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/* Translated into C++ by SciPy developers in 2024.
* Original header with Copyright information appears below.
*/
/* shichi.c
*
* Hyperbolic sine and cosine integrals
*
*
*
* SYNOPSIS:
*
* double x, Chi, Shi, shichi();
*
* shichi( x, &Chi, &Shi );
*
*
* DESCRIPTION:
*
* Approximates the integrals
*
* x
* -
* | | cosh t - 1
* Chi(x) = eul + ln x + | ----------- dt,
* | | t
* -
* 0
*
* x
* -
* | | sinh t
* Shi(x) = | ------ dt
* | | t
* -
* 0
*
* where eul = 0.57721566490153286061 is Euler's constant.
* The integrals are evaluated by power series for x < 8
* and by Chebyshev expansions for x between 8 and 88.
* For large x, both functions approach exp(x)/2x.
* Arguments greater than 88 in magnitude return INFINITY.
*
*
* ACCURACY:
*
* Test interval 0 to 88.
* Relative error:
* arithmetic function # trials peak rms
* IEEE Shi 30000 6.9e-16 1.6e-16
* Absolute error, except relative when |Chi| > 1:
* IEEE Chi 30000 8.4e-16 1.4e-16
*/
/*
* Cephes Math Library Release 2.0: April, 1987
* Copyright 1984, 1987 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#pragma once
#include "../config.h"
#include "chbevl.h"
#include "const.h"
namespace xsf {
namespace cephes {
namespace detail {
/* x exp(-x) shi(x), inverted interval 8 to 18 */
constexpr double shichi_S1[] = {
1.83889230173399459482E-17, -9.55485532279655569575E-17, 2.04326105980879882648E-16,
1.09896949074905343022E-15, -1.31313534344092599234E-14, 5.93976226264314278932E-14,
-3.47197010497749154755E-14, -1.40059764613117131000E-12, 9.49044626224223543299E-12,
-1.61596181145435454033E-11, -1.77899784436430310321E-10, 1.35455469767246947469E-9,
-1.03257121792819495123E-9, -3.56699611114982536845E-8, 1.44818877384267342057E-7,
7.82018215184051295296E-7, -5.39919118403805073710E-6, -3.12458202168959833422E-5,
8.90136741950727517826E-5, 2.02558474743846862168E-3, 2.96064440855633256972E-2,
1.11847751047257036625E0};
/* x exp(-x) shi(x), inverted interval 18 to 88 */
constexpr double shichi_S2[] = {
-1.05311574154850938805E-17, 2.62446095596355225821E-17, 8.82090135625368160657E-17,
-3.38459811878103047136E-16, -8.30608026366935789136E-16, 3.93397875437050071776E-15,
1.01765565969729044505E-14, -4.21128170307640802703E-14, -1.60818204519802480035E-13,
3.34714954175994481761E-13, 2.72600352129153073807E-12, 1.66894954752839083608E-12,
-3.49278141024730899554E-11, -1.58580661666482709598E-10, -1.79289437183355633342E-10,
1.76281629144264523277E-9, 1.69050228879421288846E-8, 1.25391771228487041649E-7,
1.16229947068677338732E-6, 1.61038260117376323993E-5, 3.49810375601053973070E-4,
1.28478065259647610779E-2, 1.03665722588798326712E0};
/* x exp(-x) chin(x), inverted interval 8 to 18 */
constexpr double shichi_C1[] = {
-8.12435385225864036372E-18, 2.17586413290339214377E-17, 5.22624394924072204667E-17,
-9.48812110591690559363E-16, 5.35546311647465209166E-15, -1.21009970113732918701E-14,
-6.00865178553447437951E-14, 7.16339649156028587775E-13, -2.93496072607599856104E-12,
-1.40359438136491256904E-12, 8.76302288609054966081E-11, -4.40092476213282340617E-10,
-1.87992075640569295479E-10, 1.31458150989474594064E-8, -4.75513930924765465590E-8,
-2.21775018801848880741E-7, 1.94635531373272490962E-6, 4.33505889257316408893E-6,
-6.13387001076494349496E-5, -3.13085477492997465138E-4, 4.97164789823116062801E-4,
2.64347496031374526641E-2, 1.11446150876699213025E0};
/* x exp(-x) chin(x), inverted interval 18 to 88 */
constexpr double shichi_C2[] = {
8.06913408255155572081E-18, -2.08074168180148170312E-17, -5.98111329658272336816E-17,
2.68533951085945765591E-16, 4.52313941698904694774E-16, -3.10734917335299464535E-15,
-4.42823207332531972288E-15, 3.49639695410806959872E-14, 6.63406731718911586609E-14,
-3.71902448093119218395E-13, -1.27135418132338309016E-12, 2.74851141935315395333E-12,
2.33781843985453438400E-11, 2.71436006377612442764E-11, -2.56600180000355990529E-10,
-1.61021375163803438552E-9, -4.72543064876271773512E-9, -3.00095178028681682282E-9,
7.79387474390914922337E-8, 1.06942765566401507066E-6, 1.59503164802313196374E-5,
3.49592575153777996871E-4, 1.28475387530065247392E-2, 1.03665693917934275131E0};
/*
* Evaluate 3F0(a1, a2, a3; z)
*
* The series is only asymptotic, so this requires z large enough.
*/
XSF_HOST_DEVICE inline double hyp3f0(double a1, double a2, double a3, double z) {
int n, maxiter;
double err, sum, term, m;
m = std::pow(z, -1.0 / 3);
if (m < 50) {
maxiter = m;
} else {
maxiter = 50;
}
term = 1.0;
sum = term;
for (n = 0; n < maxiter; ++n) {
term *= (a1 + n) * (a2 + n) * (a3 + n) * z / (n + 1);
sum += term;
if (std::abs(term) < 1e-13 * std::abs(sum) || term == 0) {
break;
}
}
err = std::abs(term);
if (err > 1e-13 * std::abs(sum)) {
return std::numeric_limits<double>::quiet_NaN();
}
return sum;
}
} // namespace detail
/* Sine and cosine integrals */
XSF_HOST_DEVICE inline int shichi(double x, double *si, double *ci) {
double k, z, c, s, a, b;
short sign;
if (x < 0.0) {
sign = -1;
x = -x;
} else {
sign = 0;
}
if (x == 0.0) {
*si = 0.0;
*ci = -std::numeric_limits<double>::infinity();
return (0);
}
if (x >= 8.0) {
goto chb;
}
if (x >= 88.0) {
goto asymp;
}
z = x * x;
/* Direct power series expansion */
a = 1.0;
s = 1.0;
c = 0.0;
k = 2.0;
do {
a *= z / k;
c += a / k;
k += 1.0;
a /= k;
s += a / k;
k += 1.0;
} while (std::abs(a / s) > detail::MACHEP);
s *= x;
goto done;
chb:
/* Chebyshev series expansions */
if (x < 18.0) {
a = (576.0 / x - 52.0) / 10.0;
k = std::exp(x) / x;
s = k * chbevl(a, detail::shichi_S1, 22);
c = k * chbevl(a, detail::shichi_C1, 23);
goto done;
}
if (x <= 88.0) {
a = (6336.0 / x - 212.0) / 70.0;
k = std::exp(x) / x;
s = k * chbevl(a, detail::shichi_S2, 23);
c = k * chbevl(a, detail::shichi_C2, 24);
goto done;
}
asymp:
if (x > 1000) {
*si = std::numeric_limits<double>::infinity();
*ci = std::numeric_limits<double>::infinity();
} else {
/* Asymptotic expansions
* http://functions.wolfram.com/GammaBetaErf/CoshIntegral/06/02/
* http://functions.wolfram.com/GammaBetaErf/SinhIntegral/06/02/0001/
*/
a = detail::hyp3f0(0.5, 1, 1, 4.0 / (x * x));
b = detail::hyp3f0(1, 1, 1.5, 4.0 / (x * x));
*si = std::cosh(x) / x * a + std::sinh(x) / (x * x) * b;
*ci = std::sinh(x) / x * a + std::cosh(x) / (x * x) * b;
}
if (sign) {
*si = -*si;
}
return 0;
done:
if (sign) {
s = -s;
}
*si = s;
*ci = detail::SCIPY_EULER + std::log(x) + c;
return (0);
}
} // namespace cephes
} // namespace xsf
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