1 /* From: @(#)k_cos.c 1.3 95/01/18 */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
6 *
7 * Developed at SunSoft, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 */
13
14
15 /*
16 * ld80 version of k_cos.c. See ../src/k_cos.c for most comments.
17 */
18
19
20 /*
21 * Domain [-0.7854, 0.7854], range ~[-2.43e-23, 2.425e-23]:
22 * |cos(x) - c(x)| < 2**-75.1
23 *
24 * The coefficients of c(x) were generated by a pari-gp script using
25 * a Remez algorithm that searches for the best higher coefficients
26 * after rounding leading coefficients to a specified precision.
27 *
28 * Simpler methods like Chebyshev or basic Remez barely suffice for
29 * cos() in 64-bit precision, because we want the coefficient of x^2
30 * to be precisely -0.5 so that multiplying by it is exact, and plain
31 * rounding of the coefficients of a good polynomial approximation only
32 * gives this up to about 64-bit precision. Plain rounding also gives
33 * a mediocre approximation for the coefficient of x^4, but a rounding
34 * error of 0.5 ulps for this coefficient would only contribute ~0.01
35 * ulps to the final error, so this is unimportant. Rounding errors in
36 * higher coefficients are even less important.
37 *
38 * In fact, coefficients above the x^4 one only need to have 53-bit
39 * precision, and this is more efficient. We get this optimization
40 * almost for free from the complications needed to search for the best
41 * higher coefficients.
42 */
43
44 #if defined(__amd64__) || defined(__i386__)
45 /* Long double constants are slow on these arches, and broken on i386. */
46 static const volatile double
47 C1hi = 0.041666666666666664, /* 0x15555555555555.0p-57 */
48 C1lo = 2.2598839032744733e-18; /* 0x14d80000000000.0p-111 */
49 #define C1 ((long double)C1hi + (long double) C1lo)
50 #else
51 static const long double
52 C1 = 0.0416666666666666666136L; /* 0xaaaaaaaaaaaaaa9b.0p-68 */
53 #endif
54
55 static const double
56 C2 = -0.0013888888888888874, /* -0x16c16c16c16c10.0p-62 */
57 C3 = 0.000024801587301571716, /* 0x1a01a01a018e22.0p-68 */
58 C4 = -0.00000027557319215507120, /* -0x127e4fb7602f22.0p-74 */
59 C5 = 0.0000000020876754400407278, /* 0x11eed8caaeccf1.0p-81 */
60 C6 = -1.1470297442401303e-11, /* -0x19393412bd1529.0p-89 */
61 C7 = 4.7383039476436467e-14; /* 0x1aac9d9af5c43e.0p-97 */
62
63 long double
__kernel_cosl(long double x,long double y)64 __kernel_cosl(long double x, long double y)
65 {
66 long double hz,z,r,w;
67
68 z = x*x;
69 r = z*(C1+z*((long double) C2+z*((long double) C3+z*((long double) C4+z*((long double) C5+z*((long double) C6+z*(long double) C7))))));
70 hz = 0.5l*z;
71 w = 1.0l-hz;
72 return w + (((1.0l-w)-hz) + (z*r-x*y));
73 }
74
