1 /* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans. 5 * 6 * Developed at SunSoft, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 */ 12 13 /* 14 * ld80 version of k_cos.c. See ../src/k_cos.c for most comments. 15 */ 16 17 #include "math_private.h" 18 19 /* 20 * Domain [-0.7854, 0.7854], range ~[-2.43e-23, 2.425e-23]: 21 * |cos(x) - c(x)| < 2**-75.1 22 * 23 * The coefficients of c(x) were generated by a pari-gp script using 24 * a Remez algorithm that searches for the best higher coefficients 25 * after rounding leading coefficients to a specified precision. 26 * 27 * Simpler methods like Chebyshev or basic Remez barely suffice for 28 * cos() in 64-bit precision, because we want the coefficient of x^2 29 * to be precisely -0.5 so that multiplying by it is exact, and plain 30 * rounding of the coefficients of a good polynomial approximation only 31 * gives this up to about 64-bit precision. Plain rounding also gives 32 * a mediocre approximation for the coefficient of x^4, but a rounding 33 * error of 0.5 ulps for this coefficient would only contribute ~0.01 34 * ulps to the final error, so this is unimportant. Rounding errors in 35 * higher coefficients are even less important. 36 * 37 * In fact, coefficients above the x^4 one only need to have 53-bit 38 * precision, and this is more efficient. We get this optimization 39 * almost for free from the complications needed to search for the best 40 * higher coefficients. 41 */ 42 static const double 43 one = 1.0; 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 + 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 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*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*C7)))))); 71 hz = 0.5*z; 72 w = one-hz; 73 return w + (((one-w)-hz) + (z*r-x*y)); 74 } 75