/src/moddable/xs/tools/fdlibm/k_cos.c
Line | Count | Source |
1 | | |
2 | | /* |
3 | | * ==================================================== |
4 | | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
5 | | * |
6 | | * Developed at SunSoft, a Sun Microsystems, Inc. business. |
7 | | * Permission to use, copy, modify, and distribute this |
8 | | * software is freely granted, provided that this notice |
9 | | * is preserved. |
10 | | * ==================================================== |
11 | | */ |
12 | | |
13 | | /* |
14 | | * __kernel_cos( x, y ) |
15 | | * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 |
16 | | * Input x is assumed to be bounded by ~pi/4 in magnitude. |
17 | | * Input y is the tail of x. |
18 | | * |
19 | | * Algorithm |
20 | | * 1. Since cos(-x) = cos(x), we need only to consider positive x. |
21 | | * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. |
22 | | * 3. cos(x) is approximated by a polynomial of degree 14 on |
23 | | * [0,pi/4] |
24 | | * 4 14 |
25 | | * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x |
26 | | * where the remez error is |
27 | | * |
28 | | * | 2 4 6 8 10 12 14 | -58 |
29 | | * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 |
30 | | * | | |
31 | | * |
32 | | * 4 6 8 10 12 14 |
33 | | * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then |
34 | | * cos(x) ~ 1 - x*x/2 + r |
35 | | * since cos(x+y) ~ cos(x) - sin(x)*y |
36 | | * ~ cos(x) - x*y, |
37 | | * a correction term is necessary in cos(x) and hence |
38 | | * cos(x+y) = 1 - (x*x/2 - (r - x*y)) |
39 | | * For better accuracy, rearrange to |
40 | | * cos(x+y) ~ w + (tmp + (r-x*y)) |
41 | | * where w = 1 - x*x/2 and tmp is a tiny correction term |
42 | | * (1 - x*x/2 == w + tmp exactly in infinite precision). |
43 | | * The exactness of w + tmp in infinite precision depends on w |
44 | | * and tmp having the same precision as x. If they have extra |
45 | | * precision due to compiler bugs, then the extra precision is |
46 | | * only good provided it is retained in all terms of the final |
47 | | * expression for cos(). Retention happens in all cases tested |
48 | | * under FreeBSD, so don't pessimize things by forcibly clipping |
49 | | * any extra precision in w. |
50 | | */ |
51 | | |
52 | | #include "math_private.h" |
53 | | |
54 | | static const double |
55 | | one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ |
56 | | C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ |
57 | | C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ |
58 | | C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ |
59 | | C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ |
60 | | C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ |
61 | | C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ |
62 | | |
63 | | double |
64 | | __kernel_cos(double x, double y) |
65 | 9.25k | { |
66 | 9.25k | double hz,z,r,w; |
67 | | |
68 | 9.25k | z = x*x; |
69 | 9.25k | w = z*z; |
70 | 9.25k | r = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6)); |
71 | 9.25k | hz = 0.5*z; |
72 | 9.25k | w = one-hz; |
73 | 9.25k | return w + (((one-w)-hz) + (z*r-x*y)); |
74 | 9.25k | } |