| // 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. |
| #[inline] |
| #[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)) |
| } |
