1*de336b0cSDavid Schultz /*
2*de336b0cSDavid Schultz * ====================================================
3*de336b0cSDavid Schultz * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4*de336b0cSDavid Schultz * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
5*de336b0cSDavid Schultz *
6*de336b0cSDavid Schultz * Developed at SunSoft, a Sun Microsystems, Inc. business.
7*de336b0cSDavid Schultz * Permission to use, copy, modify, and distribute this
8*de336b0cSDavid Schultz * software is freely granted, provided that this notice
9*de336b0cSDavid Schultz * is preserved.
10*de336b0cSDavid Schultz * ====================================================
11*de336b0cSDavid Schultz */
12*de336b0cSDavid Schultz
13*de336b0cSDavid Schultz /*
14*de336b0cSDavid Schultz * ld80 version of k_cos.c. See ../src/k_cos.c for most comments.
15*de336b0cSDavid Schultz */
16*de336b0cSDavid Schultz
17*de336b0cSDavid Schultz #include "math_private.h"
18*de336b0cSDavid Schultz
19*de336b0cSDavid Schultz /*
20*de336b0cSDavid Schultz * Domain [-0.7854, 0.7854], range ~[-2.43e-23, 2.425e-23]:
21*de336b0cSDavid Schultz * |cos(x) - c(x)| < 2**-75.1
22*de336b0cSDavid Schultz *
23*de336b0cSDavid Schultz * The coefficients of c(x) were generated by a pari-gp script using
24*de336b0cSDavid Schultz * a Remez algorithm that searches for the best higher coefficients
25*de336b0cSDavid Schultz * after rounding leading coefficients to a specified precision.
26*de336b0cSDavid Schultz *
27*de336b0cSDavid Schultz * Simpler methods like Chebyshev or basic Remez barely suffice for
28*de336b0cSDavid Schultz * cos() in 64-bit precision, because we want the coefficient of x^2
29*de336b0cSDavid Schultz * to be precisely -0.5 so that multiplying by it is exact, and plain
30*de336b0cSDavid Schultz * rounding of the coefficients of a good polynomial approximation only
31*de336b0cSDavid Schultz * gives this up to about 64-bit precision. Plain rounding also gives
32*de336b0cSDavid Schultz * a mediocre approximation for the coefficient of x^4, but a rounding
33*de336b0cSDavid Schultz * error of 0.5 ulps for this coefficient would only contribute ~0.01
34*de336b0cSDavid Schultz * ulps to the final error, so this is unimportant. Rounding errors in
35*de336b0cSDavid Schultz * higher coefficients are even less important.
36*de336b0cSDavid Schultz *
37*de336b0cSDavid Schultz * In fact, coefficients above the x^4 one only need to have 53-bit
38*de336b0cSDavid Schultz * precision, and this is more efficient. We get this optimization
39*de336b0cSDavid Schultz * almost for free from the complications needed to search for the best
40*de336b0cSDavid Schultz * higher coefficients.
41*de336b0cSDavid Schultz */
42*de336b0cSDavid Schultz static const double
43*de336b0cSDavid Schultz one = 1.0;
44*de336b0cSDavid Schultz
45*de336b0cSDavid Schultz #if defined(__amd64__) || defined(__i386__)
46*de336b0cSDavid Schultz /* Long double constants are slow on these arches, and broken on i386. */
47*de336b0cSDavid Schultz static const volatile double
48*de336b0cSDavid Schultz C1hi = 0.041666666666666664, /* 0x15555555555555.0p-57 */
49*de336b0cSDavid Schultz C1lo = 2.2598839032744733e-18; /* 0x14d80000000000.0p-111 */
50*de336b0cSDavid Schultz #define C1 ((long double)C1hi + C1lo)
51*de336b0cSDavid Schultz #else
52*de336b0cSDavid Schultz static const long double
53*de336b0cSDavid Schultz C1 = 0.0416666666666666666136L; /* 0xaaaaaaaaaaaaaa9b.0p-68 */
54*de336b0cSDavid Schultz #endif
55*de336b0cSDavid Schultz
56*de336b0cSDavid Schultz static const double
57*de336b0cSDavid Schultz C2 = -0.0013888888888888874, /* -0x16c16c16c16c10.0p-62 */
58*de336b0cSDavid Schultz C3 = 0.000024801587301571716, /* 0x1a01a01a018e22.0p-68 */
59*de336b0cSDavid Schultz C4 = -0.00000027557319215507120, /* -0x127e4fb7602f22.0p-74 */
60*de336b0cSDavid Schultz C5 = 0.0000000020876754400407278, /* 0x11eed8caaeccf1.0p-81 */
61*de336b0cSDavid Schultz C6 = -1.1470297442401303e-11, /* -0x19393412bd1529.0p-89 */
62*de336b0cSDavid Schultz C7 = 4.7383039476436467e-14; /* 0x1aac9d9af5c43e.0p-97 */
63*de336b0cSDavid Schultz
64*de336b0cSDavid Schultz long double
__kernel_cosl(long double x,long double y)65*de336b0cSDavid Schultz __kernel_cosl(long double x, long double y)
66*de336b0cSDavid Schultz {
67*de336b0cSDavid Schultz long double hz,z,r,w;
68*de336b0cSDavid Schultz
69*de336b0cSDavid Schultz z = x*x;
70*de336b0cSDavid Schultz r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*C7))))));
71*de336b0cSDavid Schultz hz = 0.5*z;
72*de336b0cSDavid Schultz w = one-hz;
73*de336b0cSDavid Schultz return w + (((one-w)-hz) + (z*r-x*y));
74*de336b0cSDavid Schultz }
75