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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,258 @@ | ||
| using Base.Math.@horner | ||
|
|
||
| # Compute the sine integral: ∫_0^x sin(t)/t dt, | ||
| # and the cosine integral: γ + log x + ∫_0^x (cos(t)-1)/t dt, | ||
| # using the rational approximants tabulated in: | ||
| # A.J. MacLeod, "Rational approximations, software and test methods for | ||
| # sine and cosine integrals," Numer. Algor. 12, pp. 259--272 (1996). | ||
| # http://dx.doi.org/10.1007/BF02142806 | ||
| # https://link.springer.com/article/10.1007/BF02142806 | ||
| # | ||
| # Note: the second zero of Ci(x) has a typo that is fixed: | ||
| # | ||
| # r1 = 3.38418 04228 51186 42639 78511 46402 in the article, but is in fact: | ||
| # | ||
| # r1 = 3.38418 04225 51186 42639 78511 46402. | ||
| # | ||
|
|
||
| function sinint(x::Float64) | ||
| t = x*x | ||
| if t ≤ 36.0 | ||
| return x * @horner(t, 1.00000000000000000000E0, | ||
| -0.44663998931312457298E-1, | ||
| 0.11209146443112369449E-2, | ||
| -0.13276124407928422367E-4, | ||
| 0.85118014179823463879E-7, | ||
| -0.29989314303147656479E-9, | ||
| 0.55401971660186204711E-12, | ||
| -0.42406353433133212926E-15) / | ||
| @horner(t, 1.00000000000000000000E0, | ||
| 0.10891556624243098264E-1, | ||
| 0.59334456769186835896E-4, | ||
| 0.21231112954641805908E-6, | ||
| 0.54747121846510390750E-9, | ||
| 0.10378561511331814674E-11, | ||
| 0.13754880327250272679E-14, | ||
| 0.10223981202236205703E-17) | ||
| elseif t ≤ 144.0 | ||
| invt = inv(t) | ||
| return copysign(π/2, x) - cos(x) * | ||
| @horner(invt, 0.99999999962173909991E0, | ||
| 0.36451060338631902917E3, | ||
| 0.44218548041288440874E5, | ||
| 0.22467569405961151887E7, | ||
| 0.49315316723035561922E8, | ||
| 0.43186795279670283193E9, | ||
| 0.11847992519956804350E10, | ||
| 0.45573267593795103181E9) / | ||
| (x * @horner(invt, 1.00000000000000000000E0, | ||
| 0.36651060273229347594E3, | ||
| 0.44927569814970692777E5, | ||
| 0.23285354882204041700E7, | ||
| 0.53117852017228262911E8, | ||
| 0.50335310667241870372E9, | ||
| 0.16575285015623175410E10, | ||
| 0.11746532837038341076E10)) - | ||
| sin(x)*invt * @horner(invt, 0.99999999920484901956E0, | ||
| 0.51385504875307321394E3, | ||
| 0.92293483452013810811E5, | ||
| 0.74071341863359841727E7, | ||
| 0.28142356162841356551E9, | ||
| 0.49280890357734623984E10, | ||
| 0.35524762685554302472E11, | ||
| 0.79194271662085049376E11, | ||
| 0.17942522624413898907E11) / | ||
| @horner(invt, 1.00000000000000000000E0, | ||
| 0.51985504708814870209E3, | ||
| 0.95292615508125947321E5, | ||
| 0.79215459679762667578E7, | ||
| 0.31977567790733781460E9, | ||
| 0.62273134702439012114E10, | ||
| 0.54570971054996441467E11, | ||
| 0.18241750166645704670E12, | ||
| 0.15407148148861454434E12) | ||
| elseif t < Inf | ||
| invt = inv(t) | ||
| return copysign(π/2, x) - cos(x) / x * (1.0 - | ||
| @horner(invt, 0.19999999999999978257E1, | ||
| 0.22206119380434958727E4, | ||
| 0.84749007623988236808E6, | ||
| 0.13959267954823943232E9, | ||
| 0.10197205463267975592E11, | ||
| 0.30229865264524075951E12, | ||
| 0.27504053804288471142E13, | ||
| 0.21818989704686874983E13) / | ||
| @horner(invt, 1.00000000000000000000E0, | ||
| 0.11223059690217167788E4, | ||
| 0.43685270974851313242E6, | ||
| 0.74654702140658116258E8, | ||
| 0.58580034751805687471E10, | ||
| 0.20157980379272098841E12, | ||
| 0.26229141857684496445E13, | ||
| 0.87852907334918467516E13)*invt) - | ||
| sin(x) * invt * (1.0 - @horner(invt, 0.59999999999999993089E1, | ||
| 0.96527746044997139158E4, | ||
| 0.56077626996568834185E7, | ||
| 0.15022667718927317198E10, | ||
| 0.19644271064733088465E12, | ||
| 0.12191368281163225043E14, | ||
| 0.31924389898645609533E15, | ||
| 0.25876053010027485934E16, | ||
| 0.12754978896268878403E16) / | ||
| @horner(invt, 1.00000000000000000000E0, | ||
| 0.16287957674166143196E4, | ||
| 0.96636303195787870963E6, | ||
| 0.26839734750950667021E9, | ||
| 0.37388510548029219241E11, | ||
| 0.26028585666152144496E13, | ||
| 0.85134283716950697226E14, | ||
| 0.11304079361627952930E16, | ||
| 0.42519841479489798424E16)*invt) | ||
| elseif isnan(x) | ||
| return NaN | ||
| else | ||
| return copysign(π/2, x) | ||
| end | ||
| end | ||
|
|
||
| function cosint(x::Float64) | ||
| t, r0, r1 = x*x, 0.616505485620716233797110404100, 3.384180422551186426397851146402 | ||
| r01, r02 = 0.6162109375, 0.29454812071623379711E-3 | ||
| r11, r12 = 3.3837890625, 0.39136005118642639785E-3 | ||
| if x < 0.0 | ||
| return throw(DomainError()) | ||
| elseif x ≤ 3.0 | ||
| return log(x/r0) + ((x - r01) - r02) * (x + r0) * | ||
| @horner(t, -0.24607411378767540707E0, | ||
| 0.72113492241301534559E-2, | ||
| -0.11867127836204767056E-3, | ||
| 0.90542655466969866243E-6, | ||
| -0.34322242412444409037E-8, | ||
| 0.51950683460656886834E-11) / | ||
| @horner(t, 1.00000000000000000000E0, | ||
| 0.12670095552700637845E-1, | ||
| 0.78168450570724148921E-4, | ||
| 0.29959200177005821677E-6, | ||
| 0.73191677761328838216E-9, | ||
| 0.94351174530907529061E-12) | ||
| elseif x ≤ 6.0 | ||
| return log(x/r1) + ((x - r11) - r12) * (x + r1) * | ||
| @horner(t, -0.15684781827145408780E0, | ||
| 0.66253165609605468916E-2, | ||
| -0.12822297297864512864E-3, | ||
| 0.12360964097729408891E-5, | ||
| -0.66450975112876224532E-8, | ||
| 0.20326936466803159446E-10, | ||
| -0.33590883135343844613E-13, | ||
| 0.23686934961435015119E-16) / | ||
| @horner(t, 1.00000000000000000000E0, | ||
| 0.96166044388828741188E-2, | ||
| 0.45257514591257035006E-4, | ||
| 0.13544922659627723233E-6, | ||
| 0.27715365686570002081E-9, | ||
| 0.37718676301688932926E-12, | ||
| 0.27706844497155995398E-15) | ||
| elseif x ≤ 12.0 | ||
| invt = inv(t) | ||
| return sin(x) * @horner(invt, 0.99999999962173909991E0, | ||
| 0.36451060338631902917E3, | ||
| 0.44218548041288440874E5, | ||
| 0.22467569405961151887E7, | ||
| 0.49315316723035561922E8, | ||
| 0.43186795279670283193E9, | ||
| 0.11847992519956804350E10, | ||
| 0.45573267593795103181E9) / | ||
| (x * @horner(invt, 1.00000000000000000000E0, | ||
| 0.36651060273229347594E3, | ||
| 0.44927569814970692777E5, | ||
| 0.23285354882204041700E7, | ||
| 0.53117852017228262911E8, | ||
| 0.50335310667241870372E9, | ||
| 0.16575285015623175410E10, | ||
| 0.11746532837038341076E10)) - | ||
| cos(x) * invt * @horner(invt, 0.99999999920484901956E0, | ||
| 0.51385504875307321394E3, | ||
| 0.92293483452013810811E5, | ||
| 0.74071341863359841727E7, | ||
| 0.28142356162841356551E9, | ||
| 0.49280890357734623984E10, | ||
| 0.35524762685554302472E11, | ||
| 0.79194271662085049376E11, | ||
| 0.17942522624413898907E11) / | ||
| @horner(invt, 1.00000000000000000000E0, | ||
| 0.51985504708814870209E3, | ||
| 0.95292615508125947321E5, | ||
| 0.79215459679762667578E7, | ||
| 0.31977567790733781460E9, | ||
| 0.62273134702439012114E10, | ||
| 0.54570971054996441467E11, | ||
| 0.18241750166645704670E12, | ||
| 0.15407148148861454434E12) | ||
| elseif x < Inf | ||
| invt = inv(t) | ||
| return sin(x)/x * (1.0 - @horner(invt, 0.19999999999999978257E1, | ||
| 0.22206119380434958727E4, | ||
| 0.84749007623988236808E6, | ||
| 0.13959267954823943232E9, | ||
| 0.10197205463267975592E11, | ||
| 0.30229865264524075951E12, | ||
| 0.27504053804288471142E13, | ||
| 0.21818989704686874983E13) / | ||
| @horner(invt, 1.00000000000000000000E0, | ||
| 0.11223059690217167788E4, | ||
| 0.43685270974851313242E6, | ||
| 0.74654702140658116258E8, | ||
| 0.58580034751805687471E10, | ||
| 0.20157980379272098841E12, | ||
| 0.26229141857684496445E13, | ||
| 0.87852907334918467516E13)*invt) - | ||
| cos(x)*invt * (1.0 - @horner(invt, 0.59999999999999993089E1, | ||
| 0.96527746044997139158E4, | ||
| 0.56077626996568834185E7, | ||
| 0.15022667718927317198E10, | ||
| 0.19644271064733088465E12, | ||
| 0.12191368281163225043E14, | ||
| 0.31924389898645609533E15, | ||
| 0.25876053010027485934E16, | ||
| 0.12754978896268878403E16) / | ||
| @horner(invt, 1.00000000000000000000E0, | ||
| 0.16287957674166143196E4, | ||
| 0.96636303195787870963E6, | ||
| 0.26839734750950667021E9, | ||
| 0.37388510548029219241E11, | ||
| 0.26028585666152144496E13, | ||
| 0.85134283716950697226E14, | ||
| 0.11304079361627952930E16, | ||
| 0.42519841479489798424E16)*invt) | ||
| elseif isnan(x) | ||
| return NaN | ||
| else | ||
| return 0.0 | ||
| end | ||
| end | ||
|
|
||
| for f in (:sinint, :cosint) | ||
| @eval begin | ||
| ($f)(x::Float32) = Float32(($f)(Float64(x))) | ||
| ($f)(x::Float16) = Float16(($f)(Float64(x))) | ||
| ($f)(x::Real) = ($f)(float(x)) | ||
| ($f)(x::AbstractFloat) = error("not implemented for ", typeof(x)) | ||
| end | ||
| end | ||
|
|
||
|
|
||
| """ | ||
| sinint(x) | ||
| Compute the sine integral function of `x`, defined by ``\\operatorname{Si}(x) := \\int_0^x\\frac{\\sin t}{t} dt`` | ||
| for real `x`. | ||
| """ | ||
| sinint | ||
|
|
||
| """ | ||
| cosint(x) | ||
| Compute the cosine integral function of `x`, defined by ``\\operatorname{Ci}(x) := \\gamma + \\log x + \\int_0^x \\frac{\\cos t - 1}{t} dt`` | ||
| for real `x > 0`, where ``\\gamma`` is the Euler-Mascheroni constant. | ||
| """ | ||
| cosint |
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