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 #include <sys/cdefs.h> 15 __FBSDID("$FreeBSD$"); 16 17 /* 18 * ld80 version of k_cos.c. See ../src/k_cos.c for most comments. 19 */ 20 21 #include "math_private.h" 22 23 /* 24 * Domain [-0.7854, 0.7854], range ~[-2.43e-23, 2.425e-23]: 25 * |cos(x) - c(x)| < 2**-75.1 26 * 27 * The coefficients of c(x) were generated by a pari-gp script using 28 * a Remez algorithm that searches for the best higher coefficients 29 * after rounding leading coefficients to a specified precision. 30 * 31 * Simpler methods like Chebyshev or basic Remez barely suffice for 32 * cos() in 64-bit precision, because we want the coefficient of x^2 33 * to be precisely -0.5 so that multiplying by it is exact, and plain 34 * rounding of the coefficients of a good polynomial approximation only 35 * gives this up to about 64-bit precision. Plain rounding also gives 36 * a mediocre approximation for the coefficient of x^4, but a rounding 37 * error of 0.5 ulps for this coefficient would only contribute ~0.01 38 * ulps to the final error, so this is unimportant. Rounding errors in 39 * higher coefficients are even less important. 40 * 41 * In fact, coefficients above the x^4 one only need to have 53-bit 42 * precision, and this is more efficient. We get this optimization 43 * almost for free from the complications needed to search for the best 44 * higher coefficients. 45 */ 46 static const double 47 one = 1.0; 48 49 #if defined(__amd64__) || defined(__i386__) 50 /* Long double constants are slow on these arches, and broken on i386. */ 51 static const volatile double 52 C1hi = 0.041666666666666664, /* 0x15555555555555.0p-57 */ 53 C1lo = 2.2598839032744733e-18; /* 0x14d80000000000.0p-111 */ 54 #define C1 ((long double)C1hi + C1lo) 55 #else 56 static const long double 57 C1 = 0.0416666666666666666136L; /* 0xaaaaaaaaaaaaaa9b.0p-68 */ 58 #endif 59 60 static const double 61 C2 = -0.0013888888888888874, /* -0x16c16c16c16c10.0p-62 */ 62 C3 = 0.000024801587301571716, /* 0x1a01a01a018e22.0p-68 */ 63 C4 = -0.00000027557319215507120, /* -0x127e4fb7602f22.0p-74 */ 64 C5 = 0.0000000020876754400407278, /* 0x11eed8caaeccf1.0p-81 */ 65 C6 = -1.1470297442401303e-11, /* -0x19393412bd1529.0p-89 */ 66 C7 = 4.7383039476436467e-14; /* 0x1aac9d9af5c43e.0p-97 */ 67 68 long double 69 __kernel_cosl(long double x, long double y) 70 { 71 long double hz,z,r,w; 72 73 z = x*x; 74 r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*C7)))))); 75 hz = 0.5*z; 76 w = one-hz; 77 return w + (((one-w)-hz) + (z*r-x*y)); 78 } 79