1 /* $OpenBSD: k_cosl.c,v 1.1 2008/12/09 20:00:35 martynas Exp $ */ 2 /* From: @(#)k_cos.c 1.3 95/01/18 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans. 7 * 8 * Developed at SunSoft, a Sun Microsystems, Inc. business. 9 * Permission to use, copy, modify, and distribute this 10 * software is freely granted, provided that this notice 11 * is preserved. 12 * ==================================================== 13 */ 14 15 /* 16 * ld80 version of k_cos.c. See ../k_cos.c for most comments. 17 */ 18 19 #include "math_private.h" 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 static const double 45 one = 1.0; 46 47 #if defined(__amd64__) || defined(__i386__) 48 /* Long double constants are slow on these arches, and broken on i386. */ 49 static const volatile double 50 C1hi = 0.041666666666666664, /* 0x15555555555555.0p-57 */ 51 C1lo = 2.2598839032744733e-18; /* 0x14d80000000000.0p-111 */ 52 #define C1 ((long double)C1hi + C1lo) 53 #else 54 static const long double 55 C1 = 0.0416666666666666666136L; /* 0xaaaaaaaaaaaaaa9b.0p-68 */ 56 #endif 57 58 static const double 59 C2 = -0.0013888888888888874, /* -0x16c16c16c16c10.0p-62 */ 60 C3 = 0.000024801587301571716, /* 0x1a01a01a018e22.0p-68 */ 61 C4 = -0.00000027557319215507120, /* -0x127e4fb7602f22.0p-74 */ 62 C5 = 0.0000000020876754400407278, /* 0x11eed8caaeccf1.0p-81 */ 63 C6 = -1.1470297442401303e-11, /* -0x19393412bd1529.0p-89 */ 64 C7 = 4.7383039476436467e-14; /* 0x1aac9d9af5c43e.0p-97 */ 65 66 long double 67 __kernel_cosl(long double x, long double y) 68 { 69 long double hz,z,r,w; 70 71 z = x*x; 72 r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*C7)))))); 73 hz = 0.5*z; 74 w = one-hz; 75 return w + (((one-w)-hz) + (z*r-x*y)); 76 } 77