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// origin: FreeBSD /usr/src/lib/msun/src/k_cos.c
//
// ====================================================
// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunSoft, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
const C1: f64 = 4.16666666666666019037e-02; /* 0x3FA55555, 0x5555554C */
const C2: f64 = -1.38888888888741095749e-03; /* 0xBF56C16C, 0x16C15177 */
const C3: f64 = 2.48015872894767294178e-05; /* 0x3EFA01A0, 0x19CB1590 */
const C4: f64 = -2.75573143513906633035e-07; /* 0xBE927E4F, 0x809C52AD */
const C5: f64 = 2.08757232129817482790e-09; /* 0x3E21EE9E, 0xBDB4B1C4 */
const C6: f64 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
// kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
// Input x is assumed to be bounded by ~pi/4 in magnitude.
// Input y is the tail of x.
//
// Algorithm
// 1. Since cos(-x) = cos(x), we need only to consider positive x.
// 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
// 3. cos(x) is approximated by a polynomial of degree 14 on
// [0,pi/4]
// 4 14
// cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
// where the remez error is
//
// | 2 4 6 8 10 12 14 | -58
// |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
// | |
//
// 4 6 8 10 12 14
// 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
// cos(x) ~ 1 - x*x/2 + r
// since cos(x+y) ~ cos(x) - sin(x)*y
// ~ cos(x) - x*y,
// a correction term is necessary in cos(x) and hence
// cos(x+y) = 1 - (x*x/2 - (r - x*y))
// For better accuracy, rearrange to
// cos(x+y) ~ w + (tmp + (r-x*y))
// where w = 1 - x*x/2 and tmp is a tiny correction term
// (1 - x*x/2 == w + tmp exactly in infinite precision).
// The exactness of w + tmp in infinite precision depends on w
// and tmp having the same precision as x. If they have extra
// precision due to compiler bugs, then the extra precision is
// only good provided it is retained in all terms of the final
// expression for cos(). Retention happens in all cases tested
// under FreeBSD, so don't pessimize things by forcibly clipping
// any extra precision in w.
#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
pub(crate) fn k_cos(x: f64, y: f64) -> f64 {
let z = x * x;
let w = z * z;
let r = z * (C1 + z * (C2 + z * C3)) + w * w * (C4 + z * (C5 + z * C6));
let hz = 0.5 * z;
let w = 1.0 - hz;
w + (((1.0 - w) - hz) + (z * r - x * y))
}
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