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 //__FBSDID("$FreeBSD: src/lib/msun/ld80/k_cosl.c,v 1.1 2008/02/17 07:32:14 das Exp $");
15
16 /*
17 * ld80 version of k_cos.c. See ../src/k_cos.c for most comments.
18 */
19
20
21 /*
22 * Domain [-0.7854, 0.7854], range ~[-2.43e-23, 2.425e-23]:
23 * |cos(x) - c(x)| < 2**-75.1
24 *
25 * The coefficients of c(x) were generated by a pari-gp script using
26 * a Remez algorithm that searches for the best higher coefficients
27 * after rounding leading coefficients to a specified precision.
28 *
29 * Simpler methods like Chebyshev or basic Remez barely suffice for
30 * cos() in 64-bit precision, because we want the coefficient of x^2
31 * to be precisely -0.5 so that multiplying by it is exact, and plain
32 * rounding of the coefficients of a good polynomial approximation only
33 * gives this up to about 64-bit precision. Plain rounding also gives
34 * a mediocre approximation for the coefficient of x^4, but a rounding
35 * error of 0.5 ulps for this coefficient would only contribute ~0.01
36 * ulps to the final error, so this is unimportant. Rounding errors in
37 * higher coefficients are even less important.
38 *
39 * In fact, coefficients above the x^4 one only need to have 53-bit
40 * precision, and this is more efficient. We get this optimization
41 * almost for free from the complications needed to search for the best
42 * higher coefficients.
43 */
44
45 #if defined(__amd64__) || defined(__i386__)
46 /* Long double constants are slow on these arches, and broken on i386. */
47 static const volatile double
48 C1hi = 0.041666666666666664, /* 0x15555555555555.0p-57 */
49 C1lo = 2.2598839032744733e-18; /* 0x14d80000000000.0p-111 */
50 #define C1 ((long double)C1hi + (long double) C1lo)
51 #else
52 static const long double
53 C1 = 0.0416666666666666666136L; /* 0xaaaaaaaaaaaaaa9b.0p-68 */
54 #endif
55
56 static const double
57 C2 = -0.0013888888888888874, /* -0x16c16c16c16c10.0p-62 */
58 C3 = 0.000024801587301571716, /* 0x1a01a01a018e22.0p-68 */
59 C4 = -0.00000027557319215507120, /* -0x127e4fb7602f22.0p-74 */
60 C5 = 0.0000000020876754400407278, /* 0x11eed8caaeccf1.0p-81 */
61 C6 = -1.1470297442401303e-11, /* -0x19393412bd1529.0p-89 */
62 C7 = 4.7383039476436467e-14; /* 0x1aac9d9af5c43e.0p-97 */
63
64 long double
__kernel_cosl(long double x,long double y)65 __kernel_cosl(long double x, long double y)
66 {
67 long double hz,z,r,w;
68
69 z = x*x;
70 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))))));
71 hz = 0.5l*z;
72 w = 1.0l-hz;
73 return w + (((1.0l-w)-hz) + (z*r-x*y));
74 }
75
