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