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/* Translated into C++ by SciPy developers in 2024.
* Original header with Copyright information appears below.
*/
/* ndtr.c
*
* Normal distribution function
*
*
*
* SYNOPSIS:
*
* double x, y, ndtr();
*
* y = ndtr( x );
*
*
*
* DESCRIPTION:
*
* Returns the area under the Gaussian probability density
* function, integrated from minus infinity to x:
*
* x
* -
* 1 | | 2
* ndtr(x) = --------- | exp( - t /2 ) dt
* sqrt(2pi) | |
* -
* -inf.
*
* = ( 1 + erf(z) ) / 2
* = erfc(z) / 2
*
* where z = x/sqrt(2). Computation is via the functions
* erf and erfc.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -13,0 30000 3.4e-14 6.7e-15
*
*
* ERROR MESSAGES:
*
* message condition value returned
* erfc underflow x > 37.519379347 0.0
*
*/
/* erf.c
*
* Error function
*
*
*
* SYNOPSIS:
*
* double x, y, erf();
*
* y = erf( x );
*
*
*
* DESCRIPTION:
*
* The integral is
*
* x
* -
* 2 | | 2
* erf(x) = -------- | exp( - t ) dt.
* sqrt(pi) | |
* -
* 0
*
* For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise
* erf(x) = 1 - erfc(x).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,1 30000 3.7e-16 1.0e-16
*
*/
/* erfc.c
*
* Complementary error function
*
*
*
* SYNOPSIS:
*
* double x, y, erfc();
*
* y = erfc( x );
*
*
*
* DESCRIPTION:
*
*
* 1 - erf(x) =
*
* inf.
* -
* 2 | | 2
* erfc(x) = -------- | exp( - t ) dt
* sqrt(pi) | |
* -
* x
*
*
* For small x, erfc(x) = 1 - erf(x); otherwise rational
* approximations are computed.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,26.6417 30000 5.7e-14 1.5e-14
*/
/*
* Cephes Math Library Release 2.2: June, 1992
* Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#pragma once
#include "../config.h"
#include "const.h"
#include "polevl.h"
namespace xsf {
namespace cephes {
namespace detail {
constexpr double ndtr_P[] = {2.46196981473530512524E-10, 5.64189564831068821977E-1, 7.46321056442269912687E0,
4.86371970985681366614E1, 1.96520832956077098242E2, 5.26445194995477358631E2,
9.34528527171957607540E2, 1.02755188689515710272E3, 5.57535335369399327526E2};
constexpr double ndtr_Q[] = {
/* 1.00000000000000000000E0, */
1.32281951154744992508E1, 8.67072140885989742329E1, 3.54937778887819891062E2, 9.75708501743205489753E2,
1.82390916687909736289E3, 2.24633760818710981792E3, 1.65666309194161350182E3, 5.57535340817727675546E2};
constexpr double ndtr_R[] = {5.64189583547755073984E-1, 1.27536670759978104416E0, 5.01905042251180477414E0,
6.16021097993053585195E0, 7.40974269950448939160E0, 2.97886665372100240670E0};
constexpr double ndtr_S[] = {
/* 1.00000000000000000000E0, */
2.26052863220117276590E0, 9.39603524938001434673E0, 1.20489539808096656605E1,
1.70814450747565897222E1, 9.60896809063285878198E0, 3.36907645100081516050E0};
constexpr double ndtr_T[] = {9.60497373987051638749E0, 9.00260197203842689217E1, 2.23200534594684319226E3,
7.00332514112805075473E3, 5.55923013010394962768E4};
constexpr double ndtr_U[] = {
/* 1.00000000000000000000E0, */
3.35617141647503099647E1, 5.21357949780152679795E2, 4.59432382970980127987E3, 2.26290000613890934246E4,
4.92673942608635921086E4};
constexpr double ndtri_UTHRESH = 37.519379347;
} // namespace detail
XSF_HOST_DEVICE inline double erf(double x);
XSF_HOST_DEVICE inline double erfc(double a) {
double p, q, x, y, z;
if (std::isnan(a)) {
set_error("erfc", SF_ERROR_DOMAIN, NULL);
return std::numeric_limits<double>::quiet_NaN();
}
if (a < 0.0) {
x = -a;
} else {
x = a;
}
if (x < 1.0) {
return 1.0 - erf(a);
}
z = -a * a;
if (z < -detail::MAXLOG) {
goto under;
}
z = std::exp(z);
if (x < 8.0) {
p = polevl(x, detail::ndtr_P, 8);
q = p1evl(x, detail::ndtr_Q, 8);
} else {
p = polevl(x, detail::ndtr_R, 5);
q = p1evl(x, detail::ndtr_S, 6);
}
y = (z * p) / q;
if (a < 0) {
y = 2.0 - y;
}
if (y != 0.0) {
return y;
}
under:
set_error("erfc", SF_ERROR_UNDERFLOW, NULL);
if (a < 0) {
return 2.0;
} else {
return 0.0;
}
}
XSF_HOST_DEVICE inline double erf(double x) {
double y, z;
if (std::isnan(x)) {
set_error("erf", SF_ERROR_DOMAIN, NULL);
return std::numeric_limits<double>::quiet_NaN();
}
if (x < 0.0) {
return -erf(-x);
}
if (std::abs(x) > 1.0) {
return (1.0 - erfc(x));
}
z = x * x;
y = x * polevl(z, detail::ndtr_T, 4) / p1evl(z, detail::ndtr_U, 5);
return y;
}
XSF_HOST_DEVICE inline double ndtr(double a) {
double x, y, z;
if (std::isnan(a)) {
set_error("ndtr", SF_ERROR_DOMAIN, NULL);
return std::numeric_limits<double>::quiet_NaN();
}
x = a * M_SQRT1_2;
z = std::abs(x);
if (z < 1.0) {
y = 0.5 + 0.5 * erf(x);
} else {
y = 0.5 * erfc(z);
if (x > 0) {
y = 1.0 - y;
}
}
return y;
}
} // namespace cephes
} // namespace xsf
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