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/* Translated into C++ by SciPy developers in 2024.
* Original header with Copyright information appears below.
*/
/* ellpe.c
*
* Complete elliptic integral of the second kind
*
*
*
* SYNOPSIS:
*
* double m, y, ellpe();
*
* y = ellpe( m );
*
*
*
* DESCRIPTION:
*
* Approximates the integral
*
*
* pi/2
* -
* | | 2
* E(m) = | sqrt( 1 - m sin t ) dt
* | |
* -
* 0
*
* Where m = 1 - m1, using the approximation
*
* P(x) - x log x Q(x).
*
* Though there are no singularities, the argument m1 is used
* internally rather than m for compatibility with ellpk().
*
* E(1) = 1; E(0) = pi/2.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0, 1 10000 2.1e-16 7.3e-17
*
*
* ERROR MESSAGES:
*
* message condition value returned
* ellpe domain x<0, x>1 0.0
*
*/
/* ellpe.c */
/* Elliptic integral of second kind */
/*
* Cephes Math Library, Release 2.1: February, 1989
* Copyright 1984, 1987, 1989 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*
* Feb, 2002: altered by Travis Oliphant
* so that it is called with argument m
* (which gets immediately converted to m1 = 1-m)
*/
#pragma once
#include "../config.h"
#include "../error.h"
#include "polevl.h"
namespace xsf {
namespace cephes {
namespace detail {
constexpr double ellpe_P[] = {1.53552577301013293365E-4, 2.50888492163602060990E-3, 8.68786816565889628429E-3,
1.07350949056076193403E-2, 7.77395492516787092951E-3, 7.58395289413514708519E-3,
1.15688436810574127319E-2, 2.18317996015557253103E-2, 5.68051945617860553470E-2,
4.43147180560990850618E-1, 1.00000000000000000299E0};
constexpr double ellpe_Q[] = {3.27954898576485872656E-5, 1.00962792679356715133E-3, 6.50609489976927491433E-3,
1.68862163993311317300E-2, 2.61769742454493659583E-2, 3.34833904888224918614E-2,
4.27180926518931511717E-2, 5.85936634471101055642E-2, 9.37499997197644278445E-2,
2.49999999999888314361E-1};
} // namespace detail
XSF_HOST_DEVICE inline double ellpe(double x) {
x = 1.0 - x;
if (x <= 0.0) {
if (x == 0.0)
return (1.0);
set_error("ellpe", SF_ERROR_DOMAIN, NULL);
return (std::numeric_limits<double>::quiet_NaN());
}
if (x > 1.0) {
return ellpe(1.0 - 1 / x) * std::sqrt(x);
}
return (polevl(x, detail::ellpe_P, 10) - std::log(x) * (x * polevl(x, detail::ellpe_Q, 9)));
}
} // namespace cephes
} // namespace xsf
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