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sincosint.jl
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sincosint.jl
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using Base.Math: @horner
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
throw(DomainError(x, "`x` must be nonnegative."))
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
@doc raw"""
sinint(x)
Compute the sine integral function of ``x``, defined by
```math
\operatorname{Si}(x)
:= \int_0^x \frac{\sin t}{t} \, \mathrm{d}t
\quad \text{for} \quad
x \in \mathbb{R} \,.
```
External links: [DLMF](https://dlmf.nist.gov/6.2.9),
[Wikipedia](https://en.wikipedia.org/wiki/Trigonometric_integral#Sine_integral).
See also: [`cosint(x)`](@ref SpecialFunctions.cosint).
# Implementation
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).
> <https://doi.org/10.1007/BF02142806>,
> <https://link.springer.com/article/10.1007/BF02142806>.
Note: the second zero of ``\text{Ci}(x)`` has a typo that is fixed:
``r_1 = 3.38418 0422\mathbf{8} 51186 42639 78511 46402`` in the article, but is in fact:
``r_1 = 3.38418 0422\mathbf{5} 51186 42639 78511 46402``.
"""
sinint
@doc raw"""
cosint(x)
Compute the cosine integral function of ``x``, defined by
```math
\operatorname{Ci}(x)
:= \gamma + \log x + \int_0^x \frac{\cos (t) - 1}{t} \, \mathrm{d}t
\quad \text{for} \quad
x > 0 \,,
```
where ``\gamma`` is the Euler-Mascheroni constant.
External links: [DLMF](https://dlmf.nist.gov/6.2.13),
[Wikipedia](https://en.wikipedia.org/wiki/Trigonometric_integral#Cosine_integral).
See also: [`sinint(x)`](@ref SpecialFunctions.sinint).
# Implementation
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).
> <https://doi.org/10.1007/BF02142806>,
> <https://link.springer.com/article/10.1007/BF02142806>.
Note: the second zero of ``\text{Ci}(x)`` has a typo that is fixed:
``r_1 = 3.38418 0422\mathbf{8} 51186 42639 78511 46402`` in the article, but is in fact:
``r_1 = 3.38418 0422\mathbf{5} 51186 42639 78511 46402``.
"""
cosint