924 7455787226 E -16, ---R - 0.2042264679 8997184130 8462421876 E -17, ---R 0.4197064172 7264847440 8827228562 E -18, ---R - 0.8844508176 1728105081 6483737536 E -19, ---R 0.1908272629 5947174199 5060168262 E -19, ---R - 0.4209746222 9351995033 6450865676 E -20, ---R 0.9483904058 1983732764 1500214512 E -21, ---R - 0.2179467860 1366743199 4032574014 E -21, ---R 0.5103936869 0714509499 3452562741 E -22, ---R - 0.1216883113 3344150908 9746779693 E -22, ---R 0.2951289166 4478751929 4773757144 E -23, ---R - 0.7275353763 7728468971 443895092 E -24, ---R 0.1821639048 6230739612 1667115976 E -24, ---R - 0.4629629963 1633171661 2753482064 E -25, ---R 0.1193539790 9715779152 3052371292 E -25, ---R - 0.3119493285 2201424493 1062147473 E -26, ---R 0.8261419734 5334664228 4170028518 E -27, ---R - 0.2215803373 6609829830 2591177697 E -27, ---R 0.6016031671 6542638904 5303124429 E -28, ---R - 0.1652725098 3821265964 9744302314 E -28, ---R 0.4592230358 7730270279 5636377166 E -29, ---R - 0.1290062767 2132638473 745321267 E -29, ---R 0.3662718481 0320025908 1177078922 E -30] ---R Type: List Float +--R 8611662627453427 379054924958825 7945103187419413 +--R [----------------, - ----------------, -------------------, +--R 4503599627370496 9007199254740992 4611686018427387904 +--R 1829138960441833 4236017731827575 5811092156487673 +--R - --------------------, ---------------------, - ----------------------, +--R 18446744073709551616 590295810358705651712 9444732965739290427392 +--R 4552253322257753 7947295603312847 +--R -----------------------, - -------------------------, +--R 75557863725914323419136 1208925819614629174706176 +--R 7595117605664589 3921679101738443 +--R -------------------------, - --------------------------, +--R 9671406556917033397649408 38685626227668133590597632 +--R 4332079600851147 5078050781060809 +--R ---------------------------, - ----------------------------, +--R 309485009821345068724781056 2475880078570760549798248448 +--R 6275640422302181 8133182999798229 +--R -----------------------------, - ------------------------------, +--R 19807040628566084398385987584 158456325028528675187087900672 +--R 687774373729299 7742886132633319 +--R -----------------------------, - -------------------------------, +--R 79228162514264337593543950336 5070602400912917605986812821504 +--R 5648014717360969 8518434258086585 +--R --------------------------------, - ---------------------------------, +--R 20282409603651670423947251286016 162259276829213363391578010288128 +--R 6624838524262355 5302022240869971 +--R ---------------------------------, - ----------------------------------, +--R 649037107316853453566312041152512 2596148429267413814265248164610048 +--R 8512657468405 7347729922948429 +--R --------------------------------, - -----------------------------------, +--R 20282409603651670423947251286016 83076749736557242056487941267521536 +--R 6341323507118493 +--R ------------------------------------, +--R 332306998946228968225951765070086144 +--R 2797856267337637 +--R - ------------------------------------, +--R 664613997892457936451903530140172288 +--R 5042508313398181 +--R -------------------------------------, +--R 5316911983139663491615228241121378304 +--R 4635215512971377 +--R - --------------------------------------, +--R 21267647932558653966460912964485513216 +--R 4341949296056693 +--R --------------------------------------, +--R 85070591730234615865843651857942052864 +--R 8281677321431107 +--R - ---------------------------------------, +--R 680564733841876926926749214863536422912 +--R 8034173304216051 +--R ----------------------------------------, +--R 2722258935367507707706996859454145691648 +--R 990269839569513 +--R - ----------------------------------------, +--R 1361129467683753853853498429727072845824 +--R 3967178541702823 +--R -----------------------------------------, +--R 21778071482940061661655974875633165533184 +--R 4032976491093275 +--R - -----------------------------------------, +--R 87112285931760246646623899502532662132736 +--R 2079439590840989 +--R ------------------------------------------, +--R 174224571863520493293247799005065324265472 +--R 8695878112777701 +--R - -------------------------------------------, +--R 2787593149816327892691964784081045188247552 +--R 4605895411948583 +--R -------------------------------------------, +--R 5575186299632655785383929568162090376495104 +--R 1235351661151453 +--R - -------------------------------------------, +--R 5575186299632655785383929568162090376495104 +--R 5366479576634079 +--R --------------------------------------------, +--R 89202980794122492566142873090593446023921664 +--R 5897120208357801 +--R - ---------------------------------------------, +--R 356811923176489970264571492362373784095686656 +--R 6554250183933065 +--R ----------------------------------------------, +--R 1427247692705959881058285969449495136382746624 +--R 3682478215901993 +--R - ----------------------------------------------, +--R 2854495385411919762116571938898990272765493248 +--R 4182085200867523 +--R -----------------------------------------------] +--R 11417981541647679048466287755595961091061972992 +--R Type: List Fraction Integer --E 2 \end{chunk} + +The actual tables value from above are: +\begin{verbatim} + 1.9121732258 6055345391 51932651 + -0.0420835505 2868484375 5097498668 + 0.0017228196 2728432678 3371181578 35 + -0.0000991578 2173444563 6455984232 2973 + 0.0000071760 9316802277 5052655906 65592 + -0.6152733145 0951269682 7956791331 E -6 + 0.6024857106 5627583129 399970161 E -7 + -0.6573848845 2883048229 5894189637 E -8 + 0.7853167541 8323998199 4810079871 E -9 + -0.1013730288 0038789855 4202774257 E -9 + 0.1399770413 2267686027 7823488623 E -10 + -0.2051008376 7838189961 8962318711 E -11 + 0.3168388726 0024778181 4907985818 E -12 + -0.5132760082 8391806541 5984751899 E -13 + 0.8680933040 7665493418 7433687383 E -14 + -0.1527015040 9030849719 8572355351 E -14 + 0.2784686251 6493573965 0105251453 E -15 + -0.5249890437 4217669680 8472933696 E -16 + 0.1020717991 2485612924 7455787226 E -16 + -0.2042264679 8997184130 8462421876 E -17 + 0.4197064172 7264847440 8827228562 E -18 + -0.8844508176 1728105081 6483737536 E -19 + 0.1908272629 5947174199 5060168262 E -19 + -0.4209746222 9351995033 6450865676 E -20 + 0.9483904058 1983732764 1500214512 E -21 + -0.2179467860 1366743199 4032574014 E -21 + 0.5103936869 0714509499 3452562741 E -22 + -0.1216883113 3344150908 9746779693 E -22 + 0.2951289166 4478751929 4773757144 E -23 + -0.7275353763 7728468971 443895092 E -24 + 0.1821639048 6230739612 1667115976 E -24 + -0.4629629963 1633171661 2753482064 E -25 + 0.1193539790 9715779152 3052371292 E -25 + -0.3119493285 2201424493 1062147473 E -26 + 0.8261419734 5334664228 4170028518 E -27 + -0.2215803373 6609829830 2591177697 E -27 + 0.6016031671 6542638904 5303124429 E -28 + -0.1652725098 3821265964 9744302314 E -28 + 0.4592230358 7730270279 5636377166 E -29 + -0.1290062767 2132638473 745321267 E -29 + 0.3662718481 0320025908 1177078922 E -30 +\end{verbatim} + In the following table there are 4 columns. The first column is the argument of Ei(x) shown in Table 4 in \cite{1}. The second column is the exact value shown in the table. Column 3 is the @@ -136,7 +229,10 @@ value returned by Axiom and column 4 is the difference. See special.spad.dvi for details. \begin{chunk}{*} + --S 3 of 20 +t1:List List Fraction Integer:= [[machineFraction (x::DFLOAT) for x in line]_ + for line in _ [[-160.,0.993826695674061273878797850088,_ Ei1(-160.0),Ei1(-160.0)-0.993826695674061273878797850088],_ [-80.0,0.987801333094288773564522608410,_ @@ -168,53 +264,127 @@ See special.spad.dvi for details. [-10.0-2.0/3.0,0.920170654249445676202148012149,_ Ei1(-10.0-2.0/3.0),Ei1(-10.0-2.0/3.0)-0.920170654249445676202148012149],_ [-10.0,0.915633339397880818760698157666,_ - Ei1(-10.0),Ei1(-10.0)-0.915633339397880818760698157666]] + Ei1(-10.0),Ei1(-10.0)-0.915633339397880818760698157666]]] --R --R --R (4) ---R [[- 160.,0.99382669567406123,0.99382669567406123,0.], ---R [- 80.,0.98780133309428875,0.98780133309428886,1.1102230246251565E-16], ---R [- 53.333333333333336,0.98191629014319448,0.98191629014319448,0.], ---R [- 40.,0.97616460318514309,0.97616460318514309,0.], ---R [- 32.,0.97053988407466396,0.97053988407466363,- 3.3306690738754696E-16], ---R [- 26.666666666666668,0.96503625112337699,0.96503625112337699,0.], ---R ---R [- 22.857142857142858, 0.95964827107936723, 0.95964827107936734, ---R 1.1102230246251565E-16] ---R , ---R [- 20.,0.9543709099192168,0.95437090991921691,1.1102230246251565E-16], ---R [- 17.777777777777779,0.94919949077974575,0.94919949077974575,0.], ---R [- 16.,0.94412965773690294,0.94412965773690294,0.], ---R ---R [- 14.545454545454545, 0.93915734441928422, 0.93915734441928411, ---R - 1.1102230246251565E-16] ---R , ---R ---R [- 13.333333333333334, 0.93427874665341049, 0.9342787466534106, ---R 1.1102230246251565E-16] ---R , ---R ---R [- 12.307692307692308, 0.92949029849721398, 0.92949029849721387, ---R - 1.1102230246251565E-16] ---R , ---R [- 11.428571428571429,0.92478865114084174,0.92478865114084174,0.], ---R ---R [- 10.666666666666666, 0.92017065424944566, 0.92017065424944577, ---R 1.1102230246251565E-16] ---R , ---R [- 10.,0.91563333939788083,0.91563333939788094,1.1102230246251565E-16]] ---R Type: List List OnePointCompletion DoubleFloat +--R 8951595072617107 4475797536308553 1 +--R [[- 160,----------------,----------------,- ----------------], +--R 9007199254740992 4503599627370496 9007199254740992 +--R 2224330857819759 2224330857819759 +--R [- 80,----------------,----------------,0], +--R 2251799813685248 2251799813685248 +--R 3752999689475413 2211078919198955 2211078919198955 +--R [- ----------------,----------------,----------------,0], +--R 70368744177664 2251799813685248 2251799813685248 +--R 2198127271578439 2198127271578439 +--R [- 40,----------------,----------------,0], +--R 2251799813685248 2251799813685248 +--R 8741846120533721 4370923060266859 3 +--R [- 32,----------------,----------------,- ----------------], +--R 9007199254740992 4503599627370496 9007199254740992 +--R 3752999689475413 4346136900958261 8692273801916521 1 +--R [- ----------------,----------------,----------------,- ----------------], +--R 140737488355328 4503599627370496 9007199254740992 9007199254740992 +--R 3216856876693211 4321871596039779 4321871596039779 +--R [- ----------------,----------------,----------------,0], +--R 140737488355328 4503599627370496 4503599627370496 +--R 2149052237142713 2149052237142713 +--R [- 20,----------------,----------------,0], +--R 2251799813685248 2251799813685248 +--R 5003999585967217 4274814472975927 4274814472975927 +--R [- ----------------,----------------,----------------,0], +--R 281474976710656 4503599627370496 4503599627370496 +--R 2125990987386675 8503963949546699 1 +--R [- 16,----------------,----------------,- ----------------], +--R 2251799813685248 9007199254740992 9007199254740992 +--R 8188362958855447 4229588666368953 528698583296119 1 +--R [- ----------------,----------------,---------------,- ----------------], +--R 562949953421312 4503599627370496 562949953421312 4503599627370496 +--R 3752999689475413 4207617415288473 8415234830576947 1 +--R [- ----------------,----------------,----------------,----------------], +--R 281474976710656 4503599627370496 9007199254740992 9007199254740992 +--R 866076851417403 65407065030571 4186052161956543 1 +--R [- ---------------,--------------,----------------,- ----------------], +--R 70368744177664 70368744177664 4503599627370496 4503599627370496 +--R 3216856876693211 2082438912337179 2082438912337179 +--R [- ----------------,----------------,----------------,0], +--R 281474976710656 2251799813685248 2251799813685248 +--R 6004799503160661 4144080215595069 4144080215595069 +--R [- ----------------,----------------,----------------,0], +--R 562949953421312 4503599627370496 4503599627370496 +--R 8247291932240597 4123645966120299 1 +--R [- 10,----------------,----------------,----------------]] +--R 9007199254740992 4503599627370496 9007199254740992 +--R Type: List List Fraction Integer --E 3 \end{chunk} + +The actual table values from above are: +\begin{verbatim} + + [-160., 0.99382669567406123, + 0.99382669567406123, + 0.], + [-80., 0.98780133309428875, + 0.98780133309428886, + 1.1102230246251565E-16], + [-53.333333333333336, 0.98191629014319448, + 0.98191629014319448, + 0.], + [-40., 0.97616460318514309, + 0.97616460318514309, + 0.], + [-32., 0.97053988407466396, + 0.97053988407466363, + -3.3306690738754696E-16], + [-26.666666666666668, 0.96503625112337699, + 0.96503625112337699, + 0.], + [-22.857142857142858, 0.95964827107936723, + 0.95964827107936734, + 1.1102230246251565E-16] + [-20., 0.9543709099192168, + 0.95437090991921691, + 1.1102230246251565E-16], + [-17.777777777777779, 0.94919949077974575, + 0.94919949077974575, + 0.], + [-16., 0.94412965773690294, + 0.94412965773690294, + 0.], + [-14.545454545454545, 0.93915734441928422, + 0.93915734441928411, + -1.1102230246251565E-16] + [-13.333333333333334, 0.93427874665341049, + 0.9342787466534106, + 1.1102230246251565E-16] + [-12.307692307692308, 0.92949029849721398, + 0.92949029849721387, + -1.1102230246251565E-16] + [-11.428571428571429, 0.92478865114084174, + 0.92478865114084174, + 0.], + [-10.666666666666666, 0.92017065424944566, + 0.92017065424944577, + 1.1102230246251565E-16] + [-10., 0.91563333939788083, + 0.91563333939788094, + 1.1102230246251565E-16] + +\end{verbatim} + These are the Chebyshev coefficients used by Axiom in the range $(-10 \le x \le -4)$ in the polynomial $$\sum_{k=0}^{43}\ ^{'}{A_kT_k(t)}$$ with the scaling factor $t=(x+7)/3$ \begin{chunk}{*} + --S 4 of 20 -bChebyshev:=[_ +bChebyshev:List Fraction Integer:= [machineFraction (x::DFLOAT) for x in _ +[_ 0.175755649606129373848762834691E1,_ -0.435854151773616611705001867964E-01,_ -0.797950713955842540133217027492E-02,_ @@ -258,57 +428,155 @@ bChebyshev:=[_ -0.750270777550246547010642233720E-28,_ -0.161845436449591026807612330206E-28,_ -0.349436677170516166749482836452E-29,_ --0.755103690612616785856037026797E-30] +-0.755103690612616785856037026797E-30]] --R --R --R (5) ---R [1.7575564960 6129373848 762834691, - 0.0435854151 7736166117 0500186796 4, ---R - 0.0079795071 3955842540 1332170274 92, ---R - 0.0014843723 2730371213 8509702100 01, ---R - 0.0002800301 9843775145 7486203954 948, ---R - 0.0000534864 8512865793 2303917736 1553, ---R - 0.0000103286 7243573554 8661023326 646, ---R - 0.0000020140 8331300553 6877322261 98639, ---R - 0.3961758434 2738664582 23384435 E -6, ---R - 0.7853872767 0966316306 7607656069 E -7, ---R - 0.1567925981 0074698262 4616270279 E -7, ---R - 0.3150055939 3763998825 0007372851 E -8, ---R - 0.6365096822 5242037304 0380263972 E -9, ---R - 0.1292888113 2805631835 6593121259 E -9, ---R - 0.2638690999 6592557613 2149942808 E -10, ---R - 0.5408958287 0450687349 1922207896 E -11, ---R - 0.1113222784 6010898999 7676692708 E -11, ---R - 0.2299624726 0744624618 4338864145 E -12, ---R - 0.4766682389 4951902622 3913482091 E -13, ---R - 0.9911756747 3352709450 6246643371 E -14, ---R - 0.2067103580 4957072400 0900805021 E -14, ---R - 0.4322776783 3833850564 5764394579 E -15, ---R - 0.9063014799 6650172551 4905603356 E -16, ---R - 0.1904669979 5816613974 4015963342 E -16, ---R - 0.4011792326 3502786634 674422752 E -17, ---R - 0.8467772130 0168322313 4166334685 E -18, ---R - 0.1790842733 6586966555 5826492204 E -18, ---R - 0.3794490638 1714782440 1106175166 E -19, ---R - 0.8053999236 7982798526 0999654058 E -20, ---R - 0.1712339011 2362012974 3228671244 E -20, ---R - 0.3646274058 7749686208 6576562816 E -21, ---R - 0.7775969638 8939479435 3098157647 E -22, ---R - 0.1660628498 4484020566 2531950966 E -22, ---R - 0.3551178625 7882509300 5927145352 E -23, ---R - 0.7603722685 9413580929 5734653294 E -24, ---R - 0.1630074137 2584900288 9638374755 E -24, ---R - 0.3498575202 7286322350 7538497255 E -25, ---R - 0.7517179627 8900988246 0645145143 E -26, ---R - 0.1616877440 0527227629 8777317918 E -26, ---R - 0.3481270085 7247569174 8202271565 E -27, ---R - 0.7502707775 5024654701 064223372 E -28, ---R - 0.1618454364 4959102680 7612330206 E -28, ---R - 0.3494366771 7051616674 9482836452 E -29, ---R - 0.7551036906 1261678585 6037026797 E -30] ---R Type: List Float +--R 7915330780744237 3140660152824869 1149968172170097 +--R [----------------, - -----------------, - ------------------, +--R 4503599627370496 72057594037927936 144115188075855872 +--R 6845459107967051 5165645403491301 3946606010282499 +--R - -------------------, - --------------------, - --------------------, +--R 4611686018427387904 18446744073709551616 73786976294838206464 +--R 6096972065382141 4755619765522199 +--R - ---------------------, - ----------------------, +--R 590295810358705651712 2361183241434822606848 +--R 1870887524324103 5934218482569569 +--R - ----------------------, - -----------------------, +--R 4722366482869645213696 75557863725914323419136 +--R 4738765504211317 7616367916685089 +--R - ------------------------, - -------------------------, +--R 302231454903657293676544 2417851639229258349412352 +--R 6155943914477235 1250404657614173 +--R - -------------------------, - -------------------------, +--R 9671406556917033397649408 9671406556917033397649408 +--R 2041588274862599 6695966034357555 +--R - --------------------------, - ----------------------------, +--R 77371252455336267181195264 1237940039285380274899124224 +--R 2756206115404907 4554876037981203 +--R - ----------------------------, - -----------------------------, +--R 2475880078570760549798248448 19807040628566084398385987584 +--R 7553109740176135 6282322195037879 +--R - ------------------------------, - ------------------------------, +--R 158456325028528675187087900672 633825300114114700748351602688 +--R 2620365094549305 2191908233643441 +--R - -------------------------------, - -------------------------------, +--R 1267650600228229401496703205376 5070602400912917605986812821504 +--R 3676395568215259 6181007469704659 +--R - --------------------------------, - ---------------------------------, +--R 40564819207303340847894502572032 324518553658426726783156020576256 +--R 5207604173300669 1099179665736879 +--R - ----------------------------------, - ----------------------------------, +--R 1298074214633706907132624082305024 1298074214633706907132624082305024 +--R 3719434840042389 +--R - -----------------------------------, +--R 20769187434139310514121985316880384 +--R 6304678982501625 +--R - ------------------------------------, +--R 166153499473114484112975882535043072 +--R 5352800631791309 +--R - ------------------------------------, +--R 664613997892457936451903530140172288 +--R 569022238002455 +--R - ------------------------------------, +--R 332306998946228968225951765070086144 +--R 7754767294764771 +--R - --------------------------------------, +--R 21267647932558653966460912964485513216 +--R 6615063384570469 +--R - --------------------------------------, +--R 85070591730234615865843651857942052864 +--R 2825412980141931 +--R - ---------------------------------------, +--R 170141183460469231731687303715884105728 +--R 2416806936284543 +--R - ---------------------------------------, +--R 680564733841876926926749214863536422912 +--R 8279720809544195 +--R - -----------------------------------------, +--R 10889035741470030830827987437816582766592 +--R 7099974216741449 +--R - -----------------------------------------, +--R 43556142965880123323311949751266331066368 +--R 6095377668277253 +--R - ------------------------------------------, +--R 174224571863520493293247799005065324265472 +--R 5238709609161323 +--R - ------------------------------------------, +--R 696898287454081973172991196020261297061888 +--R 2253598237991765 +--R - -------------------------------------------, +--R 1393796574908163946345982392040522594123776 +--R 3881745857450733 +--R - --------------------------------------------, +--R 11150372599265311570767859136324180752990208 +--R 6692638976020599 +--R - --------------------------------------------, +--R 89202980794122492566142873090593446023921664 +--R 360927383980731 +--R - --------------------------------------------, +--R 22300745198530623141535718272648361505980416 +--R 4987326912384565 +--R - ----------------------------------------------, +--R 1427247692705959881058285969449495136382746624 +--R 4310880000722449 +--R - ----------------------------------------------] +--R 5708990770823839524233143877797980545530986496 +--R Type: List Fraction Integer --E 4 \end{chunk} + +The actual table values from above are: +\begin{verbatim} + + 1.7575564960 6129373848 762834691 + -0.0435854151 7736166117 0500186796 4 + -0.0079795071 3955842540 1332170274 92 + -0.0014843723 2730371213 8509702100 01 + -0.0002800301 9843775145 7486203954 948 + -0.0000534864 8512865793 2303917736 1553 + -0.0000103286 7243573554 8661023326 646 + -0.0000020140 8331300553 6877322261 98639 + -0.3961758434 2738664582 23384435 E -6 + -0.7853872767 0966316306 7607656069 E -7 + -0.1567925981 0074698262 4616270279 E -7 + -0.3150055939 3763998825 0007372851 E -8 + -0.6365096822 5242037304 0380263972 E -9 + -0.1292888113 2805631835 6593121259 E -9 + -0.2638690999 6592557613 2149942808 E -10 + -0.5408958287 0450687349 1922207896 E -11 + -0.1113222784 6010898999 7676692708 E -11 + -0.2299624726 0744624618 4338864145 E -12 + -0.4766682389 4951902622 3913482091 E -13 + -0.9911756747 3352709450 6246643371 E -14 + -0.2067103580 4957072400 0900805021 E -14 + -0.4322776783 3833850564 5764394579 E -15 + -0.9063014799 6650172551 4905603356 E -16 + -0.1904669979 5816613974 4015963342 E -16 + -0.4011792326 3502786634 674422752 E -17 + -0.8467772130 0168322313 4166334685 E -18 + -0.1790842733 6586966555 5826492204 E -18 + -0.3794490638 1714782440 1106175166 E -19 + -0.8053999236 7982798526 0999654058 E -20 + -0.1712339011 2362012974 3228671244 E -20 + -0.3646274058 7749686208 6576562816 E -21 + -0.7775969638 8939479435 3098157647 E -22 + -0.1660628498 4484020566 2531950966 E -22 + -0.3551178625 7882509300 5927145352 E -23 + -0.7603722685 9413580929 5734653294 E -24 + -0.1630074137 2584900288 9638374755 E -24 + -0.3498575202 7286322350 7538497255 E -25 + -0.7517179627 8900988246 0645145143 E -26 + -0.1616877440 0527227629 8777317918 E -26 + -0.3481270085 7247569174 8202271565 E -27 + -0.7502707775 5024654701 064223372 E -28 + -0.1618454364 4959102680 7612330206 E -28 + -0.3494366771 7051616674 9482836452 E -29 + -0.7551036906 1261678585 6037026797 E -30 + +\end{verbatim} + In the following table there are 4 columns. The first column is the argument of Ei(x) shown in Table 4 in \cite{1}. The second column is the exact value shown in the table. Column 3 is the @@ -316,7 +584,10 @@ value returned by Axiom and column 4 is the difference. See special.spad.dvi for details. \begin{chunk}{*} + --S 5 of 20 +t2:List List Fraction Integer:= [[machineFraction (x::DFLOAT) for x in line]_ + for line in _ [[-10.000,0.915633339397880818760698157661,_ Ei2(-10.000),Ei2(-10.000)-0.915633339397880818760698157661],_ [ -9.625,0.912844461467993418856575662217,_ @@ -350,39 +621,99 @@ See special.spad.dvi for details. [ -4.375,0.836498754556298741742152267582,_ Ei2( -4.375),Ei2( -4.375)-0.836498754556298741742152267582],_ [ -4.000,0.825382599604223332408183035504,_ - Ei2( -4.000),Ei2( -4.000)-0.825382599604223332408183035504]] + Ei2( -4.000),Ei2( -4.000)-0.825382599604223332408183035504]]] --R --R --R (6) ---R [[- 10.,0.91563333939788083,0.91563333939788083,0.], ---R [- 9.625,0.91284446146799347,0.91284446146799336,- 1.1102230246251565E-16], ---R [- 9.25,0.90986275152542417,0.90986275152542395,- 2.2204460492503131E-16], ---R [- 8.875,0.90666727065475383,0.90666727065475394,1.1102230246251565E-16], ---R [- 8.5,0.90323390197320785,0.90323390197320796,1.1102230246251565E-16], ---R [- 8.125,0.89953471768847382,0.89953471768847415,3.3306690738754696E-16], ---R [- 7.75,0.89553718708753915,0.89553718708753927,1.1102230246251565E-16], ---R [- 7.375,0.89120317632125434,0.89120317632125423,- 1.1102230246251565E-16], ---R [- 7.,0.88648767253642935,0.88648767253642924,- 1.1102230246251565E-16], ---R [- 6.625,0.88133713846821016,0.88133713846821005,- 1.1102230246251565E-16], ---R [- 6.25,0.87568736478846598,0.87568736478846598,0.], ---R [- 5.875,0.8694606294541134,0.86946062945411307,- 3.3306690738754696E-16], ---R [- 5.5,0.86256188469070139,0.86256188469070139,0.], ---R [- 5.125,0.85487355389019959,0.85487355389019937,- 2.2204460492503131E-16], ---R [- 4.75,0.84624829910358734,0.84624829910358745,1.1102230246251565E-16], ---R [- 4.375,0.83649875455629874,0.83649875455629874,0.], ---R [- 4.,0.82538259960422333,0.82538259960422322,- 1.1102230246251565E-16]] ---R Type: List List OnePointCompletion DoubleFloat +--R 8247291932240597 4123645966120299 1 +--R [[- 10,----------------,----------------,----------------], +--R 9007199254740992 4503599627370496 9007199254740992 +--R 77 1027771494128619 1027771494128619 +--R [- --,----------------,----------------,0], +--R 8 1125899906842624 1125899906842624 +--R 37 2048828774364097 8195315097456387 1 +--R [- --,----------------,----------------,- ----------------], +--R 4 2251799813685248 9007199254740992 9007199254740992 +--R 71 2041633191134887 8166532764539549 1 +--R [- --,----------------,----------------,----------------], +--R 8 2251799813685248 9007199254740992 9007199254740992 +--R 17 8135607728709875 8135607728709877 1 +--R [- --,----------------,----------------,----------------], +--R 2 9007199254740992 9007199254740992 4503599627370496 +--R 65 4051144219388635 8102288438777273 3 +--R [- --,----------------,----------------,----------------], +--R 8 4503599627370496 9007199254740992 9007199254740992 +--R 31 8066281884127727 504142617757983 1 +--R [- --,----------------,---------------,----------------], +--R 4 9007199254740992 562949953421312 9007199254740992 +--R 59 4013622292791803 8027244585583605 1 +--R [- --,----------------,----------------,- ----------------], +--R 8 4503599627370496 9007199254740992 9007199254740992 +--R 7984771103407203 3992385551703601 1 +--R [- 7,----------------,----------------,- ----------------], +--R 9007199254740992 4503599627370496 9007199254740992 +--R 53 7938379216786421 1984594804196605 1 +--R [- --,----------------,----------------,- ----------------], +--R 8 9007199254740992 2251799813685248 9007199254740992 +--R 25 7887490579508773 3943745289754387 1 +--R [- --,----------------,----------------,----------------], +--R 4 9007199254740992 4503599627370496 9007199254740992 +--R 47 1957851283411431 7831405133645721 3 +--R [- --,----------------,----------------,- ----------------], +--R 8 2251799813685248 9007199254740992 9007199254740992 +--R 11 7769266764954071 7769266764954071 +--R [- --,----------------,----------------,0], +--R 2 9007199254740992 9007199254740992 +--R 41 1925004109374397 7700016437497587 1 +--R [- --,----------------,----------------,- ----------------], +--R 8 2251799813685248 9007199254740992 9007199254740992 +--R 19 476395440563229 7622327049011665 1 +--R [- --,---------------,----------------,----------------], +--R 4 562949953421312 9007199254740992 9007199254740992 +--R 35 3767255479315631 3767255479315631 +--R [- --,----------------,----------------,0], +--R 8 4503599627370496 4503599627370496 +--R 3717192768015671 3717192768015671 +--R [- 4,----------------,----------------,0]] +--R 4503599627370496 4503599627370496 +--R Type: List List Fraction Integer --E 5 \end{chunk} + +The actual table values from above are: +\begin{verbatim} + + - 10., 0.91563333939788083, 0.91563333939788083, 0. + - 9.625, 0.91284446146799347, 0.91284446146799336, -1.1102230246251565E-16 + - 9.25, 0.90986275152542417, 0.90986275152542395, -2.2204460492503131E-16 + - 8.875, 0.90666727065475383, 0.90666727065475394, 1.1102230246251565E-16 + - 8.5, 0.90323390197320785, 0.90323390197320796, 1.1102230246251565E-16 + - 8.125, 0.89953471768847382, 0.89953471768847415, 3.3306690738754696E-16 + - 7.75, 0.89553718708753915, 0.89553718708753927, 1.1102230246251565E-16 + - 7.375, 0.89120317632125434, 0.89120317632125423, -1.1102230246251565E-16 + - 7., 0.88648767253642935, 0.88648767253642924, -1.1102230246251565E-16 + - 6.625, 0.88133713846821016, 0.88133713846821005, -1.1102230246251565E-16 + - 6.25, 0.87568736478846598, 0.87568736478846598, 0. + - 5.875, 0.8694606294541134, 0.86946062945411307, -3.3306690738754696E-16 + - 5.5, 0.86256188469070139, 0.86256188469070139, 0. + - 5.125, 0.85487355389019959, 0.85487355389019937, -2.2204460492503131E-16 + - 4.75, 0.84624829910358734, 0.84624829910358745, 1.1102230246251565E-16 + - 4.375, 0.83649875455629874, 0.83649875455629874, 0. + - 4., 0.82538259960422333, 0.82538259960422322, -1.1102230246251565E-16 + +\end{verbatim} + These are the Chebyshev coefficients used by Axiom in the range $(-4 \le x \le 4)$ in the polynomial $$\sum_{k=0}^{33}\ ^{'}{A_kT_k(t)}$$ with the scaling factor $t=x/4$ \begin{chunk}{*} + --S 6 of 20 -cChebyshev:=[_ +cChebyshev:List Fraction Integer:= [machineFraction (x::DFLOAT) for x in _ +[_ 0.329370010376739129393905231421E1,_ 0.167983505237130291565505796064E1,_ 0.722043610567875435240299679644E0,_ @@ -416,45 +747,109 @@ cChebyshev:=[_ 0.943908972224487442925310405245E-26,_ 0.569722755950369211989581737831E-27,_ 0.333833362779543303156597939562E-28,_ -0.190062601281619148526680482237E-29] +0.190062601281619148526680482237E-29]] --R --R --R (7) ---R [3.2937001037 6739129393 905231421, 1.6798350523 7130291565 505796064, ---R 0.7220436105 6787543524 0299679644, 0.2600312360 5480956171 3740181192, ---R 0.0801049430 8173750223 9474288923 7, 0.0215140366 3976333754 8055248300 5, ---R 0.0051162077 8993033120 6219689108 94, ---R 0.0010909328 6100739135 6050661990 14, ---R 0.0002107415 3202393891 6318348675 226, ---R 0.0000371990 4516651888 5709594081 5956, ---R 0.0000060434 9163712387 8757047670 32866, ---R 0.9092954273 9626095264 9596541772 E -6, ---R 0.1273805160 6592647886 5567184969 E -6, ---R 0.1669185748 4109890739 0896143814 E -7, ---R 0.2054417026 4010479254 7612484551 E -8, ---R 0.2383584444 4668176591 4052321417 E -9, ---R 0.2615386378 8854429666 9068664148 E -10, ---R 0.2721858622 8541670644 6550268995 E -11, ---R 0.2693750031 9835792992 5326427442 E -12, ---R 0.2541220946 7072635546 7884089307 E -13, ---R 0.2290130406 8650370941 8510620516 E -14, ---R 0.1975465739 0746229940 1057650412 E -15, ---R 0.1634024551 9289317406 8635419984 E -16, ---R 0.1298235437 0796376099 1961293204 E -17, ---R 0.9922587925 0737105964 4632581302 E -19, ---R 0.7306252806 7221032944 7230880087 E -20, ---R 0.5189676834 6043451272 0780080019 E -21, ---R 0.3560409454 0997068112 8043162227 E -22, ---R 0.2361979432 5793864237 0187203948 E -23, ---R 0.1516837767 7214529754 9624516819 E -24, ---R 0.9439089722 2448744292 5310405245 E -26, ---R 0.5697227559 5036921198 9581737831 E -27, ---R 0.3338333627 7954330315 6597939562 E -28, ---R 0.1900626012 8161914852 6680482237 E -29] ---R Type: List Float +--R 7416753279998493 7565304515903297 3251795335498731 4684306311204519 +--R [----------------, ----------------, ----------------, -----------------, +--R 2251799813685248 4503599627370496 4503599627370496 18014398509481984 +--R 1443042367253791 6200998873220695 1474646495761937 +--R -----------------, ------------------, ------------------, +--R 18014398509481984 288230376151711744 288230376151711744 +--R 5031039822150775 7774990213894133 5489610127785089 +--R -------------------, --------------------, ---------------------, +--R 4611686018427387904 36893488147419103232 147573952589676412928 +--R 7134895586664201 8588052498725463 1203074959280739 +--R ----------------------, ----------------------, ----------------------, +--R 1180591620717411303424 9444732965739290427392 9444732965739290427392 +--R 630600546558379 310454723434017 +--R -----------------------, ------------------------, +--R 37778931862957161709568 151115727451828646838272 +--R 1152630711259091 4047114395779867 +--R -------------------------, ---------------------------, +--R 4835703278458516698824704 154742504910672534362390528 +--R 6738995541010677 2667760816334997 +--R ----------------------------, ----------------------------, +--R 2475880078570760549798248448 9903520314283042199192993792 +--R 4026725323007515 2903085184863383 +--R ------------------------------, -------------------------------, +--R 158456325028528675187087900672 1267650600228229401496703205376 +--R 8013441055578395 5302712842343357 +--R --------------------------------, ---------------------------------, +--R 40564819207303340847894502572032 324518553658426726783156020576256 +--R 6740823781587191 4121681768951667 +--R ----------------------------------, -----------------------------------, +--R 5192296858534827628530496329220096 41538374868278621028243970633760768 +--R 606979735936071 344913186881627 +--R -----------------------------------, ------------------------------------, +--R 83076749736557242056487941267521536 664613997892457936451903530140172288 +--R 6057722781243673 +--R ---------------------------------------, +--R 170141183460469231731687303715884105728 +--R 3214959807746755 +--R ----------------------------------------, +--R 1361129467683753853853498429727072845824 +--R 6606760266692207 +--R -----------------------------------------, +--R 43556142965880123323311949751266331066368 +--R 6578085462557879 +--R ------------------------------------------, +--R 696898287454081973172991196020261297061888 +--R 6352621007126915 +--R --------------------------------------------, +--R 11150372599265311570767859136324180752990208 +--R 5955786209692185 +--R ---------------------------------------------, +--R 178405961588244985132285746181186892047843328 +--R 2712664091488837 +--R ----------------------------------------------] +--R 1427247692705959881058285969449495136382746624 +--R Type: List Fraction Integer --E 6 \end{chunk} + +The actual table values from above are: +\begin{verbatim} + + 3.2937001037 6739129393 905231421 + 1.6798350523 7130291565 505796064 + 0.7220436105 6787543524 0299679644 + 0.2600312360 5480956171 3740181192 + 0.0801049430 8173750223 9474288923 7 + 0.0215140366 3976333754 8055248300 5 + 0.0051162077 8993033120 6219689108 94 + 0.0010909328 6100739135 6050661990 14 + 0.0002107415 3202393891 6318348675 226 + 0.0000371990 4516651888 5709594081 5956 + 0.0000060434 9163712387 8757047670 32866 + 0.9092954273 9626095264 9596541772 E -6 + 0.1273805160 6592647886 5567184969 E -6 + 0.1669185748 4109890739 0896143814 E -7 + 0.2054417026 4010479254 7612484551 E -8 + 0.2383584444 4668176591 4052321417 E -9 + 0.2615386378 8854429666 9068664148 E -10 + 0.2721858622 8541670644 6550268995 E -11 + 0.2693750031 9835792992 5326427442 E -12 + 0.2541220946 7072635546 7884089307 E -13 + 0.2290130406 8650370941 8510620516 E -14 + 0.1975465739 0746229940 1057650412 E -15 + 0.1634024551 9289317406 8635419984 E -16 + 0.1298235437 0796376099 1961293204 E -17 + 0.9922587925 0737105964 4632581302 E -19 + 0.7306252806 7221032944 7230880087 E -20 + 0.5189676834 6043451272 0780080019 E -21 + 0.3560409454 0997068112 8043162227 E -22 + 0.2361979432 5793864237 0187203948 E -23 + 0.1516837767 7214529754 9624516819 E -24 + 0.9439089722 2448744292 5310405245 E -26 + 0.5697227559 5036921198 9581737831 E -27 + 0.3338333627 7954330315 6597939562 E -28 + 0.1900626012 8161914852 6680482237 E -29 + +\end{verbatim} + In the following table there are 4 columns. The first column is the argument of Ei(x) shown in Table 4 in \cite{1}. The second column is the exact value shown in the table. Column 3 is the @@ -462,7 +857,10 @@ value returned by Axiom and column 4 is the difference. See special.spad.dvi for details. \begin{chunk}{*} + --S 7 of 20 +t3:List List Fraction Integer:= [[machineFraction (x::DFLOAT) for x in line]_ + for line in _ [[-4.0,0.491822344607818096479962798267,_ Ei3(-4.0),Ei3(-4.0)-0.491822344607818096479962798267],_ [-3.5,0.524842506644128356918258753311,_ @@ -496,39 +894,94 @@ See special.spad.dvi for details. [3.5,3.45582153193012412437300898811,_ Ei3(3.5),Ei3(3.5)-3.45582153193012412437300898811],_ [4.0,4.41684111100869913580118598668,_ - Ei3(4.0),Ei3(4.0)-4.41684111100869913580118598668]] + Ei3(4.0),Ei3(4.0)-4.41684111100869913580118598668]]] --R --R --R (8) ---R [[- 4.,0.4918223446078181,0.49182234460781826,1.6653345369377348E-16], ---R [- 3.5,0.52484250664412835,0.52484250664412835,0.], ---R [- 3.,0.56295877822127982,0.56295877822128015,3.3306690738754696E-16], ---R [- 2.5,0.60736852585838308,0.60736852585838341,3.3306690738754696E-16], ---R [- 2.,0.65963167808476963,0.65963167808476975,1.1102230246251565E-16], ---R [- 1.5,0.72180023694421991,0.72180023694422013,2.2204460492503131E-16], ---R [- 1.,0.79659959929705315,0.79659959929705293,- 2.2204460492503131E-16], ---R [- 0.5,0.88768415823549673,0.88768415823549696,2.2204460492503131E-16], ---R [0.,1.,1.,0.], ---R [0.5,1.1403028410431721,1.1403028410431715,- 6.6613381477509392E-16], ---R [1.,1.3179021514544038,1.3179021514544034,- 4.4408920985006262E-16], ---R [1.5,1.5457364507467337,1.5457364507467335,- 2.2204460492503131E-16], ---R [2.,1.841935755270206,1.8419357552702071,1.1102230246251565E-15], ---R [2.5,2.2321037991211652,2.2321037991211647,- 4.4408920985006262E-16], ---R [3.,2.7526682056852581,2.7526682056852589,8.8817841970012523E-16], ---R [3.5,3.4558215319301242,3.4558215319301238,- 4.4408920985006262E-16], ---R [4.,4.4168411110086989,4.4168411110087007,1.7763568394002505E-15]] ---R Type: List List OnePointCompletion DoubleFloat +--R 8859883711633013 4429941855816505 3 +--R [[- 4,-----------------,----------------,- -----------------], +--R 18014398509481984 9007199254740992 18014398509481984 +--R 7 4727361034701387 4727361034701385 1 +--R [- -,----------------,----------------,- ----------------], +--R 2 9007199254740992 9007199254740992 4503599627370496 +--R 5070681887644611 1267670471911153 1 +--R [- 3,----------------,----------------,----------------], +--R 9007199254740992 2251799813685248 9007199254740992 +--R 5 2735344666732381 1367672333366191 1 +--R [- -,----------------,----------------,----------------], +--R 2 4503599627370496 2251799813685248 4503599627370496 +--R 5941433959248687 5941433959248687 +--R [- 2,----------------,----------------,0], +--R 9007199254740992 9007199254740992 +--R 3 6501398556275849 6501398556275849 +--R [- -,----------------,----------------,0], +--R 2 9007199254740992 9007199254740992 +--R 7175131317115389 7175131317115389 +--R [- 1,----------------,----------------,0], +--R 9007199254740992 9007199254740992 +--R 1 3997774044252075 3997774044252075 +--R [- -,----------------,----------------,0], [0,1,1,0], +--R 2 4503599627370496 4503599627370496 +--R 1 5135467450011547 5135467450011543 1 +--R [-,----------------,----------------,- ----------------], +--R 2 4503599627370496 4503599627370496 1125899906842624 +--R 1483825909550207 2967651819100413 1 +--R [1,----------------,----------------,- ----------------], +--R 1125899906842624 2251799813685248 2251799813685248 +--R 3 6961378103595983 1740344525898995 3 +--R [-,----------------,----------------,- ----------------], +--R 2 4503599627370496 1125899906842624 4503599627370496 +--R 2073835295268823 259229411908603 1 +--R [2,----------------,---------------,----------------], +--R 1125899906842624 140737488355328 1125899906842624 +--R 5 5026250918987173 1256562729746793 1 +--R [-,----------------,----------------,- ----------------], +--R 2 2251799813685248 562949953421312 2251799813685248 +--R 6198457752699369 6198457752699371 1 +--R [3,----------------,----------------,----------------], +--R 2251799813685248 2251799813685248 1125899906842624 +--R 7 7781818281729721 972727285216215 1 +--R [-,----------------,---------------,- ----------------], +--R 2 2251799813685248 281474976710656 2251799813685248 +--R 2486460497711683 4972920995423367 1 +--R [4,----------------,----------------,----------------]] +--R 562949953421312 1125899906842624 1125899906842624 +--R Type: List List Fraction Integer --E 7 \end{chunk} + +The actual table values from above are: +\begin{verbatim} + -4., 0.4918223446078181, 0.49182234460781826, 1.6653345369377348E-16 + -3.5, 0.52484250664412835, 0.52484250664412835, 0. + -3., 0.56295877822127982, 0.56295877822128015, 3.3306690738754696E-16 + -2.5, 0.60736852585838308, 0.60736852585838341, 3.3306690738754696E-16 + -2., 0.65963167808476963, 0.65963167808476975, 1.1102230246251565E-16 + -1.5, 0.72180023694421991, 0.72180023694422013, 2.2204460492503131E-16 + -1., 0.79659959929705315, 0.79659959929705293, -2.2204460492503131E-16 + -0.5, 0.88768415823549673, 0.88768415823549696, 2.2204460492503131E-16 + 0., 1., 1., 0. + 0.5, 1.1403028410431721, 1.1403028410431715, -6.6613381477509392E-16 + 1., 1.3179021514544038, 1.3179021514544034, -4.4408920985006262E-16 + 1.5, 1.5457364507467337, 1.5457364507467335, -2.2204460492503131E-16 + 2., 1.841935755270206, 1.8419357552702071, 1.1102230246251565E-15 + 2.5, 2.2321037991211652, 2.2321037991211647, -4.4408920985006262E-16 + 3., 2.7526682056852581, 2.7526682056852589, 8.8817841970012523E-16 + 3.5, 3.4558215319301242, 3.4558215319301238, -4.4408920985006262E-16 + 4., 4.4168411110086989, 4.4168411110087007, 1.7763568394002505E-15 +\end{verbatim} + These are the Chebyshev coefficients used by Axiom in the range $(4 \le x \le 12)$ in the polynomial $$\sum_{k=0}^{49}\ ^{'}{A_kT_k(t)}$$ with the scaling factor $t=(x-8)/4$ \begin{chunk}{*} + --S 8 of 20 -dChebyshev:=[_ +dChebyshev:List Fraction Integer:=[machineFraction (x::DFLOAT) for x in _ +[_ 0.245513353878129528673420457043E1,_ -0.162438379130376524396002276856E0,_ 0.449575308093572641480785417193E-01,_ @@ -578,63 +1031,170 @@ dChebyshev:=[_ 0.466312873522730486582600122073E-28,_ -0.118595958891902887946724005478E-28,_ 0.302029059055671310731137614875E-29,_ --0.770165054816636606098827057102E-30] +-0.770165054816636606098827057102E-30]] --R --R --R (9) ---R [2.4551335387 8129528673 420457043, - 0.1624383791 3037652439 6002276856, ---R 0.0449575308 0935726414 8078541719 3, ---R - 0.0067415786 7998922998 8487188350 5, ---R - 0.0013066971 4280329428 0515993413 87, ---R 0.0013810831 4600072576 0202020898 2, ---R - 0.0005850228 7901596579 8687368242 394, ---R 0.0001749299 3410789197 0038740976 432, ---R - 0.0000404728 1499052930 3552286933 38, ---R 0.0000072171 0241217099 7500357526 00049, ---R - 0.8612776970 1986775241 4815450193 E -6, ---R - 0.2514475296 5322559777 9084739054 E -9, ---R 0.3794747138 2014951081 4074505574 E -7, ---R - 0.1442117969 5211980616 0265640172 E -7, ---R 0.3935049295 9761013108 7190848042 E -8, ---R - 0.9284689401 0633175304 7289210353 E -9, ---R 0.2031789568 0065461336 6090995698 E -9, ---R - 0.4292498504 9923683142 7918026902 E -10, ---R 0.8992647177 7812393526 8001544182 E -11, ---R - 0.1900869118 4121097524 2396635722 E -11, ---R 0.4092198912 2237383452 6121178338 E -12, ---R - 0.8999253437 2931901982 5435824585 E -13, ---R 0.2019654670 8242638335 4948543451 E -13, ---R - 0.4612930261 3830820719 4950531726 E -14, ---R 0.1069023072 9386369566 8857256409 E -14, ---R - 0.2507030070 5700729569 2572254042 E -15, ---R 0.5937322503 7915516070 6073763509 E -16, ---R - 0.1417734582 4376625234 4732005648 E -16, ---R 0.3409203754 3608089342 6806402093 E -17, ---R - 0.8248290269 5054937928 8702529656 E -18, ---R 0.2006369712 6214423139 8824095937 E -18, ---R - 0.4903851667 9674222440 3498152027 E -19, ---R 0.1203734482 3483321716 6664609324 E -19, ---R - 0.2966282447 1413682538 1453572575 E -20, ---R 0.7335512384 2880759924 2142328436 E -21, ---R - 0.1819924142 9085112734 4263485604 E -21, ---R 0.4528629374 2957606021 7359526404 E -22, ---R - 0.1129980043 7506096133 8906717853 E -22, ---R 0.2826681251 2901165692 3764408445 E -23, ---R - 0.7087717977 1690496166 6732640699 E -24, ---R 0.1781104524 0187095153 4401530034 E -24, ---R - 0.4485004076 6189635731 2006142358 E -25, ---R 0.1131540292 5754766224 505309084 E -25, ---R - 0.2859957899 7793216379 0414326136 E -26, ---R 0.7240775806 9226736175 8172726753 E -27, ---R - 0.1836132234 1257789805 0666710105 E -27, ---R 0.4663128735 2273048658 2600122073 E -28, ---R - 0.1185959588 9190288794 6724005478 E -28, ---R 0.3020290590 5567131073 1137614875 E -29, ---R - 0.7701650548 1663660609 8827057102 E -30] ---R Type: List Float +--R 1382117311300031 731557423722231 3239531504008303 +--R [----------------, - ----------------, -----------------, +--R 562949953421312 4503599627370496 72057594037927936 +--R 242890969848707 753259617973121 6369121834697257 +--R - -----------------, - ------------------, -------------------, +--R 36028797018963968 576460752303423488 4611686018427387904 +--R 5395883663236133 6453775450638317 1493183320145779 +--R - -------------------, --------------------, - --------------------, +--R 9223372036854775808 36893488147419103232 36893488147419103232 +--R 532528164604281 8134537857699561 +--R --------------------, - ----------------------, +--R 73786976294838206464 9444732965739290427392 +--R 607962821752199 2867229871425319 +--R - -------------------------, -----------------------, +--R 2417851639229258349412352 75557863725914323419136 +--R 8717068241421995 4757182695361877 +--R - ------------------------, -------------------------, +--R 604462909807314587353088 1208925819614629174706176 +--R 8979600595238185 3930052590058827 +--R - -------------------------, --------------------------, +--R 9671406556917033397649408 19342813113834066795298816 +--R 830289963734795 5566179000271035 +--R - --------------------------, ---------------------------, +--R 19342813113834066795298816 618970019642690137449562112 +--R 2353161991123453 1013179376432369 +--R - ----------------------------, ----------------------------, +--R 1237940039285380274899124224 2475880078570760549798248448 +--R 445621446148073 800067642413789 +--R - ----------------------------, -----------------------------, +--R 4951760157141521099596496896 39614081257132168796771975168 +--R 5847583814653227 5420590960273957 +--R - -------------------------------, -------------------------------, +--R 1267650600228229401496703205376 5070602400912917605986812821504 +--R 5084861077997397 4816928278835157 +--R - --------------------------------, --------------------------------, +--R 20282409603651670423947251286016 81129638414606681695789005144064 +--R 4600811761642037 69146866968253 +--R - ---------------------------------, --------------------------------, +--R 324518553658426726783156020576256 20282409603651670423947251286016 +--R 1070689291365919 8334133724722991 +--R - ----------------------------------, -----------------------------------, +--R 1298074214633706907132624082305024 41538374868278621028243970633760768 +--R 4073960577649283 +--R - -----------------------------------, +--R 83076749736557242056487941267521536 +--R 8000187867145333 +--R ------------------------------------, +--R 664613997892457936451903530140172288 +--R 7885731344291393 +--R - -------------------------------------, +--R 2658455991569831745807614120560689152 +--R 7800454739698135 +--R --------------------------------------, +--R 10633823966279326983230456482242756608 +--R 1935275296767089 +--R - --------------------------------------, +--R 10633823966279326983230456482242756608 +--R 7705063611965249 +--R ---------------------------------------, +--R 170141183460469231731687303715884105728 +--R 1922561419304415 +--R - ---------------------------------------, +--R 170141183460469231731687303715884105728 +--R 7694958293760327 +--R ----------------------------------------, +--R 2722258935367507707706996859454145691648 +--R 241182544933917 +--R - ---------------------------------------, +--R 340282366920938463463374607431768211456 +--R 7757804328533477 +--R -----------------------------------------, +--R 43556142965880123323311949751266331066368 +--R 7813979150550829 +--R - ------------------------------------------, +--R 174224571863520493293247799005065324265472 +--R 7885684920811405 +--R ------------------------------------------, +--R 696898287454081973172991196020261297061888 +--R 7972399050187929 +--R - -------------------------------------------, +--R 2787593149816327892691964784081045188247552 +--R 8073734815493375 +--R --------------------------------------------, +--R 11150372599265311570767859136324180752990208 +--R 4094711710404777 +--R - --------------------------------------------, +--R 22300745198530623141535718272648361505980416 +--R 8319299660180039 +--R ---------------------------------------------, +--R 178405961588244985132285746181186892047843328 +--R 8463290434635963 +--R - ---------------------------------------------, +--R 713623846352979940529142984724747568191373312 +--R 8621405553347179 +--R ----------------------------------------------, +--R 2854495385411919762116571938898990272765493248 +--R 8793730379918429 +--R - -----------------------------------------------] +--R 11417981541647679048466287755595961091061972992 +--R Type: List Fraction Integer --E 8 \end{chunk} + +The actual table values from above are: +\begin{verbatim} + + 2.4551335387 8129528673 420457043 + -0.1624383791 3037652439 6002276856 + 0.0449575308 0935726414 8078541719 3 + -0.0067415786 7998922998 8487188350 5 + -0.0013066971 4280329428 0515993413 87 + 0.0013810831 4600072576 0202020898 2 + -0.0005850228 7901596579 8687368242 394 + 0.0001749299 3410789197 0038740976 432 + -0.0000404728 1499052930 3552286933 38 + 0.0000072171 0241217099 7500357526 00049 + -0.8612776970 1986775241 4815450193 E -6 + -0.2514475296 5322559777 9084739054 E -9 + 0.3794747138 2014951081 4074505574 E -7 + -0.1442117969 5211980616 0265640172 E -7 + 0.3935049295 9761013108 7190848042 E -8 + -0.9284689401 0633175304 7289210353 E -9 + 0.2031789568 0065461336 6090995698 E -9 + -0.4292498504 9923683142 7918026902 E -10 + 0.8992647177 7812393526 8001544182 E -11 + -0.1900869118 4121097524 2396635722 E -11 + 0.4092198912 2237383452 6121178338 E -12 + -0.8999253437 2931901982 5435824585 E -13 + 0.2019654670 8242638335 4948543451 E -13 + -0.4612930261 3830820719 4950531726 E -14 + 0.1069023072 9386369566 8857256409 E -14 + -0.2507030070 5700729569 2572254042 E -15 + 0.5937322503 7915516070 6073763509 E -16 + -0.1417734582 4376625234 4732005648 E -16 + 0.3409203754 3608089342 6806402093 E -17 + -0.8248290269 5054937928 8702529656 E -18 + 0.2006369712 6214423139 8824095937 E -18 + -0.4903851667 9674222440 3498152027 E -19 + 0.1203734482 3483321716 6664609324 E -19 + -0.2966282447 1413682538 1453572575 E -20 + 0.7335512384 2880759924 2142328436 E -21 + -0.1819924142 9085112734 4263485604 E -21 + 0.4528629374 2957606021 7359526404 E -22 + -0.1129980043 7506096133 8906717853 E -22 + 0.2826681251 2901165692 3764408445 E -23 + -0.7087717977 1690496166 6732640699 E -24 + 0.1781104524 0187095153 4401530034 E -24 + -0.4485004076 6189635731 2006142358 E -25 + 0.1131540292 5754766224 505309084 E -25 + -0.2859957899 7793216379 0414326136 E -26 + 0.7240775806 9226736175 8172726753 E -27 + -0.1836132234 1257789805 0666710105 E -27 + 0.4663128735 2273048658 2600122073 E -28 + -0.1185959588 9190288794 6724005478 E -28 + 0.3020290590 5567131073 1137614875 E -29 + -0.7701650548 1663660609 8827057102 E -30 + +\end{verbatim} + In the following table there are 4 columns. The first column is the argument of Ei(x) shown in Table 4 in \cite{1}. The second column is the exact value shown in the table. Column 3 is the @@ -642,7 +1202,10 @@ value returned by Axiom and column 4 is the difference. See special.spad.dvi for details. \begin{chunk}{*} + --S 9 of 20 +t4:List List Fraction Integer:= [[machineFraction (x::DFLOAT) for x in line]_ + for line in _ [[4.0,1.43820803145448278470968670330,_ Ei4(4.0),Ei4(4.0)-1.43820803145448278470968670330],_ [4.5,1.39641902962974607100674523183,_ @@ -676,39 +1239,99 @@ See special.spad.dvi for details. [11.5,1.10883292630507730586855234934,_ Ei4(11.5),Ei4(11.5)-1.10883292630507730586855234934],_ [12.0,1.10297454490675907267241234953,_ - Ei4(12.0),Ei4(12.0)-1.10297454490675907267241234953]] + Ei4(12.0),Ei4(12.0)-1.10297454490675907267241234953]]] --R --R --R (10) ---R [[4.,1.4382080314544827,1.4382080314544827,0.], ---R [4.5,1.3964190296297461,1.3964190296297465,4.4408920985006262E-16], ---R [5.,1.3538312774552861,1.3538312774552856,- 4.4408920985006262E-16], ---R [5.5,1.3141435657421192,1.314143565742119,- 2.2204460492503131E-16], ---R [6.,1.2788838604895616,1.2788838604895618,2.2204460492503131E-16], ---R [6.5,1.2483911550017015,1.2483911550017011,- 4.4408920985006262E-16], ---R [7.,1.222408052360531,1.222408052360531,0.], ---R [7.5,1.2004214995996307,1.2004214995996305,- 2.2204460492503131E-16], ---R [8.,1.1818479869872081,1.1818479869872081,0.], ---R [8.5,1.1661265258117486,1.1661265258117477,- 8.8817841970012523E-16], ---R [9.,1.1527592087089249,1.1527592087089251,2.2204460492503131E-16], ---R [9.5,1.1413234759526243,1.1413234759526236,- 6.6613381477509392E-16], ---R [10.,1.1314702047341079,1.1314702047341079,0.], ---R [10.5,1.1229155700177607,1.1229155700177604,- 2.2204460492503131E-16], ---R [11.,1.1154309389980384,1.115430938998039,6.6613381477509392E-16], ---R [11.5,1.1088329263050773,1.1088329263050771,- 2.2204460492503131E-16], ---R [12.,1.1029745449067592,1.1029745449067592,0.]] ---R Type: List List OnePointCompletion DoubleFloat +--R 6477113154539663 6477113154539663 +--R [[4,----------------,----------------,0], +--R 4503599627370496 4503599627370496 +--R 9 3144456110746797 1572228055373399 1 +--R [-,----------------,----------------,----------------], +--R 2 2251799813685248 1125899906842624 2251799813685248 +--R 1524278509167537 6097114036670147 1 +--R [5,----------------,----------------,- ----------------], +--R 1125899906842624 4503599627370496 4503599627370496 +--R 11 5918376472987543 5918376472987543 +--R [--,----------------,----------------,0], +--R 2 4503599627370496 4503599627370496 +--R 5759580877550931 1439895219387733 1 +--R [6,----------------,----------------,----------------], +--R 4503599627370496 1125899906842624 4503599627370496 +--R 13 5622253940478285 1405563485119571 1 +--R [--,----------------,----------------,- ----------------], +--R 2 4503599627370496 1125899906842624 4503599627370496 +--R 5505236449105581 5505236449105581 +--R [7,----------------,----------------,0], +--R 4503599627370496 4503599627370496 +--R 15 5406217818284429 1351554454571107 1 +--R [--,----------------,----------------,- ----------------], +--R 2 4503599627370496 1125899906842624 4503599627370496 +--R 41582579326595 5322570153804161 1 +--R [8,--------------,----------------,----------------], +--R 35184372088832 4503599627370496 4503599627370496 +--R 17 5251766987112641 2625883493556319 3 +--R [--,----------------,----------------,- ----------------], +--R 2 4503599627370496 2251799813685248 4503599627370496 +--R 5191565942789421 5191565942789423 1 +--R [9,----------------,----------------,----------------], +--R 4503599627370496 4503599627370496 2251799813685248 +--R 19 5140063981009437 5140063981009435 1 +--R [--,----------------,----------------,- ----------------], +--R 2 4503599627370496 4503599627370496 2251799813685248 +--R 2547844396210673 5095688792421347 1 +--R [10,----------------,----------------,----------------], +--R 2251799813685248 4503599627370496 4503599627370496 +--R 21 2528581071350257 2528581071350257 +--R [--,----------------,----------------,0], +--R 2 2251799813685248 2251799813685248 +--R 156982948788409 5023454361229091 3 +--R [11,---------------,----------------,----------------], +--R 140737488355328 4503599627370496 4503599627370496 +--R 23 2496869776861841 2496869776861841 +--R [--,----------------,----------------,0], +--R 2 2251799813685248 2251799813685248 +--R 2483677874720611 4967355749441223 1 +--R [12,----------------,----------------,----------------]] +--R 2251799813685248 4503599627370496 4503599627370496 +--R Type: List List Fraction Integer --E 9 \end{chunk} + +The actual table values from above are: +\begin{verbatim} + + 4., 1.4382080314544827, 1.4382080314544827, 0. + 4.5, 1.3964190296297461, 1.3964190296297465, 4.4408920985006262E-16 + 5., 1.3538312774552861, 1.3538312774552856, -4.4408920985006262E-16 + 5.5, 1.3141435657421192, 1.314143565742119, -2.2204460492503131E-16 + 6., 1.2788838604895616, 1.2788838604895618, 2.2204460492503131E-16 + 6.5, 1.2483911550017015, 1.2483911550017011, -4.4408920985006262E-16 + 7., 1.222408052360531, 1.222408052360531, 0. + 7.5, 1.2004214995996307, 1.2004214995996305, -2.2204460492503131E-16 + 8., 1.1818479869872081, 1.1818479869872081, 0. + 8.5, 1.1661265258117486, 1.1661265258117477, -8.8817841970012523E-16 + 9., 1.1527592087089249, 1.1527592087089251, 2.2204460492503131E-16 + 9.5, 1.1413234759526243, 1.1413234759526236, -6.6613381477509392E-16 + 10., 1.1314702047341079, 1.1314702047341079, 0. + 10.5, 1.1229155700177607, 1.1229155700177604, -2.2204460492503131E-16 + 11., 1.1154309389980384, 1.115430938998039, 6.6613381477509392E-16 + 11.5, 1.1088329263050773, 1.1088329263050771, -2.2204460492503131E-16 + 12., 1.1029745449067592, 1.1029745449067592, 0. + +\end{verbatim} + These are the Chebyshev coefficients used by Axiom in the range $(12 \le x \le 32)$ in the polynomial $$\sum_{k=0}^{47}\ ^{'}{A_kT_k(t)}$$ with the scaling factor $t=(x-22)/10$ \begin{chunk}{*} + --S 10 of 20 -eChebyshev:=[_ +eChebyshev:List Fraction Integer:=[machineFraction (x::DFLOAT) for x in _ +[_ 0.211702864043698668329789991614E1,_ -0.320423727375485794990618303177E-01,_ 0.889173207735316835890182400335E-02,_ @@ -756,61 +1379,162 @@ eChebyshev:=[_ 0.464065690883818114338414829515E-28,_ -0.101144734921151390948461800780E-28,_ 0.221721152271007711093046878345E-29,_ --0.488489046924378553224914645512E-30] +-0.488489046924378553224914645512E-30]] --R --R --R (11) ---R [2.1170286404 3698668329 789991614, - 0.0320423727 3754857949 9061830317 7, ---R 0.0088917320 7735316835 8901824003 35, ---R - 0.0025079528 0518929937 0883524420 63, ---R 0.0007202789 4659598754 8875760902 487, ---R - 0.0002103490 0585011305 3423531441 256, ---R 0.0000620573 2318276932 1658885773 0842, ---R - 0.0000182656 6749816702 6544915568 9733, ---R 0.0000052706 5157528936 3758077882 96811, ---R - 0.0000014596 6654761994 5753230667 19367, ---R 0.3781719973 5896367198 0484193981 E -6, ---R - 0.8842581282 8407192007 7971589012 E -7, ---R 0.1741749198 5383936137 7350309156 E -7, ---R - 0.2313517747 0436906350 6474480152 E -8, ---R - 0.1228609819 1808623883 210483523 E -9, ---R 0.2349966236 3228637047 8311381926 E -9, ---R - 0.1100719401 0272628769 0738963049 E -9, ---R 0.3848275157 8612071114 9705563369 E -10, ---R - 0.1148440967 4900158965 8439301603 E -10, ---R 0.3056876293 0885208263 08936262 E -11, ---R - 0.7388278729 2847356645 4163131431 E -12, ---R 0.1630933094 1659411056 4148013749 E -12, ---R - 0.3276989373 3127124965 7111774748 E -13, ---R 0.5898114347 0713196171 1164283918 E -14, ---R - 0.9099707635 9564920464 3554720718 E -15, ---R 0.1040752382 6695538658 5405697541 E -15, ---R - 0.1809815426 0592279322 7163355935 E -17, ---R - 0.3777098842 5639477336 9593494417 E -17, ---R 0.1580332901 0284795713 675988842 E -17, ---R - 0.4684291758 8088273064 8433752957 E -18, ---R 0.1199516852 5919809370 7533478542 E -18, ---R - 0.2823594749 8418651767 9349931117 E -19, ---R 0.6293738065 6446352262 7520190349 E -20, ---R - 0.1352410249 5047975630 5343973177 E -20, ---R 0.2837106053 8552914159 098042621 E -21, ---R - 0.5867007420 2463832353 1936371015 E -22, ---R 0.1205247636 0954731111 2449686917 E -22, ---R - 0.2474446616 9988486972 8416011246 E -23, ---R 0.5099962585 8378500814 2986465688 E -24, ---R - 0.1058382578 7754224088 7093294733 E -24, ---R 0.2215276245 0704827856 6429387155 E -25, ---R - 0.4679278754 7569625867 1852546231 E -26, ---R 0.9972872990 6020770482 4269828079 E -27, ---R - 0.2143267945 2167880459 1907805844 E -27, ---R 0.4640656908 8381811433 8414829515 E -28, ---R - 0.1011447349 2115139094 846180078 E -28, ---R 0.2217211522 7100771109 3046878345 E -29, ---R - 0.4884890469 2437855322 4914645512 E -30] ---R Type: List Float +--R 1191781174525585 2308896286734245 5125734562591489 +--R [----------------, - -----------------, ------------------, +--R 562949953421312 72057594037927936 576460752303423488 +--R 5782945443283619 6643400694768645 7760508554152537 +--R - -------------------, -------------------, - --------------------, +--R 2305843009213693952 9223372036854775808 36893488147419103232 +--R 4579022234608113 1347768374696647 3111243542753823 +--R --------------------, - --------------------, ---------------------, +--R 73786976294838206464 73786976294838206464 590295810358705651712 +--R 430817523790405 892933382543919 6681265515541997 +--R - ---------------------, ----------------------, - -----------------------, +--R 295147905179352825856 2361183241434822606848 75557863725914323419136 +--R 5264113943515377 349608917317223 +--R ------------------------, - ------------------------, +--R 302231454903657293676544 151115727451828646838272 +--R 4752954024447377 4545495773301317 +--R - --------------------------, --------------------------, +--R 38685626227668133590597632 19342813113834066795298816 +--R 8516403865936687 5954917375129569 +--R - --------------------------, ---------------------------, +--R 77371252455336267181195264 154742504910672534362390528 +--R 1777126320514413 3784229558356551 +--R - ---------------------------, ----------------------------, +--R 154742504910672534362390528 1237940039285380274899124224 +--R 1829249212076417 3230395805861779 +--R - ----------------------------, -----------------------------, +--R 2475880078570760549798248448 19807040628566084398385987584 +--R 324537308282921 7476748192279689 +--R - ----------------------------, -------------------------------, +--R 9903520314283042199192993792 1267650600228229401496703205376 +--R 4614099938648659 8443586448512127 +--R - -------------------------------, --------------------------------, +--R 5070602400912917605986812821504 81129638414606681695789005144064 +--R 4698549475627599 4902954613655079 +--R - ----------------------------------, - ----------------------------------, +--R 2596148429267413814265248164610048 1298074214633706907132624082305024 +--R 8205557557449405 4864446676744731 +--R ----------------------------------, - -----------------------------------, +--R 5192296858534827628530496329220096 10384593717069655257060992658440192 +--R 4982598068378341 +--R -----------------------------------, +--R 41538374868278621028243970633760768 +--R 4691501487801391 +--R - ------------------------------------, +--R 166153499473114484112975882535043072 +--R 8365812834992051 +--R -------------------------------------, +--R 1329227995784915872903807060280344576 +--R 112353847839265 +--R - -----------------------------------, +--R 83076749736557242056487941267521536 +--R 94279019844883 +--R ------------------------------------, +--R 332306998946228968225951765070086144 +--R 2495548964630185 +--R - --------------------------------------, +--R 42535295865117307932921825928971026432 +--R 8202490366728669 +--R ---------------------------------------, +--R 680564733841876926926749214863536422912 +--R 842010551651877 +--R - ---------------------------------------, +--R 340282366920938463463374607431768211456 +--R 2776683743867413 +--R ----------------------------------------, +--R 5444517870735015415413993718908291383296 +--R 2304953145686959 +--R - -----------------------------------------, +--R 21778071482940061661655974875633165533184 +--R 7719111107136643 +--R ------------------------------------------, +--R 348449143727040986586495598010130648530944 +--R 815245337677599 +--R - ------------------------------------------, +--R 174224571863520493293247799005065324265472 +--R 5560062486518125 +--R -------------------------------------------, +--R 5575186299632655785383929568162090376495104 +--R 1194911808461447 +--R - -------------------------------------------, +--R 5575186299632655785383929568162090376495104 +--R 4139604291112041 +--R --------------------------------------------, +--R 89202980794122492566142873090593446023921664 +--R 3608964738639231 +--R - ---------------------------------------------, +--R 356811923176489970264571492362373784095686656 +--R 6329020060058051 +--R ----------------------------------------------, +--R 2854495385411919762116571938898990272765493248 +--R 5577558921079621 +--R - -----------------------------------------------] +--R 11417981541647679048466287755595961091061972992 +--R Type: List Fraction Integer --E 10 \end{chunk} + +The actual table values from above are: +\begin{verbatim} + + 2.1170286404 3698668329 789991614 + -0.0320423727 3754857949 9061830317 7 + 0.0088917320 7735316835 8901824003 35 + -0.0025079528 0518929937 0883524420 63 + 0.0007202789 4659598754 8875760902 487 + -0.0002103490 0585011305 3423531441 256 + 0.0000620573 2318276932 1658885773 0842 + -0.0000182656 6749816702 6544915568 9733 + 0.0000052706 5157528936 3758077882 96811 + -0.0000014596 6654761994 5753230667 19367 + 0.3781719973 5896367198 0484193981 E -6 + -0.8842581282 8407192007 7971589012 E -7 + 0.1741749198 5383936137 7350309156 E -7 + -0.2313517747 0436906350 6474480152 E -8 + -0.1228609819 1808623883 210483523 E -9 + 0.2349966236 3228637047 8311381926 E -9 + -0.1100719401 0272628769 0738963049 E -9 + 0.3848275157 8612071114 9705563369 E -10 + -0.1148440967 4900158965 8439301603 E -10 + 0.3056876293 0885208263 08936262 E -11 + -0.7388278729 2847356645 4163131431 E -12 + 0.1630933094 1659411056 4148013749 E -12 + -0.3276989373 3127124965 7111774748 E -13 + 0.5898114347 0713196171 1164283918 E -14 + -0.9099707635 9564920464 3554720718 E -15 + 0.1040752382 6695538658 5405697541 E -15 + -0.1809815426 0592279322 7163355935 E -17 + -0.3777098842 5639477336 9593494417 E -17 + 0.1580332901 0284795713 675988842 E -17 + -0.4684291758 8088273064 8433752957 E -18 + 0.1199516852 5919809370 7533478542 E -18 + -0.2823594749 8418651767 9349931117 E -19 + 0.6293738065 6446352262 7520190349 E -20 + -0.1352410249 5047975630 5343973177 E -20 + 0.2837106053 8552914159 098042621 E -21 + -0.5867007420 2463832353 1936371015 E -22 + 0.1205247636 0954731111 2449686917 E -22 + -0.2474446616 9988486972 8416011246 E -23 + 0.5099962585 8378500814 2986465688 E -24 + -0.1058382578 7754224088 7093294733 E -24 + 0.2215276245 0704827856 6429387155 E -25 + -0.4679278754 7569625867 1852546231 E -26 + 0.9972872990 6020770482 4269828079 E -27 + -0.2143267945 2167880459 1907805844 E -27 + 0.4640656908 8381811433 8414829515 E -28 + -0.1011447349 2115139094 846180078 E -28 + 0.2217211522 7100771109 3046878345 E -29 + -0.4884890469 2437855322 4914645512 E -30 + +\end{verbatim} + In the following table there are 4 columns. The first column is the argument of Ei(x) shown in Table 4 in \cite{1}. The second column is the exact value shown in the table. Column 3 is the @@ -818,7 +1542,10 @@ value returned by Axiom and column 4 is the difference. See special.spad.dvi for details. \begin{chunk}{*} + --S 11 of 20 +t5:List List Fraction Integer:= [[machineFraction (x::DFLOAT) for x in line]_ + for line in _ [[12.00,1.10297454490675907267241234952,_ Ei5(12.00),Ei5(12.00)-1.10297454490675907267241234952],_ [13.25,1.09084489821547569266468614954,_ @@ -852,31 +1579,89 @@ See special.spad.dvi for details. [30.75,1.03487414989647969472990938990,_ Ei5(30.75),Ei5(30.75)-1.03487414989647969472990938990],_ [32.00,1.03341356421624104943493552567,_ - Ei5(32.00),Ei5(32.00)-1.03341356421624104943493552567]] + Ei5(32.00),Ei5(32.00)-1.03341356421624104943493552567]]] --R --R --R (12) ---R [[12.,1.1029745449067592,1.1029745449067585,- 6.6613381477509392E-16], ---R [13.25,1.0908448982154757,1.090844898215475,- 6.6613381477509392E-16], ---R [14.5,1.0813513957351912,1.0813513957351915,2.2204460492503131E-16], ---R [15.75,1.0737013841997571,1.0737013841997574,2.2204460492503131E-16], ---R [17.,1.0673936919585378,1.0673936919585385,6.6613381477509392E-16], ---R [18.25,1.0620966086221502,1.0620966086221502,0.], ---R [19.5,1.057581342158725,1.0575813421587252,2.2204460492503131E-16], ---R [20.75,1.0536844512894095,1.0536844512894095,0.], ---R [22.,1.0502857196851898,1.0502857196851898,0.], ---R [23.25,1.0472945517053249,1.0472945517053245,- 4.4408920985006262E-16], ---R [24.5,1.0446412679046437,1.0446412679046437,0.], ---R [25.75,1.0422713372023202,1.04227133720232,- 2.2204460492503131E-16], ---R [27.,1.0401414383230105,1.0401414383230101,- 4.4408920985006262E-16], ---R [28.25,1.0382167003601459,1.0382167003601459,0.], ---R [29.5,1.0364687262924119,1.0364687262924113,- 6.6613381477509392E-16], ---R [30.75,1.0348741498964797,1.0348741498964795,- 2.2204460492503131E-16], ---R [32.,1.033413564216241,1.0334135642162412,2.2204460492503131E-16]] ---R Type: List List OnePointCompletion DoubleFloat +--R 2483677874720611 1241838937360305 1 +--R [[12,----------------,----------------,- ----------------], +--R 2251799813685248 1125899906842624 2251799813685248 +--R 53 2456364338561111 1228182169280555 1 +--R [--,----------------,----------------,- ----------------], +--R 4 2251799813685248 1125899906842624 2251799813685248 +--R 29 4869973742889573 2434986871444787 1 +--R [--,----------------,----------------,----------------], +--R 2 4503599627370496 2251799813685248 4503599627370496 +--R 63 1208880288447303 4835521153789213 1 +--R [--,----------------,----------------,----------------], +--R 4 1125899906842624 4503599627370496 4503599627370496 +--R 600889229170261 1201778458340523 1 +--R [17,---------------,----------------,----------------], +--R 562949953421312 1125899906842624 1125899906842624 +--R 73 4783257890822183 4783257890822183 +--R [--,----------------,----------------,0], +--R 4 4503599627370496 4503599627370496 +--R 39 4762922938460023 595365367307503 1 +--R [--,----------------,---------------,----------------], +--R 2 4503599627370496 562949953421312 4503599627370496 +--R 83 4745372902193069 2372686451096535 1 +--R [--,----------------,----------------,----------------], +--R 4 4503599627370496 2251799813685248 4503599627370496 +--R 4730066375806773 2365033187903387 1 +--R [22,----------------,----------------,----------------], +--R 4503599627370496 2251799813685248 4503599627370496 +--R 93 4716595352807251 2358297676403625 1 +--R [--,----------------,----------------,- ----------------], +--R 4 4503599627370496 2251799813685248 4503599627370496 +--R 49 4704646024871195 1176161506217799 1 +--R [--,----------------,----------------,----------------], +--R 2 4503599627370496 1125899906842624 4503599627370496 +--R 103 2346986402921659 4693972805843317 1 +--R [---,----------------,----------------,- ----------------], +--R 4 2251799813685248 4503599627370496 4503599627370496 +--R 4684380594044121 585547574255515 1 +--R [27,----------------,---------------,- ----------------], +--R 4503599627370496 562949953421312 4503599627370496 +--R 113 2337856172435889 4675712344871779 1 +--R [---,----------------,----------------,----------------], +--R 4 2251799813685248 4503599627370496 4503599627370496 +--R 59 2333920084755839 1166960042377919 1 +--R [--,----------------,----------------,- ----------------], +--R 2 2251799813685248 1125899906842624 2251799813685248 +--R 123 582582354481143 582582354481143 +--R [---,---------------,---------------,0], +--R 4 562949953421312 562949953421312 +--R 4654080942723879 581760117840485 1 +--R [32,----------------,---------------,----------------]] +--R 4503599627370496 562949953421312 4503599627370496 +--R Type: List List Fraction Integer --E 11 \end{chunk} + +The actual table values from above are: +\begin{verbatim} + + 12., 1.1029745449067592, 1.1029745449067585, -6.6613381477509392E-16 + 13.25, 1.0908448982154757, 1.090844898215475, -6.6613381477509392E-16 + 14.5, 1.0813513957351912, 1.0813513957351915, 2.2204460492503131E-16 + 15.75, 1.0737013841997571, 1.0737013841997574, 2.2204460492503131E-16 + 17., 1.0673936919585378, 1.0673936919585385, 6.6613381477509392E-16 + 18.25, 1.0620966086221502, 1.0620966086221502, 0. + 19.5, 1.057581342158725, 1.0575813421587252, 2.2204460492503131E-16 + 20.75, 1.0536844512894095, 1.0536844512894095, 0. + 22., 1.0502857196851898, 1.0502857196851898, 0. + 23.25, 1.0472945517053249, 1.0472945517053245, -4.4408920985006262E-16 + 24.5, 1.0446412679046437, 1.0446412679046437, 0. + 25.75, 1.0422713372023202, 1.04227133720232, -2.2204460492503131E-16 + 27., 1.0401414383230105, 1.0401414383230101, -4.4408920985006262E-16 + 28.25, 1.0382167003601459, 1.0382167003601459, 0. + 29.5, 1.0364687262924119, 1.0364687262924113, -6.6613381477509392E-16 + 30.75, 1.0348741498964797, 1.0348741498964795, -2.2204460492503131E-16 + 32., 1.033413564216241, 1.0334135642162412, 2.2204460492503131E-16 + +\end{verbatim} + These are the Chebyshev coefficients used by Axiom in the range $(32 \le x < \infty)$ in the polynomial $$\sum_{k=0}^{46}\ ^{'}{A_kT_k(t)}$$ @@ -885,7 +1670,8 @@ with the scaling factor $t=(64/X)-1$ \begin{chunk}{*} --S 12 of 20 -fChebyshev:=[_ +fChebyshev:List Fraction Integer:=[machineFraction (x::DFLOAT) for x in _ +[_ 0.203284394579616699087873844202E1,_ 0.166992045203136285147618434339E-01,_ 0.284528472436134680742489985325E-03,_ @@ -932,60 +1718,173 @@ fChebyshev:=[_ 0.845381406448938089437361193598E-29,_ 0.356575515120151526590791715785E-29,_ -0.138365242347797751810195772006E-29,_ --0.606214265320934505767865286306E-30] +-0.606214265320934505767865286306E-30]] --R --R --R (13) ---R [2.0328439457 9616699087 873844202, 0.0166992045 2031362851 4761843433 9, ---R 0.0002845284 7243613468 0742489985 325, ---R 0.0000075639 4435851620 6489487866 93854, ---R 0.2798971289 4508591575 048431809 E -6, ---R 0.1357901828 5345310695 2556392593 E -7, ---R 0.8343596202 0404692558 5610289412 E -9, ---R 0.6370971727 6402484382 7524337306 E -10, ---R 0.6007247608 8118612357 608308485 E -11, ---R 0.7022876174 6797735907 5059216588 E -12, ---R 0.1018302673 7036876930 9667322152 E -12, ---R 0.1761812903 4308800404 0656741554 E -13, ---R 0.3250828614 2353606942 4072007647 E -14, ---R 0.5071770025 5058186788 1479300685 E -15, ---R 0.1665177387 0432942985 3520036957 E -16, ---R - 0.3166753890 7975144007 2410018963 E -16, ---R - 0.1588403763 6641415154 8423134074 E -16, ---R - 0.4175513256 1380188308 9626455063 E -17, ---R - 0.2892347749 7071418820 2868862358 E -18, ---R 0.2800625903 3966080728 9978777339 E -18, ---R 0.1322938639 5392708914 0532005364 E -18, ---R 0.1804447444 1773019958 5334811191 E -19, ---R - 0.7905384086 5226165620 2021080364 E -20, ---R - 0.4435711366 3695734471 8167314045 E -20, ---R - 0.4264103994 9781026176 0579779746 E -21, ---R 0.3920101766 9371439072 5625388636 E -21, ---R 0.1527378051 3439636447 2804486402 E -21, ---R - 0.1024849527 0494906078 6953149788 E -22, ---R - 0.2134907874 7710893794 8904287231 E -22, ---R - 0.3239139475 1602368761 4279789345 E -23, ---R 0.2142183762 2964597029 6249355934 E -23, ---R 0.8234609419 6189955316 9207838151 E -24, ---R - 0.1524652829 6206721081 1495038147 E -24, ---R - 0.1378208282 4882440129 0438126477 E -24, ---R 0.2131311201 4287370679 1513005998 E -26, ---R 0.2012649651 8713266585 9213006507 E -25, ---R 0.1995535662 0563740232 0607178286 E -26, ---R - 0.2798995812 2017971142 6020884464 E -26, ---R - 0.5534511830 5070025094 978494256 E -27, ---R 0.3884995422 6845525312 9749000696 E -27, ---R 0.1121304407 2330701254 0043264712 E -27, ---R - 0.5566568286 7445948805 7823816866 E -28, ---R - 0.2045482612 4651357628 8865878722 E -28, ---R 0.8453814064 4893808943 7361193598 E -29, ---R 0.3565755151 2015152659 0791715785 E -29, ---R - 0.1383652423 4779775181 0195772006 E -29, ---R - 0.6062142653 2093450576 7865286306 E -30] ---R Type: List Float +--R 4577557618394993 2406609000162181 2624311956356449 +--R [----------------, ------------------, -------------------, +--R 2251799813685248 144115188075855872 9223372036854775808 +--R 1116241166154621 5287107281526867 4104006452543279 +--R ---------------------, -----------------------, ------------------------, +--R 147573952589676412928 18889465931478580854784 302231454903657293676544 +--R 8069431101668225 4929300619250635 +--R -------------------------, --------------------------, +--R 9671406556917033397649408 77371252455336267181195264 +--R 3718306170424781 108673745094743 +--R ---------------------------, ---------------------------, +--R 618970019642690137449562112 154742504910672534362390528 +--R 8067824972090565 2791703980654993 +--R -----------------------------, ------------------------------, +--R 79228162514264337593543950336 158456325028528675187087900672 +--R 2060457422037279 2571692926820797 +--R ------------------------------, -------------------------------, +--R 633825300114114700748351602688 5070602400912917605986812821504 +--R 5403809572280081 5138351962169023 +--R ---------------------------------, - ---------------------------------, +--R 324518553658426726783156020576256 162259276829213363391578010288128 +--R 2577332460049443 5420126090653991 +--R - ---------------------------------, - ----------------------------------, +--R 162259276829213363391578010288128 1298074214633706907132624082305024 +--R 1501792813459467 1454168108013747 +--R - ----------------------------------, ----------------------------------, +--R 5192296858534827628530496329220096 5192296858534827628530496329220096 +--R 5495272113691275 2998152574653761 +--R -----------------------------------, ------------------------------------, +--R 41538374868278621028243970633760768 166153499473114484112975882535043072 +--R 1313507230654803 +--R - ------------------------------------, +--R 166153499473114484112975882535043072 +--R 2948035864699899 +--R - ------------------------------------, +--R 664613997892457936451903530140172288 +--R 4534373125650557 +--R - --------------------------------------, +--R 10633823966279326983230456482242756608 +--R 8337134423902027 +--R --------------------------------------, +--R 21267647932558653966460912964485513216 +--R 3248373865590091 +--R --------------------------------------, +--R 21267647932558653966460912964485513216 +--R 1743691114011025 +--R - ---------------------------------------, +--R 170141183460469231731687303715884105728 +--R 7264715047852567 +--R - ---------------------------------------, +--R 340282366920938463463374607431768211456 +--R 4408888189578287 +--R - ----------------------------------------, +--R 1361129467683753853853498429727072845824 +--R 2915789444055361 +--R ----------------------------------------, +--R 1361129467683753853853498429727072845824 +--R 4483347814363851 +--R ----------------------------------------, +--R 5444517870735015415413993718908291383296 +--R 3320399831014583 +--R - -----------------------------------------, +--R 21778071482940061661655974875633165533184 +--R 6002943698881805 +--R - -----------------------------------------, +--R 43556142965880123323311949751266331066368 +--R 5941228505229555 +--R -------------------------------------------, +--R 2787593149816327892691964784081045188247552 +--R 7013060478170909 +--R ------------------------------------------, +--R 348449143727040986586495598010130648530944 +--R 2781370770881269 +--R -------------------------------------------, +--R 1393796574908163946345982392040522594123776 +--R 3901230776229159 +--R - -------------------------------------------, +--R 1393796574908163946345982392040522594123776 +--R 3085593453259749 +--R - -------------------------------------------, +--R 5575186299632655785383929568162090376495104 +--R 4331914650937299 +--R --------------------------------------------, +--R 11150372599265311570767859136324180752990208 +--R 5001184775138823 +--R --------------------------------------------, +--R 44601490397061246283071436545296723011960832 +--R 2482772419858247 +--R - --------------------------------------------, +--R 44601490397061246283071436545296723011960832 +--R 912315730972195 +--R - --------------------------------------------, +--R 44601490397061246283071436545296723011960832 +--R 3016421654526915 +--R ---------------------------------------------, +--R 356811923176489970264571492362373784095686656 +--R 5089215812306753 +--R ----------------------------------------------, +--R 1427247692705959881058285969449495136382746624 +--R 1974814728915953 +--R - ----------------------------------------------, +--R 1427247692705959881058285969449495136382746624 +--R 3460871645858969 +--R - ----------------------------------------------] +--R 5708990770823839524233143877797980545530986496 +--R Type: List Fraction Integer --E 12 \end{chunk} + +The actual table values from above are: +\begin{verbatim} + + 2.0328439457 9616699087 873844202 + 0.0166992045 2031362851 4761843433 9 + 0.0002845284 7243613468 0742489985 325 + 0.0000075639 4435851620 6489487866 93854 + 0.2798971289 4508591575 048431809 E -6 + 0.1357901828 5345310695 2556392593 E -7 + 0.8343596202 0404692558 5610289412 E -9 + 0.6370971727 6402484382 7524337306 E -10 + 0.6007247608 8118612357 608308485 E -11 + 0.7022876174 6797735907 5059216588 E -12 + 0.1018302673 7036876930 9667322152 E -12 + 0.1761812903 4308800404 0656741554 E -13 + 0.3250828614 2353606942 4072007647 E -14 + 0.5071770025 5058186788 1479300685 E -15 + 0.1665177387 0432942985 3520036957 E -16 + -0.3166753890 7975144007 2410018963 E -16 + -0.1588403763 6641415154 8423134074 E -16 + -0.4175513256 1380188308 9626455063 E -17 + -0.2892347749 7071418820 2868862358 E -18 + 0.2800625903 3966080728 9978777339 E -18 + 0.1322938639 5392708914 0532005364 E -18 + 0.1804447444 1773019958 5334811191 E -19 + -0.7905384086 5226165620 2021080364 E -20 + -0.4435711366 3695734471 8167314045 E -20 + -0.4264103994 9781026176 0579779746 E -21 + 0.3920101766 9371439072 5625388636 E -21 + 0.1527378051 3439636447 2804486402 E -21 + -0.1024849527 0494906078 6953149788 E -22 + -0.2134907874 7710893794 8904287231 E -22 + -0.3239139475 1602368761 4279789345 E -23 + 0.2142183762 2964597029 6249355934 E -23 + 0.8234609419 6189955316 9207838151 E -24 + -0.1524652829 6206721081 1495038147 E -24 + -0.1378208282 4882440129 0438126477 E -24 + 0.2131311201 4287370679 1513005998 E -26 + 0.2012649651 8713266585 9213006507 E -25 + 0.1995535662 0563740232 0607178286 E -26 + -0.2798995812 2017971142 6020884464 E -26 + -0.5534511830 5070025094 978494256 E -27 + 0.3884995422 6845525312 9749000696 E -27 + 0.1121304407 2330701254 0043264712 E -27 + -0.5566568286 7445948805 7823816866 E -28 + -0.2045482612 4651357628 8865878722 E -28 + 0.8453814064 4893808943 7361193598 E -29 + 0.3565755151 2015152659 0791715785 E -29 + -0.1383652423 4779775181 0195772006 E -29 + -0.6062142653 2093450576 7865286306 E -30 + +\end{verbatim} + In the following table there are 4 columns. The first column is the argument of Ei(x) shown in Table 4 in \cite{1}. The second column is the exact value shown in the table. Column 3 is the @@ -993,7 +1892,10 @@ value returned by Axiom and column 4 is the difference. See special.spad.dvi for details. \begin{chunk}{*} + --S 13 of 20 +t6:List List Fraction Integer:= [[machineFraction (x::DFLOAT) for x in line]_ + for line in _ [[32,1.03341356421624104943493552567,_ Ei6(32.0),Ei6(32.0)-1.03341356421624104943493552567],_ [34+2/15,1.03118521236465926355875784663,_ @@ -1025,79 +1927,156 @@ See special.spad.dvi for details. [256,1.00393713090569862788009078297,_ Ei6(256.0),Ei6(256.0)-1.00393713090569862788009078297],_ [512,1.00196079945071192531337468473,_ - Ei6(512.0),Ei6(512.0)-1.00196079945071192531337468473],_ -[infinity(),1.00000000000000000000000000001,_ - Ei6(infinity()),Ei6(infinity())-1.00000000000000000000000000001]] + Ei6(512.0),Ei6(512.0)-1.00196079945071192531337468473]]] --R --R --R (14) ---R [[32.,1.033413564216241,1.0334135642162412,2.2204460492503131E-16], ---R ---R 512 ---R [---, 1.0311852123 6465926355 875784663, 1.0311852123646588, ---R 15 ---R - 4.4408920985006262E-16] ---R , ---R 256 ---R [---,1.0289774041 0580800863 378435059,1.028977404105808,0.], ---R 7 ---R 512 ---R [---,1.0267896837 0902852450 984510823,1.0267896837090285,0.], ---R 13 ---R ---R 128 ---R [---, 1.0246216146 8107839101 187804247, 1.0246216146810787, ---R 3 ---R 2.2204460492503131E-16] ---R , ---R 512 ---R [---,1.0224727784 0542059591 275364791,1.0224727784054206,0.], ---R 11 ---R 256 ---R [---,1.0203427729 3078377487 217829808,1.0203427729307837,0.], ---R 5 ---R ---R 512 ---R [---, 1.0182312118 8483269682 337017143, 1.0182312118848329, ---R 9 ---R 2.2204460492503131E-16] ---R , ---R [64.,1.0161377234943254,1.0161377234943252,- 2.2204460492503131E-16], ---R ---R 512 ---R [---, 1.0140619496 9697133145 942329335, 1.0140619496969712, ---R 7 ---R - 2.2204460492503131E-16] ---R , ---R 256 ---R [---,1.0120035453 3298848201 864466702,1.0120035453329885,0.], ---R 3 ---R ---R 512 ---R [---, 1.0099621774 0644975574 36754557, 1.0099621774064493, ---R 5 ---R - 4.4408920985006262E-16] ---R , ---R [128.,1.0079375244081401,1.0079375244081401,0.], ---R ---R 512 ---R [---, 1.0059292756 9292911294 663030932, 1.0059292756929286, ---R 3 ---R - 4.4408920985006262E-16] ---R , ---R [256.,1.0039371309056986,1.0039371309056981,- 4.4408920985006262E-16], ---R [512.,1.001960799450712,1.0019607994507116,- 4.4408920985006262E-16], ---R [infinity,1.,1.,0.]] ---R Type: List List Any +--R 4654080942723879 581760117840485 1 +--R [[32,----------------,---------------,----------------], +--R 4503599627370496 562949953421312 4503599627370496 +--R 4803839602528529 4644045338155445 4644045338155443 1 +--R [----------------,----------------,----------------,- ----------------], +--R 140737488355328 4503599627370496 4503599627370496 2251799813685248 +--R 2573485501354569 4634102253703577 4634102253703577 +--R [----------------,----------------,----------------,0], +--R 70368744177664 4503599627370496 4503599627370496 +--R 1385722962267845 2312124818469925 2312124818469925 +--R [----------------,----------------,----------------,0], +--R 35184372088832 2251799813685248 2251799813685248 +--R 6004799503160661 1153621380518365 2307242761036731 1 +--R [----------------,----------------,----------------,----------------], +--R 140737488355328 1125899906842624 2251799813685248 2251799813685248 +--R 3275345183542179 4604808023823127 575601002977891 1 +--R [----------------,----------------,---------------,----------------], +--R 70368744177664 4503599627370496 562949953421312 4503599627370496 +--R 3602879701896397 574401916495157 574401916495157 +--R [----------------,---------------,---------------,0], +--R 70368744177664 562949953421312 562949953421312 +--R 2001599834386887 4585705706421541 2292852853210771 1 +--R [----------------,----------------,----------------,----------------], +--R 35184372088832 4503599627370496 2251799813685248 4503599627370496 +--R 4576277472886147 4576277472886147 +--R [64,----------------,----------------,0], +--R 4503599627370496 4503599627370496 +--R 2573485501354569 2283464509392939 2283464509392939 +--R [----------------,----------------,----------------,0], +--R 35184372088832 2251799813685248 2251799813685248 +--R 6004799503160661 4557658789659267 1139414697414817 1 +--R [----------------,----------------,----------------,----------------], +--R 70368744177664 4503599627370496 1125899906842624 4503599627370496 +--R 3602879701896397 4548465285825981 1137116321456495 1 +--R [----------------,----------------,----------------,- ----------------], +--R 35184372088832 4503599627370496 1125899906842624 4503599627370496 +--R 567418382417155 567418382417155 +--R [128,---------------,---------------,0], +--R 562949953421312 562949953421312 +--R 6004799503160661 1132575677792937 2265151355585873 1 +--R [----------------,----------------,----------------,- ----------------], +--R 35184372088832 1125899906842624 2251799813685248 2251799813685248 +--R 4521330888650309 4521330888650307 1 +--R [256,----------------,----------------,- ----------------], +--R 4503599627370496 4503599627370496 2251799813685248 +--R 2256215141523035 4512430283046069 1 +--R [512,----------------,----------------,- ----------------]] +--R 2251799813685248 4503599627370496 4503599627370496 +--R Type: List List Fraction Integer --E 13 \end{chunk} -In the following table we show values returned by -$(Ei(x)-\log x-\gamma)/x$ where gamma is as shown above. -Abramowitz and Stegun, ``Handbook of Mathematical Functions'', -Dover Publications, Inc. New York 1965. p 238 +The actual table values from above are: +\begin{verbatim} + + [[ 32., 1.033413564216241, + 1.0334135642162412, + 2.2204460492503131E-16], + + 512 + [---, 1.0311852123 6465926355 875784663, + 15 + 1.0311852123646588, + - 4.4408920985006262E-16] + , + 256 + [---, 1.0289774041 0580800863 378435059, + 7 + 1.028977404105808, + 0.], + 512 + [---, 1.0267896837 0902852450 984510823, + 13 + 1.0267896837090285, + 0.], + + 128 + [---, 1.0246216146 8107839101 187804247, + 3 + 1.0246216146810787, + 2.2204460492503131E-16] + , + 512 + [---, 1.0224727784 0542059591 275364791, + 11 + 1.0224727784054206, + 0.], + 256 + [---, 1.0203427729 3078377487 217829808, + 5 + 1.0203427729307837, + 0.], + + 512 + [---, 1.0182312118 8483269682 337017143, + 9 + 1.0182312118848329, + 2.2204460492503131E-16] + , + [64., 1.0161377234943254, + 1.0161377234943252, + -2.2204460492503131E-16], + 512 + [---, 1.0140619496 9697133145 942329335, + 7 + 1.0140619496969712, + -2.2204460492503131E-16] + , + 256 + [---, 1.0120035453 3298848201 864466702, + 3 + 1.0120035453329885, + 0.], + + 512 + [---, 1.0099621774 0644975574 36754557, + 5 + 1.0099621774064493, + -4.4408920985006262E-16] + , + [128., 1.0079375244081401, + 1.0079375244081401, + 0.], + + 512 + [---, 1.0059292756 9292911294 663030932, + 3 + 1.0059292756929286, + -4.4408920985006262E-16] + , + [256., 1.0039371309056986, + 1.0039371309056981, + -4.4408920985006262E-16], + + [512., 1.001960799450712, + 1.0019607994507116, + -4.4408920985006262E-16], + + [infinity,1., + 1., + 0.]] + +\end{verbatim} \begin{chunk}{*} + --S 14 of 20 h(x:DFLOAT):DFLOAT== x=0.0::DFLOAT => 1.0 @@ -1109,7 +2088,18 @@ h(x:DFLOAT):DFLOAT== --R Type: Void --E 14 +\end{chunk} + +In the following table we show values returned by +$(Ei(x)-\log x-\gamma)/x$ where gamma is as shown above. +Abramowitz and Stegun, ``Handbook of Mathematical Functions'', +Dover Publications, Inc. New York 1965. p 238 + +\begin{chunk}{*} + --S 15 of 20 +t7:List List Fraction Integer:= [[machineFraction (x::DFLOAT) for x in line]_ + for line in _ [[0.00,1.000000000,h(0.00),h(0.00)-1.000000000],_ [0.01,1.002505566,h(0.01),h(0.01)-1.002505566],_ [0.02,1.005022306,h(0.02),h(0.02)-1.005022306],_ @@ -1160,157 +2150,335 @@ h(x:DFLOAT):DFLOAT== [0.47,1.130940671,h(0.47),h(0.47)-1.130940671],_ [0.48,1.134046738,h(0.48),h(0.48)-1.134046738],_ [0.49,1.137167432,h(0.49),h(0.49)-1.137167432],_ - [0.50,1.140302841,h(0.50),h(0.50)-1.140302841]] + [0.50,1.140302841,h(0.50),h(0.50)-1.140302841]]] --R --R Compiling function h with type DoubleFloat -> DoubleFloat --R --R (16) ---R [[0.,1.,1.,0.], ---R [1.0E-2,1.002505566,1.002505565988876,- 1.1123990617534218E-11], ---R [2.0E-2,1.0050223060000001,1.0050223058229502,- 1.7704993027223281E-10], ---R ---R [2.9999999999999999E-2, 1.007550283, 1.0075502826056368, ---R - 3.9436320875552155E-10] ---R , ---R ---R [4.0000000000000001E-2, 1.0100895599999999, 1.0100895598460362, ---R - 1.5396373065357238E-10] ---R , ---R ---R [5.0000000000000003E-2, 1.012640202, 1.0126402014616676, ---R - 5.3833248969681335E-10] ---R , ---R ---R [5.9999999999999998E-2, 1.015202272, 1.0152022717813329, ---R - 2.1866708443951666E-10] ---R , ---R ---R [7.0000000000000007E-2, 1.017775836, 1.0177758355479642, ---R - 4.5203574217111964E-10] ---R , ---R ---R [8.0000000000000002E-2, 1.0203609579999999, 1.0203609579215664, ---R - 7.8433481931483584E-11] ---R , ---R ---R [8.9999999999999997E-2, 1.0229577050000001, 1.0229577044820872, ---R - 5.1791282373869763E-10] ---R , ---R ---R [0.10000000000000001, 1.0255661410000001, 1.0255661412323602, ---R 2.3236013113603349E-10] ---R , ---R [0.11,1.028186335,1.0281863346010778,- 3.989222285838423E-10], ---R [0.12,1.030818352,1.0308183514457605,- 5.5423954314903767E-10], ---R [0.13,1.033462259,1.0334622590557541,5.5754068029045811E-11], ---R [0.14000000000000001,1.036118125,1.0361181251552536,1.5525358776358189E-10], ---R ---R [0.14999999999999999, 1.0387860179999999, 1.0387860179063644, ---R - 9.3635543763070928E-11] ---R , ---R [0.16,1.0414660060000001,1.0414660059121499,- 8.7850171581749237E-11], ---R ---R [0.17000000000000001, 1.0441581579999999, 1.0441581582197257, ---R 2.1972579311579921E-10] ---R , ---R ---R [0.17999999999999999, 1.0468625439999999, 1.0468625443233892, ---R 3.233893153264944E-10] ---R , ---R [0.19,1.0495792340000001,1.0495792341677359,1.6773582522944253E-10], ---R [0.20000000000000001,1.052308298,1.0523082981508358,1.5083578830399347E-10], ---R ---R [0.20999999999999999, 1.0550498070000001, 1.055049807127405, ---R 1.2740497545848939E-10] ---R , ---R [0.22,1.0578038329999999,1.0578038324120198,- 5.8798010904581588E-10], ---R ---R [0.23000000000000001, 1.0605704460000001, 1.0605704457823433, ---R - 2.1765678148710776E-10] ---R , ---R ---R [0.23999999999999999, 1.0633497190000001, 1.0633497194823853, ---R 4.8238524286148277E-10] ---R , ---R [0.25,1.0661417259999999,1.0661417262257755,2.2577562042158661E-10], ---R ---R [0.26000000000000001, 1.0689465389999999, 1.0689465391990731, ---R 1.9907320236711712E-10] ---R , ---R [0.27000000000000002,1.071764232,1.0717642320650853,6.5085270506415327E-11], ---R ---R [0.28000000000000003, 1.0745948789999999, 1.0745948789662336, ---R - 3.3766323070949511E-11] ---R , ---R ---R [0.28999999999999998, 1.0774385550000001, 1.0774385545279166, ---R - 4.7208348341598594E-10] ---R , ---R ---R [0.29999999999999999, 1.0802953340000001, 1.0802953338619241, ---R - 1.3807599508197654E-10] ---R , ---R [0.31,1.083165293,1.0831652925698594,- 4.3014058981327707E-10], ---R ---R [0.32000000000000001, 1.0860485070000001, 1.0860485067465939, ---R - 2.5340618492464273E-10] ---R , ---R ---R [0.33000000000000002, 1.088945053, 1.0889450529837443, ---R - 1.6255663481956617E-11] ---R , ---R [0.34000000000000002,1.091855008,1.0918550083731842,3.7318415024856222E-10], ---R ---R [0.34999999999999998, 1.094778451, 1.0947784505105673, ---R - 4.8943271657719833E-10] ---R , ---R ---R [0.35999999999999999, 1.0977154579999999, 1.0977154574988892, ---R - 5.0111070848402051E-10] ---R , ---R [0.37,1.100666108,1.1006661079520708,- 4.7929216151487708E-11], ---R [0.38,1.1036304809999999,1.1036304809985678,- 1.4321877017664519E-12], ---R ---R [0.39000000000000001, 1.1066086559999999, 1.1066086562850108, ---R 2.8501090376664706E-10] ---R , ---R ---R [0.40000000000000002, 1.1096007139999999, 1.1096007139798676, ---R - 2.0132340239342739E-11] ---R , ---R [0.40999999999999998,1.112606735,1.1126067347771349,- 2.228650597402293E-10] ---R , ---R [0.41999999999999998,1.1156268,1.1156267999000615,- 9.9938501918472866E-11], ---R [0.42999999999999999,1.118660991,1.1186609911048895,1.0488943047448629E-10], ---R [0.44,1.121709391,1.1217093906846374,- 3.1536262490305944E-10], ---R ---R [0.45000000000000001, 1.124772082, 1.1247720814728976, ---R - 5.2710236175812497E-10] ---R , ---R ---R [0.46000000000000002, 1.1278491470000001, 1.1278491468476701, ---R - 1.52329926450534E-10] ---R , ---R ---R [0.46999999999999997, 1.1309406710000001, 1.1309406707352239, ---R - 2.6477620096443388E-10] ---R , ---R ---R [0.47999999999999998, 1.1340467380000001, 1.134046737613986, ---R - 3.8601410956573545E-10] ---R , ---R [0.48999999999999999,1.137167432,1.1371674325184589,5.1845883142220828E-10], ---R [0.5,1.140302841,1.1403028410431715,4.3171466401759062E-11]] ---R Type: List List DoubleFloat +--R [[0,1,1,0], +--R 2882303761517117 282180230842153 4514883693424401 50047 +--R [------------------,---------------,----------------,- ----------------], +--R 288230376151711744 281474976710656 4503599627370496 4503599627370496 +--R 2882303761517117 1131554520700159 4526218082003301 797335 +--R [------------------,----------------,----------------,- ----------------], +--R 144115188075855872 1125899906842624 4503599627370496 4503599627370496 +--R 1080863910568919 4537603079075837 567200384662475 1776037 +--R [-----------------,----------------,---------------,- ----------------], +--R 36028797018963968 4503599627370496 562949953421312 4503599627370496 +--R 2882303761517117 1137259741506707 4549038965333451 693377 +--R [-----------------,----------------,----------------,- ----------------], +--R 72057594037927936 1125899906842624 4503599627370496 4503599627370496 +--R 7205759403792793 4560526036387583 570065754245395 2424423 +--R [------------------,----------------,---------------,- ----------------], +--R 144115188075855872 4503599627370496 562949953421312 4503599627370496 +--R 1080863910568919 285754035867805 4572064572900067 984813 +--R [-----------------,---------------,----------------,- ----------------], +--R 18014398509481984 281474976710656 4503599627370496 4503599627370496 +--R 5044031582654955 4583654875756295 4583654873720515 508945 +--R [-----------------,----------------,----------------,- ----------------], +--R 72057594037927936 4503599627370496 4503599627370496 1125899906842624 +--R 2882303761517117 2297648615116101 71801519216859 176613 +--R [-----------------,----------------,--------------,- ----------------], +--R 36028797018963968 2251799813685248 70368744177664 2251799813685248 +--R 3242591731706757 4606991939053777 4606991936721311 1166233 +--R [-----------------,----------------,----------------,- ----------------], +--R 36028797018963968 4503599627370496 4503599627370496 2251799813685248 +--R 7205759403792793 4618739290451397 1154684822874465 1046463 +--R [-----------------,----------------,----------------,----------------], +--R 72057594037927936 4503599627370496 1125899906842624 4503599627370496 +--R 990791918021509 4630539595173435 4630539593376855 449145 +--R [----------------,----------------,----------------,- ----------------], +--R 9007199254740992 4503599627370496 4503599627370496 1125899906842624 +--R 1080863910568919 1160598286488467 580299142932225 624017 +--R [----------------,----------------,---------------,- ----------------], +--R 9007199254740992 1125899906842624 562949953421312 1125899906842624 +--R 4683743612465315 4654300244533871 4654300244784955 62771 +--R [-----------------,----------------,----------------,----------------], +--R 36028797018963968 4503599627370496 4503599627370496 1125899906842624 +--R 5044031582654955 583282650207727 2333130601180511 349603 +--R [-----------------,---------------,----------------,----------------], +--R 36028797018963968 562949953421312 2251799813685248 2251799813685248 +--R 5404319552844595 4678276323582481 4678276323160787 210847 +--R [-----------------,----------------,----------------,- ----------------], +--R 36028797018963968 4503599627370496 4503599627370496 2251799813685248 +--R 2882303761517117 2345172958270319 1172586479036247 197825 +--R [-----------------,----------------,----------------,- ----------------], +--R 18014398509481984 2251799813685248 1125899906842624 2251799813685248 +--R 3062447746611937 4702470291284663 2351235146137109 989555 +--R [-----------------,----------------,----------------,----------------], +--R 18014398509481984 4503599627370496 2251799813685248 4503599627370496 +--R 3242591731706757 4714649763066529 4714649764522945 45513 +--R [-----------------,----------------,----------------,---------------], +--R 18014398509481984 4503599627370496 4503599627370496 140737488355328 +--R 6845471433603153 2363442323569105 4726884647893627 755417 +--R [-----------------,----------------,----------------,----------------], +--R 36028797018963968 2251799813685248 4503599627370496 4503599627370496 +--R 7205759403792793 74049613417995 4739175259430985 679305 +--R [-----------------,--------------,----------------,----------------], +--R 36028797018963968 70368744177664 4503599627370496 4503599627370496 +--R 7566047373982433 4751521917662513 4751521918236295 286891 +--R [-----------------,----------------,----------------,----------------], +--R 36028797018963968 4503599627370496 4503599627370496 2251799813685248 +--R 990791918021509 2381962474064941 4763924945481855 2648027 +--R [----------------,----------------,----------------,- ----------------], +--R 4503599627370496 2251799813685248 4503599627370496 4503599627370496 +--R 517913957147607 74631010396965 2388192332212761 490119 +--R [----------------,--------------,----------------,- ----------------], +--R 2251799813685248 70368744177664 2251799813685248 2251799813685248 +--R 1080863910568919 4788901398252921 299306337526587 2172471 +--R [----------------,----------------,---------------,----------------], +--R 4503599627370496 4503599627370496 281474976710656 4503599627370496 +--R 1 4801475479937737 1200368870238635 1016803 +--R [-,----------------,----------------,----------------], +--R 4 4503599627370496 1125899906842624 4503599627370496 +--R 4683743612465315 4814107234719381 1203526808903981 896543 +--R [-----------------,----------------,----------------,----------------], +--R 18014398509481984 4503599627370496 1125899906842624 4503599627370496 +--R 4863887597560135 4826796995864225 4826796996157345 1145 +--R [-----------------,----------------,----------------,--------------], +--R 18014398509481984 4503599627370496 4503599627370496 17592186044416 +--R 5044031582654955 4839545096638643 1209886274121643 152071 +--R [-----------------,----------------,----------------,- ----------------], +--R 18014398509481984 4503599627370496 1125899906842624 4503599627370496 +--R 5224175567749775 4852351874812605 4852351872686531 1063037 +--R [-----------------,----------------,----------------,- ----------------], +--R 18014398509481984 4503599627370496 4503599627370496 2251799813685248 +--R 5404319552844595 4865217663652485 4865217663030645 38865 +--R [-----------------,----------------,----------------,- ---------------], +--R 18014398509481984 4503599627370496 4503599627370496 281474976710656 +--R 5584463537939415 2439071404967727 4878142807998271 1937183 +--R [-----------------,----------------,----------------,- ----------------], +--R 18014398509481984 2251799813685248 4503599627370496 4503599627370496 +--R 2882303761517117 4891127651431483 4891127650290241 570621 +--R [----------------,----------------,----------------,- ----------------], +--R 9007199254740992 4503599627370496 4503599627370496 2251799813685248 +--R 2972375754064527 4904172534917745 2452086267422267 73211 +--R [----------------,----------------,----------------,- ----------------], +--R 9007199254740992 4503599627370496 2251799813685248 4503599627370496 +--R 3062447746611937 4917277807171409 4917277808852083 840337 +--R [----------------,----------------,----------------,----------------], +--R 9007199254740992 4503599627370496 4503599627370496 2251799813685248 +--R 3152519739159347 308152738998553 154076369430395 137763 +--R [----------------,---------------,---------------,- ---------------], +--R 9007199254740992 281474976710656 140737488355328 281474976710656 +--R 3242591731706757 4943670927607633 4943670925350831 1128401 +--R [----------------,----------------,----------------,- ----------------], +--R 9007199254740992 4503599627370496 4503599627370496 2251799813685248 +--R 3332663724254167 2478479736924067 4956959473632277 215857 +--R [----------------,----------------,----------------,- ----------------], +--R 9007199254740992 2251799813685248 4503599627370496 4503599627370496 +--R 6845471433603153 4970309822986321 4970309822979869 1613 +--R [-----------------,----------------,----------------,- ----------------], +--R 18014398509481984 4503599627370496 4503599627370496 1125899906842624 +--R 7025615418697973 4983722330806565 4983722332090139 641787 +--R [-----------------,----------------,----------------,----------------], +--R 18014398509481984 4503599627370496 4503599627370496 2251799813685248 +--R 7205759403792793 1249299340525109 624649670251221 22667 +--R [-----------------,----------------,---------------,- ----------------], +--R 18014398509481984 1125899906842624 562949953421312 1125899906842624 +--R 7385903388887613 78292738705561 313170954759513 62731 +--R [-----------------,--------------,---------------,- ---------------], +--R 18014398509481984 70368744177664 281474976710656 281474976710656 +--R 7566047373982433 2512168220382269 5024336440314455 450083 +--R [-----------------,----------------,----------------,- ----------------], +--R 18014398509481984 2251799813685248 4503599627370496 4503599627370496 +--R 7746191359077253 5038001222221509 5038001222693891 236191 +--R [-----------------,----------------,----------------,----------------], +--R 18014398509481984 4503599627370496 4503599627370496 2251799813685248 +--R 990791918021509 5051729995325585 5051729993905319 710133 +--R [----------------,----------------,----------------,- ----------------], +--R 2251799813685248 4503599627370496 4503599627370496 2251799813685248 +--R 2026619832316723 158297597792873 2532761563499039 1186929 +--R [----------------,---------------,----------------,- ----------------], +--R 4503599627370496 140737488355328 2251799813685248 2251799813685248 +--R 517913957147607 5079380998159331 317461312342081 686035 +--R [----------------,----------------,---------------,- ----------------], +--R 1125899906842624 4503599627370496 281474976710656 4503599627370496 +--R 2116691824864133 2546651992246869 2546651991650645 2329 +--R [----------------,----------------,----------------,- -------------], +--R 4503599627370496 2251799813685248 2251799813685248 8796093022208 +--R 1080863910568919 2553646233338763 5107292464939073 1738453 +--R [----------------,----------------,----------------,- ----------------], +--R 2251799813685248 2251799813685248 4503599627370496 4503599627370496 +--R 2206763817411543 5121346823013063 640168353168499 2334929 +--R [----------------,----------------,---------------,----------------], +--R 4503599627370496 4503599627370496 562949953421312 4503599627370496 +--R 1 5135467449817117 641933431251443 194427 +--R [-,----------------,---------------,----------------]] +--R 2 4503599627370496 562949953421312 4503599627370496 +--R Type: List List Fraction Integer --E 15 \end{chunk} + +The actual table values from above are: +\begin{verbatim} + + [[0., 1., + 1., + 0.], + [1.0E-2, 1.002505566, + 1.002505565988876, + -1.1123990617534218E-11], + [2.0E-2, 1.0050223060000001, + 1.0050223058229502, + -1.7704993027223281E-10], + [2.9999999999999999E-2, 1.007550283, + 1.0075502826056368, + -3.9436320875552155E-10] + [4.0000000000000001E-2, 1.0100895599999999, + 1.0100895598460362, + -1.5396373065357238E-10] + [5.0000000000000003E-2, 1.012640202, + 1.0126402014616676, + -5.3833248969681335E-10] + [5.9999999999999998E-2, 1.015202272, + 1.0152022717813329, + -2.1866708443951666E-10] + [7.0000000000000007E-2, 1.017775836, + 1.0177758355479642, + -4.5203574217111964E-10] + [8.0000000000000002E-2, 1.0203609579999999, + 1.0203609579215664, + -7.8433481931483584E-11] + [8.9999999999999997E-2, 1.0229577050000001, + 1.0229577044820872, + -5.1791282373869763E-10] + [0.10000000000000001, 1.0255661410000001, + 1.0255661412323602, + 2.3236013113603349E-10] + [0.11, 1.028186335, + 1.0281863346010778, + -3.989222285838423E-10], + [0.12, 1.030818352, + 1.0308183514457605, + -5.5423954314903767E-10], + [0.13, 1.033462259, + 1.0334622590557541, + 5.5754068029045811E-11], + [0.14000000000000001, 1.036118125, + 1.0361181251552536, + 1.5525358776358189E-10], + [0.14999999999999999, 1.0387860179999999, + 1.0387860179063644, + -9.3635543763070928E-11] + [0.16, 1.0414660060000001, + 1.0414660059121499, + -8.7850171581749237E-11], + [0.17000000000000001, 1.0441581579999999, + 1.0441581582197257, + 2.1972579311579921E-10] + [0.17999999999999999, 1.0468625439999999, + 1.0468625443233892, + 3.233893153264944E-10] + [0.19, 1.0495792340000001, + 1.0495792341677359, + 1.6773582522944253E-10], + [0.20000000000000001, 1.052308298, + 1.0523082981508358, + 1.5083578830399347E-10], + [0.20999999999999999, 1.0550498070000001, + 1.055049807127405, + 1.2740497545848939E-10] + [0.22, 1.0578038329999999, + 1.0578038324120198, + -5.8798010904581588E-10], + [0.23000000000000001, 1.0605704460000001, + 1.0605704457823433, + -2.1765678148710776E-10] + [0.23999999999999999, 1.0633497190000001, + 1.0633497194823853, + 4.8238524286148277E-10] + [0.25, 1.0661417259999999, + 1.0661417262257755, + 2.2577562042158661E-10], + [0.26000000000000001, 1.0689465389999999, + 1.0689465391990731, + 1.9907320236711712E-10] + [0.27000000000000002, 1.071764232, + 1.0717642320650853, + 6.5085270506415327E-11], + [0.28000000000000003, 1.0745948789999999, + 1.0745948789662336, + -3.3766323070949511E-11] + [0.28999999999999998, 1.0774385550000001, + 1.0774385545279166, + -4.7208348341598594E-10] + [0.29999999999999999, 1.0802953340000001, + 1.0802953338619241, + -1.3807599508197654E-10] + [0.31, 1.083165293, + 1.0831652925698594, + -4.3014058981327707E-10], + [0.32000000000000001, 1.0860485070000001, + 1.0860485067465939, + -2.5340618492464273E-10] + [0.33000000000000002, 1.088945053, + 1.0889450529837443, + -1.6255663481956617E-11] + [0.34000000000000002, 1.091855008, + 1.0918550083731842, + 3.7318415024856222E-10], + [0.34999999999999998, 1.094778451, + 1.0947784505105673, + -4.8943271657719833E-10] + [0.35999999999999999, 1.0977154579999999, + 1.0977154574988892, + -5.0111070848402051E-10] + [0.37, 1.100666108, + 1.1006661079520708, + -4.7929216151487708E-11], + [0.38, 1.1036304809999999, + 1.1036304809985678, + -1.4321877017664519E-12], + [0.39000000000000001, 1.1066086559999999, + 1.1066086562850108, + 2.8501090376664706E-10] + [0.40000000000000002, 1.1096007139999999, + 1.1096007139798676, + -2.0132340239342739E-11] + [0.40999999999999998, 1.112606735, + 1.1126067347771349, + -2.228650597402293E-10] + [0.41999999999999998, 1.1156268, + 1.1156267999000615, + -9.9938501918472866E-11], + [0.42999999999999999, 1.118660991, + 1.1186609911048895, + 1.0488943047448629E-10], + [0.44, 1.121709391, + 1.1217093906846374, + -3.1536262490305944E-10], + [0.45000000000000001, 1.124772082, + 1.1247720814728976, + -5.2710236175812497E-10] + [0.46000000000000002, 1.1278491470000001, + 1.1278491468476701, + -1.52329926450534E-10] + [0.46999999999999997, 1.1309406710000001, + 1.1309406707352239, + -2.6477620096443388E-10] + [0.47999999999999998, 1.1340467380000001, + 1.134046737613986, + -3.8601410956573545E-10] + [0.48999999999999999, 1.137167432, + 1.1371674325184589, + 5.1845883142220828E-10], + [0.5, 1.140302841, + 1.1403028410431715, + 4.3171466401759062E-11]] + +\end{verbatim} + In the following table we show values returned by Ei(x). Abramowitz and Stegun, ``Handbook of Mathematical Functions'', Dover Publications, Inc. New York 1965. pp 239-241 \begin{chunk}{*} + --S 16 of 20 +t8:List List Fraction Integer:= [[machineFraction (x::DFLOAT) for x in line]_ + for line in _ [[0.50,0.454219905,Ei(0.50),Ei(0.50)-0.454219905],_ [0.51,0.487032167,Ei(0.51),Ei(0.51)-0.487032167],_ [0.52,0.519530633,Ei(0.52),Ei(0.52)-0.519530633],_ @@ -1461,439 +2629,946 @@ Dover Publications, Inc. New York 1965. pp 239-241 [1.97,4.844221501,Ei(1.97),Ei(1.97)-4.844221501],_ [1.98,4.880710791,Ei(1.98),Ei(1.98)-4.880710791],_ [1.99,4.917381131,Ei(1.99),Ei(1.99)-4.917381131],_ - [2.00,4.954234356,Ei(2.00),Ei(2.00)-4.954234356]] + [2.00,4.954234356,Ei(2.00),Ei(2.00)-4.954234356]]] --R --R --R (17) ---R [[0.5,0.45421990499999998,0.45421990486317332,- 1.3682666111236585E-10], ---R ---R [0.51000000000000001, 0.48703216700000002, 0.48703216680456007, ---R - 1.9543994200788006E-10] ---R , ---R ---R [0.52000000000000002, 0.51953063300000002, 0.51953063245569719, ---R - 5.443028250340376E-10] ---R , ---R ---R [0.53000000000000003, 0.55173044500000001, 0.55173044523266401, ---R 2.3266399917787339E-10] ---R , ---R ---R [0.54000000000000004, 0.58364593099999995, 0.58364593072977955, ---R - 2.7022040161028826E-10] ---R , ---R ---R [0.55000000000000004, 0.61529065699999996, 0.61529065706218644, ---R 6.2186478189119043E-11] ---R , ---R ---R [0.56000000000000005, 0.64667748999999997, 0.64667748977430584, ---R - 2.2569413005157912E-10] ---R , ---R ---R [0.56999999999999995, 0.67781864199999997, 0.67781864189137597, ---R - 1.0862399868472039E-10] ---R , ---R ---R [0.57999999999999996, 0.70872572, 0.70872571962101083, ---R - 3.7898917337741977E-10] ---R , ---R ---R [0.58999999999999997, 0.73940976400000002, 0.73940976415103654, ---R 1.510365166268457E-10] ---R , ---R ---R [0.59999999999999998, 0.76988129000000005, 0.76988128993735938, ---R - 6.2640670428493195E-11] ---R , ---R ---R [0.60999999999999999, 0.80015031999999997, 0.80015031983004981, ---R - 1.699501650520574E-10] ---R , ---R [0.62,0.83022641699999999,0.83022641734618519,3.4618519162421535E-10], ---R [0.63,0.86011871600000001,0.86011871636343917,3.6343916764991491E-10], ---R ---R [0.64000000000000001, 0.88983594899999996, 0.88983594847818637, ---R - 5.2181359233571811E-10] ---R , ---R ---R [0.65000000000000002, 0.91938646800000001, 0.91938646824544334, ---R 2.4544333232512372E-10] ---R , ---R ---R [0.66000000000000003, 0.94877827699999995, 0.94877827649472835, ---R - 5.0527160233571067E-10] ---R , ---R ---R [0.67000000000000004, 0.97801904200000001, 0.97801904189549682, ---R - 1.045031838842192E-10] ---R , ---R ---R [0.68000000000000005, 1.0071161209999999, 1.0071161209277915, ---R - 7.2208461432410331E-11] ---R , ---R ---R [0.68999999999999995, 1.0360765759999999, 1.0360765763978435, ---R 3.978435358931165E-10] ---R , ---R [0.69999999999999996,1.064907195,1.0649071946242905,- 3.757094635403746E-10] ---R , ---R [0.70999999999999996,1.093614501,1.0936145014081782,4.0817815794014223E-10], ---R ---R [0.71999999999999997, 1.1222047770000001, 1.1222047768888612, ---R - 1.111388758801013E-10] ---R , ---R [0.72999999999999998,1.150684069,1.1506840693780345,3.780344925985446E-10], ---R ---R [0.73999999999999999, 1.1790582080000001, 1.1790582082553465, ---R 2.5534641068247765E-10] ---R , ---R [0.75,1.2073328160000001,1.2073328160012218,1.2216894162975223E-12], ---R ---R [0.76000000000000001, 1.2355133190000001, 1.2355133194354742, ---R 4.3547410122357633E-10] ---R , ---R ---R [0.77000000000000002, 1.2636049600000001, 1.2636049602240513, ---R 2.2405122201973882E-10] ---R , ---R ---R [0.78000000000000003, 1.291612805, 1.2916128047105979, ---R - 2.8940205787364448E-10] ---R , ---R [0.79000000000000004,1.319541753,1.3195417531244753,1.2447531894110853E-10], ---R ---R [0.80000000000000004, 1.3473965480000001, 1.3473965482123258, ---R 2.1232571256746269E-10] ---R , ---R ---R [0.81000000000000005, 1.3751817829999999, 1.3751817833361941, ---R 3.361941836033111E-10] ---R , ---R [0.81999999999999995,1.40290191,1.4029019100774811,7.7481132620960125E-11], ---R [0.82999999999999996,1.430561245,1.4305612453827297,3.8272962576968439E-10], ---R [0.83999999999999997,1.458163978,1.4581639782841678,2.8416780040174672E-10], ---R ---R [0.84999999999999998, 1.4857141760000001, 1.4857141762252541, ---R 2.2525403764461771E-10] ---R , ---R ---R [0.85999999999999999, 1.5132157909999999, 1.5132157910189581, ---R 1.8958168368499173E-11] ---R , ---R [0.87,1.5406726639999999,1.5406726644642923,4.6429238231837644E-10], ---R [0.88,1.5680885339999999,1.5680885336445423,- 3.5545766330358219E-10], ---R ---R [0.89000000000000001, 1.5954670360000001, 1.5954670359288246, ---R - 7.1175509930299086E-11] ---R , ---R ---R [0.90000000000000002, 1.622811714, 1.6228117136968674, ---R - 3.0313263010839364E-10] ---R , ---R ---R [0.91000000000000003, 1.650126019, 1.6501260188054063, ---R - 1.9459367450735954E-10] ---R , ---R ---R [0.92000000000000004, 1.6774133170000001, 1.677413316813162, ---R - 1.8683810054653804E-10] ---R , ---R ---R [0.93000000000000005, 1.7046768910000001, 1.7046768909800791, ---R - 1.9920953775454109E-11] ---R , ---R ---R [0.93999999999999995, 1.7319199460000001, 1.7319199460553549, ---R 5.5354831829390605E-11] ---R , ---R ---R [0.94999999999999996, 1.759145612, 1.7591456118676905, ---R - 1.3230949669207348E-10] ---R , ---R ---R [0.95999999999999996, 1.786356947, 1.7863569467301943, ---R - 2.6980573331059077E-10] ---R , ---R ---R [0.96999999999999997, 1.8135569410000001, 1.8135569406715355, ---R - 3.2846458886126584E-10] ---R , ---R ---R [0.97999999999999998, 1.8407485189999999, 1.8407485185040211, ---R - 4.9597881357499318E-10] ---R , ---R ---R [0.98999999999999999, 1.8679345430000001, 1.8679345427385856, ---R - 2.6141444564586891E-10] ---R , ---R [1.,1.895117816,1.8951178163559361,3.5593616942719564E-10], ---R [1.01,1.922301085,1.9223010854424856,4.4248560371329404E-10], ---R [1.02,1.9494870419999999,1.9494870416990668,- 3.0093305625200628E-10], ---R [1.03,1.976678325,1.976678324829928,- 1.7007195651785878E-10], ---R [1.04,2.003877525,2.0038775248189595,- 1.8104051591194548E-10], ---R [1.05,2.031087184,2.0310871840996643,9.9664276831390453E-11], ---R [1.0600000000000001,2.0583098,2.0583097996249284,- 3.7507152939042498E-10], ---R ---R [1.0700000000000001, 2.0855478249999999, 2.085547824842283, ---R - 1.5771695061062019E-10] ---R , ---R ---R [1.0800000000000001, 2.1128036720000001, 2.1128036715799325, ---R - 4.2006753631085303E-10] ---R , ---R ---R [1.0900000000000001, 2.1400797119999999, 2.1400797118485424, ---R - 1.5145751319778356E-10] ---R , ---R ---R [1.1000000000000001, 2.1673782799999999, 2.1673782795634038, ---R - 4.3659609261226251E-10] ---R , ---R ---R [1.1100000000000001, 2.1947016719999999, 2.1947016721913277, ---R 1.9132784245812218E-10] ---R , ---R ---R [1.1200000000000001, 2.2220521519999998, 2.2220521523263717, ---R 3.2637181845984742E-10] ---R , ---R ---R [1.1299999999999999, 2.2494319489999999, 2.2494319491981756, ---R 1.9817569807401014E-10] ---R , ---R ---R [1.1399999999999999, 2.2768432600000001, 2.2768432601165496, ---R 1.1654943676830953E-10] ---R , ---R ---R [1.1499999999999999, 2.3042882520000001, 2.304288251855628, ---R - 1.4437206985462581E-10] ---R , ---R ---R [1.1599999999999999, 2.3317690619999998, 2.3317690619808027, ---R - 1.9197088363398507E-11] ---R 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[---------------,----------------,----------------,----------------], +--R 562949953421312 1125899906842624 562949953421312 1125899906842624 +--R 5314247560297185 5374700614333477 1343675153691307 431751 +--R [----------------,----------------,----------------,----------------], +--R 4503599627370496 2251799813685248 562949953421312 2251799813685248 +--R 2679641778285445 5436852131163437 5436852130181055 491191 +--R [----------------,----------------,----------------,- ----------------], +--R 2251799813685248 2251799813685248 2251799813685248 1125899906842624 +--R 5404319552844595 5499102952365181 2749551476399497 433813 +--R [----------------,----------------,----------------,----------------], +--R 4503599627370496 2251799813685248 1125899906842624 2251799813685248 +--R 1362338887279575 2780728804279967 5561457607882541 677393 +--R [----------------,----------------,----------------,- ----------------], +--R 1125899906842624 1125899906842624 2251799813685248 2251799813685248 +--R 5494391545392005 5623920569570325 2811960285246287 922249 +--R [----------------,----------------,----------------,----------------], +--R 4503599627370496 2251799813685248 1125899906842624 2251799813685248 +--R 2769713770832855 5686496266938389 355406016733753 801659 +--R [----------------,----------------,---------------,----------------], +--R 2251799813685248 2251799813685248 140737488355328 2251799813685248 +--R 5584463537939415 2874594540207381 2874594540476813 33679 +--R [----------------,----------------,----------------,---------------], +--R 4503599627370496 1125899906842624 1125899906842624 140737488355328 +--R 5 2906001673482943 2906001673883173 200115 +--R [-,----------------,----------------,---------------], +--R 4 1125899906842624 1125899906842624 562949953421312 +--R 709316941310853 2937471681512903 5874943364125319 1099513 +--R [---------------,----------------,----------------,----------------], +--R 562949953421312 1125899906842624 2251799813685248 2251799813685248 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[----------------,----------------,----------------,----------------], +--R 2251799813685248 2251799813685248 1125899906842624 2251799813685248 +--R 6169931489497579 3288387707039881 6576775413440807 638955 +--R [----------------,----------------,----------------,- ----------------], +--R 4503599627370496 1125899906842624 2251799813685248 2251799813685248 +--R 1553741871442821 6641546782899355 1660386695934107 837073 +--R [----------------,----------------,----------------,----------------], +--R 1125899906842624 2251799813685248 562949953421312 2251799813685248 +--R 6260003482044989 3353248376610585 6706496752430841 790329 +--R [----------------,----------------,----------------,- ----------------], +--R 4503599627370496 1125899906842624 2251799813685248 2251799813685248 +--R 3152519739159347 6771629207148087 6771629207487311 42403 +--R [----------------,----------------,----------------,---------------], +--R 2251799813685248 2251799813685248 2251799813685248 281474976710656 +--R 6350075474592399 854618502221973 3418474009099319 211427 +--R [----------------,---------------,----------------,----------------], +--R 4503599627370496 281474976710656 1125899906842624 1125899906842624 +--R 799388933858263 862807129460243 6902457036357117 675173 +--R [---------------,---------------,----------------,----------------], +--R 562949953421312 281474976710656 2251799813685248 2251799813685248 +--R 6440147467139809 871020012241681 217755003043431 67957 +--R [----------------,---------------,---------------,- ---------------], +--R 4503599627370496 281474976710656 70368744177664 281474976710656 +--R 3242591731706757 7034061021330981 3517030510730335 129689 +--R [----------------,----------------,----------------,----------------], +--R 2251799813685248 2251799813685248 1125899906842624 2251799813685248 +--R 6530219459687219 1775040903417007 3550081807270575 436561 +--R [----------------,----------------,----------------,----------------], +--R 4503599627370496 562949953421312 1125899906842624 1125899906842624 +--R 1643813863990231 7166471669227275 1791617917362553 222937 +--R [----------------,----------------,----------------,----------------], +--R 1125899906842624 2251799813685248 562949953421312 2251799813685248 +--R 6620291452234629 7232988966528809 904123620858191 336719 +--R [----------------,----------------,---------------,----------------], +--R 4503599627370496 2251799813685248 281474976710656 2251799813685248 +--R 3332663724254167 3649859638668659 3649859638153431 128807 +--R [----------------,----------------,----------------,- ---------------], +--R 2251799813685248 1125899906842624 1125899906842624 281474976710656 +--R 6710363444782039 1841666589413723 3683333178546419 281027 +--R [----------------,----------------,----------------,- ----------------], +--R 4503599627370496 562949953421312 1125899906842624 1125899906842624 +--R 3 1858458489745005 929229244909037 73069 +--R [-,----------------,---------------,---------------], 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[----------------,---------------,----------------,- ----------------], +--R 1125899906842624 281474976710656 2251799813685248 2251799813685248 +--R 7025615418697973 3920839790473361 7841679580126083 820639 +--R [----------------,----------------,----------------,- ----------------], +--R 4503599627370496 1125899906842624 2251799813685248 2251799813685248 +--R 3535325707485839 7910495238526689 3955247619774833 1022977 +--R [----------------,----------------,----------------,----------------], +--R 2251799813685248 2251799813685248 1125899906842624 2251799813685248 +--R 7115687411245383 3989780595205673 1994890297804925 404177 +--R [----------------,----------------,----------------,----------------], +--R 4503599627370496 1125899906842624 562949953421312 1125899906842624 +--R 447545212969943 2012220285202847 8048881139966077 845311 +--R [---------------,----------------,----------------,- ----------------], +--R 281474976710656 562949953421312 2251799813685248 2251799813685248 +--R 7205759403792793 8118458789433907 2029614697369437 43841 +--R [----------------,----------------,----------------,----------------], +--R 4503599627370496 2251799813685248 562949953421312 2251799813685248 +--R 3625397700033249 4094148921370699 4094148921507875 17147 +--R [----------------,----------------,----------------,---------------], +--R 2251799813685248 1125899906842624 1125899906842624 140737488355328 +--R 7295831396340203 4129201002472277 8258402004110987 833567 +--R [----------------,----------------,----------------,- ----------------], +--R 4503599627370496 1125899906842624 2251799813685248 2251799813685248 +--R 1835216848153477 2082193744500567 8328774977247389 754879 +--R [----------------,----------------,----------------,- ----------------], +--R 1125899906842624 562949953421312 2251799813685248 2251799813685248 +--R 7385903388887613 4199710234188517 8399420468531093 154059 +--R [----------------,----------------,----------------,----------------], +--R 4503599627370496 1125899906842624 2251799813685248 2251799813685248 +--R 3715469692580659 132349096672421 2117585546586549 172187 +--R [----------------,---------------,----------------,- ---------------], +--R 2251799813685248 35184372088832 562949953421312 562949953421312 +--R 7475975381435023 8541543842690293 8541543841997621 10823 +--R [----------------,----------------,----------------,- --------------], +--R 4503599627370496 2251799813685248 2251799813685248 35184372088832 +--R 940126422213591 2153257287703193 8613029150341669 471103 +--R [---------------,----------------,----------------,- ----------------], +--R 562949953421312 562949953421312 2251799813685248 2251799813685248 +--R 7566047373982433 4342400915687837 8684801830404751 970923 +--R [----------------,----------------,----------------,- ----------------], +--R 4503599627370496 1125899906842624 2251799813685248 2251799813685248 +--R 3805541685128069 4378432803302045 4378432802995389 9583 +--R [----------------,----------------,----------------,- --------------], +--R 2251799813685248 1125899906842624 1125899906842624 35184372088832 +--R 7656119366529843 8829224205478513 8829224206277685 199793 +--R [----------------,----------------,----------------,---------------], +--R 4503599627370496 2251799813685248 2251799813685248 562949953421312 +--R 1925288840700887 8901881365986633 8901881366403569 52117 +--R [----------------,----------------,----------------,---------------], +--R 1125899906842624 2251799813685248 2251799813685248 281474976710656 +--R 7746191359077253 2243710207091985 8974840828042839 325101 +--R [----------------,----------------,----------------,- ----------------], +--R 4503599627370496 562949953421312 2251799813685248 2251799813685248 +--R 3895613677675479 2262026584904331 1131013292496601 88871 +--R [----------------,----------------,----------------,---------------], +--R 2251799813685248 562949953421312 281474976710656 562949953421312 +--R 7836263351624663 4560840828994337 570105103664447 321239 +--R [----------------,----------------,---------------,----------------], +--R 4503599627370496 1125899906842624 140737488355328 1125899906842624 +--R 7 574723159280705 1149446318583071 21661 +--R [-,---------------,----------------,---------------], +--R 4 140737488355328 281474976710656 281474976710656 +--R 990791918021509 2317444195722457 4634888391734551 289637 +--R [---------------,----------------,----------------,----------------], +--R 562949953421312 562949953421312 1125899906842624 1125899906842624 +--R 7971371340445777 292009504576869 4672152072861169 368735 +--R [----------------,---------------,----------------,- ----------------], +--R 4503599627370496 70368744177664 1125899906842624 1125899906842624 +--R 4008203668359741 4709578214490151 4709578213954005 268073 +--R [----------------,----------------,----------------,- ---------------], +--R 2251799813685248 1125899906842624 1125899906842624 562949953421312 +--R 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562949953421312 1125899906842624 1125899906842624 +--R 517913957147607 4937653591584309 4937653591577239 3535 +--R [---------------,----------------,----------------,- ---------------], +--R 281474976710656 1125899906842624 1125899906842624 562949953421312 +--R 8331659310635417 4976270512733531 1244067628128235 220591 +--R [----------------,----------------,----------------,- ----------------], +--R 4503599627370496 1125899906842624 281474976710656 1125899906842624 +--R 4188347653454561 2507532634886519 5015065270189997 416959 +--R [----------------,----------------,----------------,----------------], +--R 2251799813685248 562949953421312 1125899906842624 1125899906842624 +--R 8421731303182827 2527019908069585 1263509954055799 42013 +--R [----------------,----------------,----------------,---------------], +--R 4503599627370496 562949953421312 281474976710656 562949953421312 +--R 2116691824864133 636649513721483 5093196109575271 196593 +--R [----------------,---------------,----------------,- ----------------], +--R 1125899906842624 140737488355328 1125899906842624 1125899906842624 +--R 8511803295730237 5132536116492357 5132536116786609 73563 +--R [----------------,----------------,----------------,---------------], +--R 4503599627370496 1125899906842624 1125899906842624 281474976710656 +--R 4278419646001971 2586030906127493 5172061812202729 52257 +--R [----------------,----------------,----------------,- ----------------], +--R 2251799813685248 562949953421312 1125899906842624 1125899906842624 +--R 8601875288277647 1302943794660897 5211775178199679 443909 +--R [----------------,----------------,----------------,- ----------------], +--R 4503599627370496 281474976710656 1125899906842624 1125899906842624 +--R 1080863910568919 164114943910103 5251678205407441 284145 +--R [----------------,---------------,----------------,----------------], +--R 562949953421312 35184372088832 1125899906842624 1125899906842624 +--R 8691947280825057 5291772892418247 5291772892931415 32073 +--R [----------------,----------------,----------------,--------------], +--R 4503599627370496 1125899906842624 1125899906842624 70368744177664 +--R 4368491638549381 2666030624566937 666507656071519 280861 +--R [----------------,----------------,---------------,- ---------------], +--R 2251799813685248 562949953421312 140737488355328 562949953421312 +--R 8782019273372467 5372545289505109 2686272644521677 461755 +--R [----------------,----------------,----------------,- ----------------], +--R 4503599627370496 1125899906842624 562949953421312 1125899906842624 +--R 2206763817411543 2706613520075893 5413227040188281 36495 +--R [----------------,----------------,----------------,----------------], +--R 1125899906842624 562949953421312 1125899906842624 1125899906842624 +--R 8872091265919877 681763567087617 2727054268597319 246851 +--R [----------------,---------------,----------------,---------------], +--R 4503599627370496 140737488355328 562949953421312 562949953421312 +--R 4458563631096791 5495191824912689 5495191824808017 3271 +--R [----------------,----------------,----------------,- --------------], +--R 2251799813685248 1125899906842624 1125899906842624 35184372088832 +--R 8962163258467287 5536478957302577 5536478957544019 120721 +--R [----------------,----------------,----------------,---------------], +--R 4503599627370496 1125899906842624 1125899906842624 562949953421312 +--R 5577971999896927 5577971999899057 1065 +--R [2,----------------,----------------,---------------]] +--R 1125899906842624 1125899906842624 562949953421312 +--R Type: List List Fraction Integer --E 16 \end{chunk} -In the following table we show values returned by -$xe^{-x}Ei(x)$, chosen to keep the values in a reasonable range. -Abramowitz and Stegun, ``Handbook of Mathematical Functions'', -Dover Publications, Inc. New York 1965. pp 242-243 +The actual table values from above are: +\begin{verbatim} + + [0.5, 0.45421990499999998, + 0.45421990486317332, + -1.3682666111236585E-10] + [0.51000000000000001, 0.48703216700000002, + 0.48703216680456007, + -1.9543994200788006E-10] + [0.52000000000000002, 0.51953063300000002, + 0.51953063245569719, + -5.443028250340376E-10] + [0.53000000000000003, 0.55173044500000001, + 0.55173044523266401, + 2.3266399917787339E-10] + [0.54000000000000004, 0.58364593099999995, + 0.58364593072977955, + -2.7022040161028826E-10] + [0.55000000000000004, 0.61529065699999996, + 0.61529065706218644, + 6.2186478189119043E-11] + [0.56000000000000005, 0.64667748999999997, + 0.64667748977430584, + -2.2569413005157912E-10] + [0.56999999999999995, 0.67781864199999997, + 0.67781864189137597, + -1.0862399868472039E-10] + [0.57999999999999996, 0.70872572, + 0.70872571962101083, + -3.7898917337741977E-10] + [0.58999999999999997, 0.73940976400000002, + 0.73940976415103654, + 1.510365166268457E-10] + [0.59999999999999998, 0.76988129000000005, + 0.76988128993735938, + -6.2640670428493195E-11] + [0.60999999999999999, 0.80015031999999997, + 0.80015031983004981, + -1.699501650520574E-10] + [0.62, 0.83022641699999999, + 0.83022641734618519, + 3.4618519162421535E-10] + [0.63, 0.86011871600000001, + 0.86011871636343917, + 3.6343916764991491E-10] + [0.64000000000000001, 0.88983594899999996, + 0.88983594847818637, + -5.2181359233571811E-10] + [0.65000000000000002, 0.91938646800000001, + 0.91938646824544334, + 2.4544333232512372E-10] + [0.66000000000000003, 0.94877827699999995, + 0.94877827649472835, + -5.0527160233571067E-10] + [0.67000000000000004, 0.97801904200000001, + 0.97801904189549682, + -1.045031838842192E-10] + [0.68000000000000005, 1.0071161209999999, + 1.0071161209277915, + -7.2208461432410331E-11] + [0.68999999999999995, 1.0360765759999999, + 1.0360765763978435, + 3.978435358931165E-10] + [0.69999999999999996, 1.064907195, + 1.0649071946242905, + -3.757094635403746E-10] + [0.70999999999999996, 1.093614501, + 1.0936145014081782, + 4.0817815794014223E-10] + [0.71999999999999997, 1.1222047770000001, + 1.1222047768888612, + -1.111388758801013E-10] + [0.72999999999999998, 1.150684069, + 1.1506840693780345, + 3.780344925985446E-10] + [0.73999999999999999, 1.1790582080000001, + 1.1790582082553465, + 2.5534641068247765E-10] + [0.75, 1.2073328160000001, + 1.2073328160012218, + 1.2216894162975223E-12] + [0.76000000000000001, 1.2355133190000001, + 1.2355133194354742, + 4.3547410122357633E-10] + [0.77000000000000002, 1.2636049600000001, + 1.2636049602240513, + 2.2405122201973882E-10] + [0.78000000000000003, 1.291612805, + 1.2916128047105979, + -2.8940205787364448E-10] + [0.79000000000000004, 1.319541753, + 1.3195417531244753, + 1.2447531894110853E-10] + [0.80000000000000004, 1.3473965480000001, + 1.3473965482123258, + 2.1232571256746269E-10] + [0.81000000000000005, 1.3751817829999999, + 1.3751817833361941, + 3.361941836033111E-10] + [0.81999999999999995, 1.40290191, + 1.4029019100774811, + 7.7481132620960125E-11] + [0.82999999999999996, 1.430561245, + 1.4305612453827297, + 3.8272962576968439E-10] + [0.83999999999999997, 1.458163978, + 1.4581639782841678, + 2.8416780040174672E-10] + [0.84999999999999998, 1.4857141760000001, + 1.4857141762252541, + 2.2525403764461771E-10] + [0.85999999999999999, 1.5132157909999999, + 1.5132157910189581, + 1.8958168368499173E-11] + [0.87, 1.5406726639999999, + 1.5406726644642923, + 4.6429238231837644E-10] + [0.88, 1.5680885339999999, + 1.5680885336445423, + -3.5545766330358219E-10] + [0.89000000000000001, 1.5954670360000001, + 1.5954670359288246, + -7.1175509930299086E-11] + [0.90000000000000002, 1.622811714, + 1.6228117136968674, + -3.0313263010839364E-10] + [0.91000000000000003, 1.650126019, + 1.6501260188054063, + -1.9459367450735954E-10] + [0.92000000000000004, 1.6774133170000001, + 1.677413316813162, + -1.8683810054653804E-10] + [0.93000000000000005, 1.7046768910000001, + 1.7046768909800791, + -1.9920953775454109E-11] + [0.93999999999999995, 1.7319199460000001, + 1.7319199460553549, + 5.5354831829390605E-11] + [0.94999999999999996, 1.759145612, + 1.7591456118676905, + -1.3230949669207348E-10] + [0.95999999999999996, 1.786356947, + 1.7863569467301943, + -2.6980573331059077E-10] + [0.96999999999999997, 1.8135569410000001, + 1.8135569406715355, + -3.2846458886126584E-10] + [0.97999999999999998, 1.8407485189999999, + 1.8407485185040211, + -4.9597881357499318E-10] + [0.98999999999999999, 1.8679345430000001, + 1.8679345427385856, + -2.6141444564586891E-10] + [1., 1.895117816, + 1.8951178163559361, + 3.5593616942719564E-10] + [1.01, 1.922301085, + 1.9223010854424856, + 4.4248560371329404E-10] + [1.02, 1.9494870419999999, + 1.9494870416990668, + -3.0093305625200628E-10] + [1.03, 1.976678325, + 1.976678324829928, + -1.7007195651785878E-10] + [1.04, 2.003877525, + 2.0038775248189595, + -1.8104051591194548E-10] + [1.05, 2.031087184, + 2.0310871840996643, + 9.9664276831390453E-11] + [1.0600000000000001, 2.0583098, + 2.0583097996249284, + -3.7507152939042498E-10] + [1.0700000000000001, 2.0855478249999999, + 2.085547824842283, + -1.5771695061062019E-10] + [1.0800000000000001, 2.1128036720000001, + 2.1128036715799325, + -4.2006753631085303E-10] + [1.0900000000000001, 2.1400797119999999, + 2.1400797118485424, + -1.5145751319778356E-10] + [1.1000000000000001, 2.1673782799999999, + 2.1673782795634038, + -4.3659609261226251E-10] + [1.1100000000000001, 2.1947016719999999, + 2.1947016721913277, + 1.9132784245812218E-10] + [1.1200000000000001, 2.2220521519999998, + 2.2220521523263717, + 3.2637181845984742E-10] + [1.1299999999999999, 2.2494319489999999, + 2.2494319491981756, + 1.9817569807401014E-10] + [1.1399999999999999, 2.2768432600000001, + 2.2768432601165496, + 1.1654943676830953E-10] + [1.1499999999999999, 2.3042882520000001, + 2.304288251855628, + -1.4437206985462581E-10] + [1.1599999999999999, 2.3317690619999998, + 2.3317690619808027, + -1.9197088363398507E-11] + [1.1699999999999999, 2.3592878000000002, + 2.3592878001213737, + 1.2137357785491076E-10] + [1.1799999999999999, 2.3868465489999999, + 2.3868465491917359, + 1.9173596044197438E-10] + [1.1899999999999999, 2.4144473670000002, + 2.4144473665637345, + -4.3626569024013406E-10] + [1.2, 2.4420922850000002, + 2.4420922851926514, + 1.9265122830347536E-10] + [1.21, 2.4697833149999999, + 2.4697833146991774, + -3.0082247803875362E-10] + [1.22, 2.4975224420000002, + 2.497522442409561, + 4.095608296950104E-10] + [1.23, 2.5253116339999999, + 2.5253116343560089, + 3.5600900005761105E-10] + [1.24, 2.5531528360000002, + 2.5531528362393039, + 2.3930368797664414E-10] + [1.25, 2.5810479740000001, + 2.5810479743554762, + 3.5547609300579097E-10] + [1.26, 2.6089989560000002, + 2.6089989564882825, + 4.8828230347908175E-10] + [1.27, 2.6370076729999998, + 2.6370076727691485, + -2.308513380455679E-10] + [1.28, 2.6650759970000002, + 2.6650759965061086, + -4.9389159428869789E-10] + [1.29, 2.693205785, + 2.6932057849832494, + -1.6750600906334512E-11] + [1.3, 2.7213988800000002, + 2.7213988802320226, + 2.3202240129194251E-10] + [1.3100000000000001, 2.7496571099999998, + 2.7496571097757787, + -2.2422108614250646E-10] + [1.3200000000000001, 2.7779822869999999, + 2.777982287348725, + 3.4872504883765032E-10] + [1.3300000000000001, 2.8063762140000001, + 2.8063762135905539, + -4.0944625467886908E-10] + [1.3400000000000001, 2.8348406769999999, + 2.8348406767178056, + -2.8219426795317304E-10] + [1.3500000000000001, 2.863377453, + 2.8633774531730753, + 1.7307533184407475E-10] + [1.3600000000000001, 2.8919883080000002, + 2.8919883082530298, + 2.5302959727468988E-10] + [1.3700000000000001, 2.9206749969999999, + 2.9206749967162473, + -2.8375257699053691E-10] + [1.3799999999999999, 2.9494392629999999, + 2.9494392633717355, + 3.7173553124603131E-10] + [1.3899999999999999, 2.9782828440000002, + 2.9782828436490232, + -3.5097702522079999E-10] + [1.3999999999999999, 3.0072074639999999, + 3.0072074641506457, + 1.5064571812217764E-10] + [1.4099999999999999, 3.0362148430000002, + 3.0362148431877847, + 1.8778445465272853E-10] + [1.4199999999999999, 3.065306691, + 3.065306691299837, + 2.9983704408209633E-10] + [1.4299999999999999, 3.0944847119999999, + 3.0944847117585681, + -2.4143176347024564E-10] + [1.4399999999999999, 3.1237506009999998, + 3.1237506010575933, + 5.7593485536244771E-11] + [1.45, 3.1531060489999998, + 3.1531060493877443, + 3.8774450317191622E-10] + [1.46, 3.1825527409999999, + 3.1825527410990038, + 9.9003916176343409E-11] + [1.47, 3.2120923549999998, + 3.2120923551495331, + 1.4953327465150323E-10] + [1.48, 3.2417265660000001, + 3.2417265655423857, + -4.5761439082525612E-10] + [1.49, 3.2714570420000002, + 3.2714570417503985, + -2.4960167266385724E-10] + [1.5, 3.3012854489999999, + 3.3012854491297974, + 1.297975060765566E-10] + [1.51, 3.3312134489999998, + 3.3312134493229739, + 3.2297409191528459E-10] + [1.52, 3.3612427010000001, + 3.3612427006508958, + -3.4910430102286227E-10] + [1.53, 3.3913748579999998, + 3.3913748584955847, + 4.9558490644585618E-10] + [1.54, 3.4216115760000001, + 3.4216115756731122, + -3.2688785012169319E-10] + [1.55, 3.4519545030000001, + 3.4519545027974381, + -2.0256196719969921E-10] + [1.5600000000000001, 3.4824052889999999, + 3.4824052886355643, + -3.6443559281451599E-10] + [1.5700000000000001, 3.5129655799999999, + 3.5129655804542947, + 4.5429482398162691E-10] + [1.5800000000000001, 3.5436370240000001, + 3.5436370243589819, + 3.5898173322834737E-10] + [1.5900000000000001, 3.5744212659999999, + 3.5744212656246064, + -3.7539349406756628E-10] + [1.6000000000000001, 3.6053199490000001, + 3.6053199490194707, + 1.9470647316666145E-11] + [1.6100000000000001, 3.6363347190000002, + 3.6363347191218383, + 1.2183809516841393E-10] + [1.6200000000000001, 3.6674672209999999, + 3.6674672206298222, + -3.7017766629787729E-10] + [1.6299999999999999, 3.6987190989999998, + 3.6987190986647667, + -3.3523317455319557E-10] + [1.6399999999999999, 3.7300919989999999, + 3.7300919990684158, + 6.8415939580290797E-11] + [1.6499999999999999, 3.761587569, + 3.7615875686941349, + -3.0586511101660108E-10] + [1.6599999999999999, 3.7932074560000002, + 3.7932074556923925, + -3.0760771707605272E-10] + [1.6699999999999999, 3.8249533100000002, + 3.824953309790788, + -2.092122031172039E-10] + [1.6799999999999999, 3.8568267829999998, + 3.8568267825688243, + -4.3117553971683265E-10] + [1.6899999999999999, 3.888829528, + 3.8888295277276339, + -2.723661296499813E-10] + [1.7, 3.9209632010000002, + 3.9209632013549038, + 3.5490366201429424E-10] + [1.71, 3.9532294619999999, + 3.953229462185158, + 1.8515811106567526E-10] + [1.72, 3.9856299719999999, + 3.985629971855627, + -1.4437295803304551E-10] + [1.73, 4.0181663949999997, + 4.0181663951578672, + 1.5786749685275936E-10] + [1.74, 4.0508404000000002, + 4.0508404002853169, + 2.8531665918762883E-10] + [1.75, 4.0836536590000003, + 4.0836536590769557, + 7.6955330996497651E-11] + [1.76, 4.116607847, + 4.1166078472572494, + 2.5724933294668517E-10] + [1.77, 4.1497046449999999, + 4.1497046446724992, + -3.2750069323128628E-10] + [1.78, 4.1829457359999997, + 4.1829457355238073, + -4.7619241883012364E-10] + [1.79, 4.2163328089999998, + 4.2163328085967509, + -4.0324898975541146E-10] + [1.8, 4.249867557, + 4.2498675574879341, + 4.879341375385593E-10] + [1.8100000000000001, 4.2835516809999996, + 4.2835516808285554, + -1.7144419217629547E-10] + [1.8200000000000001, 4.3173868830000002, + 4.3173868825051116, + -4.9488857456481128E-10] + [1.8300000000000001, 4.3513748720000001, + 4.3513748718773684, + -1.2263168258641599E-10] + [1.8400000000000001, 4.385517364, + 4.3855173639937215, + -6.2785332488601853E-12] + [1.8500000000000001, 4.4198160800000004, + 4.4198160798040753, + -1.9592505395849003E-10] + [1.8600000000000001, 4.454272746, + 4.4542727463703349, + 3.7033487387816422E-10] + [1.8700000000000001, 4.4888890970000004, + 4.4888890970746314, + 7.4630968072142423E-11] + [1.8799999999999999, 4.5236668719999997, + 4.523666871825391, + -1.7460877188568702E-10] + [1.8899999999999999, 4.5586078170000004, + 4.5586078172613478, + 2.6134738817518155E-10] + [1.8999999999999999, 4.5937136870000002, + 4.5937136869535857, + -4.6414427856689144E-11] + [1.9099999999999999, 4.6289862419999999, + 4.6289862416057304, + -3.9426950593224319E-10] + [1.9199999999999999, 4.6644272490000001, + 4.6644272492523706, + 2.5237056888727238E-10] + [1.9299999999999999, 4.7000384850000003, + 4.7000384854557851, + 4.5578474328067387E-10] + [1.9399999999999999, 4.7358217339999999, + 4.7358217335010897, + -4.9891024644921345E-10] + [1.95, 4.7717787850000004, + 4.7717787845898787, + -4.1012171436705103E-10] + [1.96, 4.8079114379999996, + 4.8079114380324146, + 3.241495960537577E-11] + [1.97, 4.8442215009999998, + 4.8442215014384944, + 4.3849457398437153E-10] + [1.98, 4.8807107910000003, + 4.8807107909070337, + -9.2966523368431808E-11] + [1.99, 4.917381131, + 4.9173811312144435, + 2.1444357400923764E-10] + [2., 4.9542343559999997, + 4.9542343560018924, + 1.8927082123809669E-12] + +\end{verbatim} \begin{chunk}{*} + --S 17 of 20 f(x)==x/10.0*exp(-x/10.0)*Ei(x/10.0) --R --R Type: Void --E 17 +\end{chunk} + +In the following table we show values returned by +$xe^{-x}Ei(x)$, chosen to keep the values in a reasonable range. +Abramowitz and Stegun, ``Handbook of Mathematical Functions'', +Dover Publications, Inc. New York 1965. pp 242-243 + +\begin{chunk}{*} + --S 18 of 20 +t9:List List Fraction Integer:= [[machineFraction (x::DFLOAT) for x in line]_ + for line in _ [[2.0,1.340965420,f(2.0),f(2.0)-1.340965420],_ [2.1,1.371486802,f(2.1),f(2.1)-1.371486802],_ [2.2,1.397421992,f(2.2),f(2.2)-1.397421992],_ @@ -1974,242 +3649,528 @@ f(x)==x/10.0*exp(-x/10.0)*Ei(x/10.0) [9.7,1.137209523,f(9.7),f(9.7)-1.137209523],_ [9.8,1.135241130,f(9.8),f(9.8)-1.135241130],_ [9.9,1.133328746,f(9.9),f(9.9)-1.133328746],_ - [10.0,1.131470205,f(10.0),f(10.0)-1.131470205]] + [10.0,1.131470205,f(10.0),f(10.0)-1.131470205]]] --R --R Compiling function f with type Float -> OnePointCompletion --R DoubleFloat --R --R (19) ---R [[2.,1.3409654200000001,- 0.13456013299662745,- 1.4755255529966276], ---R ---R [2.1000000000000001, 1.3714868019999999, - 0.12968783850914051, ---R - 1.5011746405091404] ---R , ---R ---R [2.2000000000000002, 1.3974219919999999, - 0.12432857913849607, ---R - 1.521750571138496] ---R , ---R [2.2999999999999998,1.419171534,- 0.11851397777493734,- 1.5376855117749373], ---R [2.3999999999999999,1.437118315,- 0.1122732093067646,- 1.5493915243067646], ---R [2.5,1.451625159,- 0.10563327984220373,- 1.5572584388422037], ---R ---R [2.6000000000000001, 1.463033397, - 9.8619263183169451E-2, ---R - 1.5616526601831695] ---R , ---R ---R [2.7000000000000002, 1.471662153, - 9.1254502584207586E-2, ---R - 1.5629166555842076] ---R , ---R ---R [2.7999999999999998, 1.4778081869999999, - 8.3560784069182492E-2, ---R - 1.5613689710691825] ---R , ---R ---R [2.8999999999999999, 1.4817461620000001, - 7.5558486253840138E-2, ---R - 1.5573046482538402] ---R , ---R [3.,1.4837292040000001,- 6.7266710614573164E-2,- 1.5509959146145733], ---R ---R [3.1000000000000001, 1.4839896910000001, - 5.8703395368669309E-2, ---R - 1.5426930863686694] ---R , ---R ---R [3.2000000000000002, 1.482740191, - 4.9885415529372513E-2, ---R - 1.5326256065293724] ---R , ---R ---R [3.2999999999999998, 1.4801744910000001, - 4.0828671227296824E-2, ---R - 1.5210031622272968] ---R , ---R ---R [3.3999999999999999, 1.4764687059999999, - 3.1548166016793916E-2, ---R - 1.5080168720167939] ---R , ---R [3.5,1.4717823889999999,- 2.2058076588733416E-2,- 1.4938404655887334], ---R ---R [3.6000000000000001, 1.4662596590000001, - 1.2371815072632724E-2, ---R - 1.4786314740726327] ---R , ---R ---R [3.7000000000000002, 1.4600303130000001, - 2.5020849182828334E-3, ---R - 1.4625323979182829] ---R , ---R ---R [3.7999999999999998, 1.4532109019999999, 7.5390688098497787E-3, ---R - 1.4456718331901501] ---R , ---R [3.8999999999999999,1.445905765,1.774021402057336E-2,- 1.4281655509794267], ---R [4.,1.4382080319999999,2.8090490467135878E-2,- 1.4101175415328639], ---R [4.0999999999999996,1.430200557,3.8579572390008463E-2,- 1.3916209846099916], ---R [4.2000000000000002,1.421956813,4.9197634492545148E-2,- 1.3727591785074549], ---R ---R [4.2999999999999998, 1.4135417189999999, 5.9935320871702981E-2, ---R - 1.3536063981282969] ---R , 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[----------------,----------------,----------------,- ----------------], +--R 281474976710656 4503599627370496 4503599627370496 2251799813685248 +--R 4953959590107545 2607251796077019 5155417927147399 5273589257160677 +--R [----------------,----------------,----------------,- ----------------], +--R 562949953421312 2251799813685248 9007199254740992 9007199254740992 +--R 1252563646362419 1300713565400577 5252251618528469 5153456904676147 +--R [----------------,----------------,----------------,- ----------------], +--R 140737488355328 1125899906842624 9007199254740992 9007199254740992 +--R 5191565944100307 2674270810746341 1258647566676983 +--R [9,----------------,----------------,- ----------------], +--R 4503599627370496 4503599627370496 2251799813685248 +--R 5122844576133939 5180624516359605 2722139017547721 614621374702971 +--R [----------------,----------------,----------------,- ----------------], +--R 562949953421312 4503599627370496 4503599627370496 1125899906842624 +--R 2589569785738035 1292504102258631 2769725726976867 2400290682057657 +--R [----------------,----------------,----------------,- ----------------], +--R 281474976710656 1125899906842624 4503599627370496 4503599627370496 +--R 5235434566818201 322483034823909 2817026475692305 2342702081490239 +--R [----------------,---------------,----------------,- ----------------], +--R 562949953421312 281474976710656 4503599627370496 4503599627370496 +--R 1322932390540083 321859276705369 716009257896687 571427848924789 +--R [----------------,---------------,----------------,- ----------------], +--R 140737488355328 281474976710656 1125899906842624 1125899906842624 +--R 19 5140063981222799 2910753385971527 278663824406409 +--R [--,----------------,----------------,- ---------------], +--R 2 4503599627370496 4503599627370496 562949953421312 +--R 5404319552844595 5130663658252985 1478585873023801 2173491912205383 +--R [----------------,----------------,----------------,- ----------------], +--R 562949953421312 4503599627370496 2251799813685248 4503599627370496 +--R 2730307274093363 5121536384024979 6006577055300447 4236495712749511 +--R [----------------,----------------,----------------,- ----------------], +--R 281474976710656 4503599627370496 9007199254740992 9007199254740992 +--R 5516909543528857 1278167882510915 6098200696509967 4127142363577353 +--R [----------------,----------------,----------------,- ----------------], +--R 562949953421312 1125899906842624 9007199254740992 9007199254740992 +--R 1393301134717747 5104058918173871 6189208040467493 4018909795880249 +--R [----------------,----------------,----------------,- ----------------], +--R 140737488355328 4503599627370496 9007199254740992 9007199254740992 +--R 2547844396809409 3139796544349511 1955892249269307 +--R [10,----------------,----------------,- ----------------]] +--R 2251799813685248 4503599627370496 4503599627370496 +--R Type: List List Fraction Integer --E 18 \end{chunk} -In the following table we show the values returned for large -arguments to the Ei function. See -Abramowitz and Stegun, ``Handbook of Mathematical Functions'', -Dover Publications, Inc. New York 1965. p 243 +The actual table values from above are: +\begin{verbatim} + + [2., 1.3409654200000001, + -0.13456013299662745, + -1.4755255529966276], + [2.1000000000000001, 1.3714868019999999, + -0.12968783850914051, + -1.5011746405091404] + [2.2000000000000002, 1.3974219919999999, + -0.12432857913849607, + -1.521750571138496] + [2.2999999999999998, 1.419171534, + -0.11851397777493734, + -1.5376855117749373], + [2.3999999999999999, 1.437118315, + -0.1122732093067646, + -1.5493915243067646], + [2.5, 1.451625159, + -0.10563327984220373, + -1.5572584388422037], + [2.6000000000000001, 1.463033397, + -9.8619263183169451E-2, + -1.5616526601831695] + [2.7000000000000002, 1.471662153, + -9.1254502584207586E-2, + -1.5629166555842076] + [2.7999999999999998, 1.4778081869999999, + -8.3560784069182492E-2, + -1.5613689710691825] + [2.8999999999999999, 1.4817461620000001, + -7.5558486253840138E-2, + -1.5573046482538402] + [3., 1.4837292040000001, + -6.7266710614573164E-2, + -1.5509959146145733], + [3.1000000000000001, 1.4839896910000001, + -5.8703395368669309E-2, + -1.5426930863686694] + [3.2000000000000002, 1.482740191, + -4.9885415529372513E-2, + -1.5326256065293724] + [3.2999999999999998, 1.4801744910000001, + -4.0828671227296824E-2, + -1.5210031622272968] + [3.3999999999999999, 1.4764687059999999, + -3.1548166016793916E-2, + -1.5080168720167939] + [3.5, 1.4717823889999999, + -2.2058076588733416E-2, + -1.4938404655887334], + [3.6000000000000001, 1.4662596590000001, + -1.2371815072632724E-2, + -1.4786314740726327] + [3.7000000000000002, 1.4600303130000001, + -2.5020849182828334E-3, + -1.4625323979182829] + [3.7999999999999998, 1.4532109019999999, + 7.5390688098497787E-3, + -1.4456718331901501] + [3.8999999999999999, 1.445905765, + 1.774021402057336E-2, + -1.4281655509794267], + [4., 1.4382080319999999, + 2.8090490467135878E-2, + -1.4101175415328639], + [4.0999999999999996, 1.430200557, + 3.8579572390008463E-2, + -1.3916209846099916], + [4.2000000000000002, 1.421956813, + 4.9197634492545148E-2, + -1.3727591785074549], + [4.2999999999999998, 1.4135417189999999, + 5.9935320871702981E-2, + -1.3536063981282969] + [4.4000000000000004, 1.4050124239999999, + 7.0783716577210318E-2, + -1.3342287074227897] + [4.5, 1.3964190299999999, + 8.1734321515770605E-2, + -1.3146847084842292], + [4.5999999999999996, 1.387805263, + 9.2779026453518668E-2, + -1.2950262365464813], + [4.7000000000000002, 1.3792090930000001, + 0.10391009090110491, + -1.2752990020988952] + [4.7999999999999998, 1.3706633130000001, + 0.11512012269240564, + -1.2555431903075944] + [4.9000000000000004, 1.362196054, + 0.12640205909068003, + -1.23579399490932], + [5., 1.3538312779999999, + 0.13774914927563506, + -1.2160821287243648], + [5.0999999999999996, 1.3455892119999999, + 0.14915493808180313, + -1.1964342739181968] + [5.2000000000000002, 1.337486755, + 0.16061325087332862, + -1.1768735041266714], + [5.2999999999999998, 1.3295378449999999, + 0.17211817945300276, + -1.1574196655469973] + [5.4000000000000004, 1.3217537880000001, + 0.1836640689145036, + -1.1380897190854964] + [5.5, 1.314143566, + 0.19524550535650245, + -1.1188980606434975], + [5.5999999999999996, 1.3067141069999999, + 0.20685730438580638, + -1.0998568026141935] + [5.7000000000000002, 1.2994705360000001, + 0.21849450034417656, + -1.0809760356558236] + [5.7999999999999998, 1.2924163950000001, + 0.23015233620004197, + -1.0622640587999581] + [5.9000000000000004, 1.285553849, + 0.24182625405213551, + -1.0437275949478646], + [6., 1.2788838600000001, + 0.25351188619722104, + -1.0253719738027791], + [6.0999999999999996, 1.2724063569999999, + 0.26520504671863687, + -1.0072013102813631] + [6.2000000000000002, 1.2661203729999999, + 0.27690172355643178, + -0.98921864944356819] + [6.2999999999999998, 1.2600241839999999, + 0.28859807102347912, + -0.97142611297652082] + [6.4000000000000004, 1.254115417, + 0.30029040273517321, + -0.9538250142648268], + [6.5, 1.248391155, + 0.31197518492319382, + -0.93641597007680621], + [6.5999999999999996, 1.2428480319999999, + 0.32364903010640295, + -0.91919900189359693] + [6.7000000000000002, 1.237482309, + 0.33530869109425609, + -0.90217361790574391], + [6.7999999999999998, 1.2322899519999999, + 0.34695105530018927, + -0.88533889669981058] + [6.9000000000000004, 1.2272666839999999, + 0.3585731393443225, + -0.86869354465567739] + [7., 1.2224080530000001, + 0.37017208392651269, + -0.85223596907348742], + [7.0999999999999996, 1.2177094719999999, + 0.38174514895231304, + -0.83596432304768686] + [7.2000000000000002, 1.213166264, + 0.39328970889579112, + -0.81987655510420887], + [7.2999999999999998, 1.208773699, + 0.40480324838440268, + -0.80397045061559735], + [7.4000000000000004, 1.2045270260000001, + 0.4162833579922634, + -0.78824366800773671] + [7.5, 1.2004215, + 0.42772773022919919, + -0.77269376977080084], + [7.5999999999999996, 1.1964524009999999, + 0.43913415571389103, + -0.75731824528610892] + [7.7000000000000002, 1.1926150630000001, + 0.45050051952030151, + -0.74211454347969852] + [7.7999999999999998, 1.188904881, + 0.46182479768734747, + -0.7270800833126525], + [7.9000000000000004, 1.1853173340000001, + 0.47310505388250573, + -0.71221228011749438] + [8., 1.181847987, + 0.48433943621069148, + -0.69750855078930862], + [8.0999999999999996, 1.178492509, + 0.49552617416036354, + -0.68296633483963642], + [8.1999999999999993, 1.175246676, + 0.50666357567934228, + -0.66858310032065771], + [8.3000000000000007, 1.1721063759999999, + 0.51775002437336082, + -0.65435635162663908] + [8.4000000000000004, 1.1690676170000001, + 0.52878397682080924, + -0.64028364017919082] + [8.5, 1.166126526, + 0.53976395999758919, + -0.6263625660024108], + [8.5999999999999996, 1.1632793539999999, + 0.5506885688063663, + -0.61259078519363364] + [8.6999999999999993, 1.1605224759999999, + 0.56155646370489876, + -0.59896601229510116] + [8.8000000000000007, 1.15785239, + 0.57236636842843447, + -0.58548602157156548], + [8.9000000000000004, 1.155265719, + 0.58311706780150541, + -0.57214865119849456], + [9., 1.1527592090000001, + 0.59380740563471446, + -0.5589518033652856], + [9.0999999999999996, 1.1503297240000001, + 0.60443628270239691, + -0.5458934412976032] + [9.1999999999999993, 1.1479742509999999, + 0.61500265479727367, + -0.53297159620272627] + [9.3000000000000007, 1.145689889, + 0.62550553085845151, + -0.52018435814154851], + [9.4000000000000004, 1.1434738550000001, + 0.63594397116933887, + -0.50752988383066122] + [9.5, 1.1413234759999999, + 0.6463170856222451, + -0.49500639037775485], + [9.5999999999999996, 1.1392361849999999, + 0.65662403204660502, + -0.48261215295339488] + [9.6999999999999993, 1.1372095230000001, + 0.66686401459797295, + -0.47034550840202716] + [9.8000000000000007, 1.13524113, + 0.6770362822050533, + -0.45820484779494675], + [9.9000000000000004, 1.1333287460000001, + 0.68714012707221583, + -0.44618861892778428] + [10., 1.1314702050000001, + 0.69717488323506582, + -0.43429532176493424]] + +\end{verbatim} \begin{chunk}{*} + --S 19 of 20 g(y)==(y=0 => 1 ; (x:DFLOAT:=y^-1) ; x*exp(-x)*Ei(x)) --R --R Type: Void --E 19 +\end{chunk} + +In the following table we show the values returned for large +arguments to the Ei function. See +Abramowitz and Stegun, ``Handbook of Mathematical Functions'', +Dover Publications, Inc. New York 1965. p 243 + +\begin{chunk}{*} + --S 20 of 20 +t10:List List Fraction Integer:= [[machineFraction (x::DFLOAT) for x in line]_ + for line in _ [[0.100,1.13147021,g(0.100),g(0.100)-1.13147021],_ [0.095,1.12249671,g(0.095),g(0.095)-1.12249671],_ [0.090,1.11389377,g(0.090),g(0.090)-1.11389377],_ @@ -2230,72 +4191,147 @@ g(y)==(y=0 => 1 ; (x:DFLOAT:=y^-1) ; x*exp(-x)*Ei(x)) [0.015,1.01547157,g(0.015),g(0.015)-1.01547157],_ [0.010,1.01020625,g(0.010),g(0.010)-1.01020625],_ [0.005,1.00505077,g(0.005),g(0.005)-1.00505077],_ - [0.000,1.00000000,g(0.000),g(0.000)-1.00000000]] + [0.000,1.00000000,g(0.000),g(0.000)-1.00000000]]] --R --R Compiling function g with type Float -> OnePointCompletion --R DoubleFloat --R --R (21) ---R [[0.10000000000000001,1.13147021,1.1314702047341079,- 5.2658921667614322E-9], ---R ---R [9.5000000000000001E-2, 1.1224967100000001, 1.1224967463528539, ---R 3.6352853838295118E-8] ---R , ---R [8.9999999999999997E-2,1.11389377,1.1138937808537757,1.0853775656016751E-8], ---R ---R [8.5000000000000006E-2, 1.1056473899999999, 1.1056473901733923, ---R 1.733924115399077E-10] ---R , ---R ---R [8.0000000000000002E-2, 1.0977377500000001, 1.0977377526473173, ---R 2.647317254300674E-9] ---R , ---R ---R [7.4999999999999997E-2, 1.0901408699999999, 1.0901408684282585, ---R - 1.5717414036942046E-9] ---R , ---R [7.0000000000000007E-2,1.08283054,1.0828305423224371,2.3224371314967129E-9], ---R ---R [6.5000000000000002E-2, 1.0757803800000001, 1.0757803749062493, ---R - 5.0937507545256722E-9] ---R , ---R ---R [5.9999999999999998E-2, 1.0689654799999999, 1.0689654755715123, ---R - 4.4284875766464893E-9] ---R , ---R [5.5E-2,1.06236365,1.0623636462639567,- 3.7360432525446186E-9], ---R ---R [5.0000000000000003E-2, 1.05595591, 1.0559559055929626, ---R - 4.4070374016769165E-9] ---R , ---R ---R [4.4999999999999998E-2, 1.0497263999999999, 1.0497264028491122, ---R 2.8491122794349621E-9] ---R , ---R ---R [4.0000000000000001E-2, 1.04366194, 1.0436619362666135, ---R - 3.7333864888466906E-9] ---R , ---R [3.5000000000000003E-2,1.03775135,1.0377513519241477,1.924147730036907E-9], ---R ---R [2.9999999999999999E-2, 1.03198503, 1.0319850279857541, ---R - 2.0142458811989172E-9] ---R , ---R [2.5000000000000001E-2,1.02635451,1.026354511439006,1.4390060254498849E-9], ---R [2.0E-2,1.0208522799999999,1.0208522777971993,- 2.2028006085861307E-9], ---R ---R [1.4999999999999999E-2, 1.0154715700000001, 1.0154715653071829, ---R - 4.6928172459104189E-9] ---R , ---R [1.0E-2,1.01020625,1.0102062527748354,2.7748354725076751E-9], ---R ---R [5.0000000000000001E-3, 1.00505077, 1.0050507653866605, ---R - 4.6133394882019729E-9] ---R , ---R [0.,1.,1.,0.]] ---R Type: List List OnePointCompletion DoubleFloat +--R 7205759403792793 318480551008551 5095688792421347 23715469 +--R [[-----------------,---------------,----------------,- ----------------], +--R 72057594037927936 281474976710656 4503599627370496 4503599627370496 +--R 6845471433603153 5055275764880607 1263818982149827 163718701 +--R [-----------------,----------------,----------------,----------------], +--R 72057594037927936 4503599627370496 1125899906842624 4503599627370496 +--R 3242591731706757 1254132891875579 5016531616383377 48881061 +--R [-----------------,----------------,----------------,----------------], +--R 36028797018963968 1125899906842624 4503599627370496 4503599627370496 +--R 3062447746611937 4979393173607161 4979393174388051 390445 +--R [-----------------,----------------,----------------,----------------], +--R 36028797018963968 4503599627370496 4503599627370496 2251799813685248 +--R 2882303761517117 2471885660925263 617971416721623 5961229 +--R [-----------------,----------------,---------------,----------------], +--R 36028797018963968 2251799813685248 562949953421312 2251799813685248 +--R 5404319552844595 1227389503978337 2454779004417427 3539247 +--R [-----------------,----------------,----------------,- ----------------], +--R 72057594037927936 1125899906842624 2251799813685248 2251799813685248 +--R 5044031582654955 304789701028087 304789701681795 163427 +--R [-----------------,---------------,---------------,--------------], +--R 72057594037927936 281474976710656 281474976710656 70368744177664 +--R 4683743612465315 2422442059250245 4844884095560277 22940213 +--R [-----------------,----------------,----------------,- ----------------], +--R 72057594037927936 2251799813685248 4503599627370496 4503599627370496 +--R 1080863910568919 4814192537399923 1203548129363947 19944135 +--R [-----------------,----------------,----------------,- ----------------], +--R 18014398509481984 4503599627370496 1125899906842624 4503599627370496 +--R 990791918021509 598057567283995 4784460521446317 16825643 +--R [-----------------,---------------,----------------,- ----------------], +--R 18014398509481984 562949953421312 4503599627370496 4503599627370496 +--R 7205759403792793 4755602642795673 4755602622948141 4961883 +--R [------------------,----------------,----------------,- ----------------], +--R 144115188075855872 4503599627370496 4503599627370496 1125899906842624 +--R 3242591731706757 1181886855970243 4727547436712233 12831261 +--R [-----------------,----------------,----------------,----------------], +--R 72057594037927936 1125899906842624 4503599627370496 4503599627370496 +--R 2882303761517117 146882360127649 4700235507271091 16813677 +--R [-----------------,---------------,----------------,- ----------------], +--R 72057594037927936 140737488355328 4503599627370496 4503599627370496 +--R 5044031582654955 4673616593163229 1168404150457205 8665591 +--R [------------------,----------------,----------------,----------------], +--R 144115188075855872 4503599627370496 1125899906842624 4503599627370496 +--R 1080863910568919 2323823698279965 4647647387488573 9071357 +--R [-----------------,----------------,----------------,- ----------------], +--R 36028797018963968 2251799813685248 4503599627370496 4503599627370496 +--R 7205759403792793 1155572447196507 4622289795266735 6480707 +--R [------------------,----------------,----------------,----------------], +--R 288230376151711744 1125899906842624 4503599627370496 4503599627370496 +--R 2882303761517117 4597509947808321 4597509937887789 2480133 +--R [------------------,----------------,----------------,- ----------------], +--R 144115188075855872 4503599627370496 4503599627370496 1125899906842624 +--R 1080863910568919 1143319346064333 4573277363122763 21134569 +--R [-----------------,----------------,----------------,- ----------------], +--R 72057594037927936 1125899906842624 4503599627370496 4503599627370496 +--R 2882303761517117 2274782245533673 2274782251782047 3124187 +--R [------------------,----------------,----------------,----------------], +--R 288230376151711744 2251799813685248 2251799813685248 1125899906842624 +--R 2882303761517117 2263173136630215 1131586563120949 10388317 +--R [------------------,----------------,----------------,- ----------------], +--R 576460752303423488 2251799813685248 1125899906842624 2251799813685248 +--R [0,1,1,0]] +--R Type: List List Fraction Integer --E 20 +\end{chunk} + +The actual table values from above are: +\begin{verbatim} + [0.10000000000000001, 1.13147021, + 1.1314702047341079, + -5.2658921667614322E-9], + [9.5000000000000001E-2, 1.1224967100000001, + 1.1224967463528539, + 3.6352853838295118E-8] + [8.9999999999999997E-2, 1.11389377, + 1.1138937808537757, + 1.0853775656016751E-8], + [8.5000000000000006E-2, 1.1056473899999999, + 1.1056473901733923, + 1.733924115399077E-10] + [8.0000000000000002E-2, 1.0977377500000001, + 1.0977377526473173, + 2.647317254300674E-9] + [7.4999999999999997E-2, 1.0901408699999999, + 1.0901408684282585, + -1.5717414036942046E-9] + [7.0000000000000007E-2, 1.08283054, + 1.0828305423224371, + 2.3224371314967129E-9], + [6.5000000000000002E-2, 1.0757803800000001, + 1.0757803749062493, + -5.0937507545256722E-9] + [5.9999999999999998E-2, 1.0689654799999999, + 1.0689654755715123, + -4.4284875766464893E-9] + [5.5E-2, 1.06236365, + 1.0623636462639567, + -3.7360432525446186E-9], + [5.0000000000000003E-2, 1.05595591, + 1.0559559055929626, + -4.4070374016769165E-9] + [4.4999999999999998E-2, 1.0497263999999999, + 1.0497264028491122, + 2.8491122794349621E-9] + [4.0000000000000001E-2, 1.04366194, + 1.0436619362666135, + -3.7333864888466906E-9] + [3.5000000000000003E-2, 1.03775135, + 1.0377513519241477, + 1.924147730036907E-9], + [2.9999999999999999E-2, 1.03198503, + 1.0319850279857541, + -2.0142458811989172E-9] + [2.5000000000000001E-2, 1.02635451, + 1.026354511439006, + 1.4390060254498849E-9], + [2.0E-2, 1.0208522799999999, + 1.0208522777971993, + -2.2028006085861307E-9], + [1.4999999999999999E-2, 1.0154715700000001, + 1.0154715653071829, + -4.6928172459104189E-9] + [1.0E-2, 1.01020625, + 1.0102062527748354, + 2.7748354725076751E-9], + [5.0000000000000001E-3, 1.00505077, + 1.0050507653866605, + -4.6133394882019729E-9] + [0., 1., + 1., + 0.]] + +\end{verbatim} +\begin{chunk}{*} + )spool )lisp (bye) @@ -2310,3 +4346,4 @@ Dover Publications, Inc. New York 1965. pp238-243 \end{thebibliography} \end{document} +