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/* ellpk.c
*
* Complete elliptic integral of the first kind
*
*
*
* SYNOPSIS:
*
* double m1, y, ellpk();
*
* y = ellpk( m1 );
*
*
*
* DESCRIPTION:
*
* Approximates the integral
*
*
*
* pi/2
* -
* | |
* | dt
* K(m) = | ------------------
* | 2
* | | sqrt( 1 - m sin t )
* -
* 0
*
* where m = 1 - m1, using the approximation
*
* P(x) - log x Q(x).
*
* The argument m1 is used internally rather than m so that the logarithmic
* singularity at m = 1 will be shifted to the origin; this
* preserves maximum accuracy.
*
* K(0) = pi/2.
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,1 30000 2.5e-16 6.8e-17
*
* ERROR MESSAGES:
*
* message condition value returned
* ellpk domain x<0, x>1 0.0
*
*/
/* ellpk.c */
/*
* Cephes Math Library, Release 2.0: April, 1987
* Copyright 1984, 1987 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#![allow(clippy::excessive_precision)]
use super::consts::MACHEP;
use super::polevl::polevl;
static P: [f64; 11] = [
1.37982864606273237150E-4,
2.28025724005875567385E-3,
7.97404013220415179367E-3,
9.85821379021226008714E-3,
6.87489687449949877925E-3,
6.18901033637687613229E-3,
8.79078273952743772254E-3,
1.49380448916805252718E-2,
3.08851465246711995998E-2,
9.65735902811690126535E-2,
1.38629436111989062502E0
];
static Q: [f64; 11] = [
2.94078955048598507511E-5,
9.14184723865917226571E-4,
5.94058303753167793257E-3,
1.54850516649762399335E-2,
2.39089602715924892727E-2,
3.01204715227604046988E-2,
3.73774314173823228969E-2,
4.88280347570998239232E-2,
7.03124996963957469739E-2,
1.24999999999870820058E-1,
4.99999999999999999821E-1
];
/// Natural log of 4.0
static C1: f64 = 1.3862943611198906188E0;
//$$K(m) = \int_{0}^{\pi / 2} \frac{d\theta}{\sqrt{1 - m\,\sin^2(\theta)}}$$
pub fn ellpk(x: f64) -> f64 {
//! Complete elliptic integral of the first kind evaluated at `1.0 - x`
//!
//! ## Description:
//!
//! Approximates the integral
//!
#![doc=include_str!("ellpk.svg")]
//!
//! where `m = 1 - x`, using the approximation `P(x) - ln(x) * Q(x)`
//!
//! The argument `x` is used internally rather than `m` so that the logarithmic singularity at `m = 1` will be shifted to the origin; this preserves maximum accuracy.
//!
//! ## Accuracy:
//!
//! Relative Error:
//!<table>
//! <tr>
//! <th>Arithmetic</th>
//! <th>Domain</th>
//! <th># Trials</th>
//! <th>Peak</th>
//! <th>RMS</th>
//! </tr>
//! <tr>
//! <td>IEEE</td>
//! <td>0, 1</td>
//! <td>30000</td>
//! <td>2.5e-16</td>
//! <td>6.8e-17</td>
//! </tr>
//!</table>
//!
//! ## Examples
//!
//! ```rust
//! use spec_math::cephes64::ellpk;
//!
//! let x = 0.0;
//!
//! assert_eq!(ellpk(1.0 - x), std::f64::consts::PI * 0.5);
//! ```
if x < 0.0 {
//sf_error("ellpk", SF_ERROR_DOMAIN, NULL);
f64::NAN
} else if x > 1.0 {
if x.is_infinite() {
0.0
} else {
ellpk(1.0 / x) / x.sqrt()
}
} else if x > MACHEP {
polevl(x, &P, 10) - x.ln() * polevl(x, &Q, 10)
} else if x == 0.0 {
//sf_error("ellpk", SF_ERROR_SINGULAR, NULL);
f64::INFINITY
} else {
C1 - 0.5 * x.ln()
}
}
#[cfg(test)]
mod ellpk_tests {
use super::*;
#[test]
fn ellpk_trivials() {
assert_eq!(ellpk(-f64::INFINITY).is_nan(), true);
assert_eq!(ellpk(-10.0).is_nan(), true);
assert_eq!(ellpk(-1.0).is_nan(), true);
assert_eq!(ellpk(-1e-10).is_nan(), true);
assert_eq!(ellpk(0.0), f64::INFINITY);
assert_eq!(ellpk(f64::INFINITY), 0.0);
}
#[test]
fn ellpk_large() {
assert_eq!(ellpk(1.1), 1.5335928197134567);
assert_eq!(ellpk(1.5), 1.415737208425956);
assert_eq!(ellpk(2.0), 1.3110287771460598);
assert_eq!(ellpk(10.0), 0.8152643095897213);
assert_eq!(ellpk(100.0), 0.3695637362989875);
assert_eq!(ellpk(1e10), 0.000128992198263876);
}
#[test]
fn ellpk_medium() {
assert_eq!(ellpk(1.15e-16), 19.73709413388468);
assert_eq!(ellpk(1e-10), 12.8992198263876);
assert_eq!(ellpk(0.5), 1.8540746773013719);
assert_eq!(ellpk(0.9), 1.6124413487202192);
assert_eq!(ellpk(1.0), 1.5707963267948966);
}
#[test]
fn ellpk_small() {
assert_eq!(ellpk(1.1e-16), 19.759320015170093); // Should actually be 19.759320015170094
assert_eq!(ellpk(1.0e-20), 24.412145291060348); // Should actually round to 24.412145291060347
}
}