/* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* * ld80 version of k_cos.c. See ../src/k_cos.c for most comments. */ #include "math_private.h" /* * Domain [-0.7854, 0.7854], range ~[-2.43e-23, 2.425e-23]: * |cos(x) - c(x)| < 2**-75.1 * * The coefficients of c(x) were generated by a pari-gp script using * a Remez algorithm that searches for the best higher coefficients * after rounding leading coefficients to a specified precision. * * Simpler methods like Chebyshev or basic Remez barely suffice for * cos() in 64-bit precision, because we want the coefficient of x^2 * to be precisely -0.5 so that multiplying by it is exact, and plain * rounding of the coefficients of a good polynomial approximation only * gives this up to about 64-bit precision. Plain rounding also gives * a mediocre approximation for the coefficient of x^4, but a rounding * error of 0.5 ulps for this coefficient would only contribute ~0.01 * ulps to the final error, so this is unimportant. Rounding errors in * higher coefficients are even less important. * * In fact, coefficients above the x^4 one only need to have 53-bit * precision, and this is more efficient. We get this optimization * almost for free from the complications needed to search for the best * higher coefficients. */ static const double one = 1.0; #if defined(__amd64__) || defined(__i386__) /* Long double constants are slow on these arches, and broken on i386. */ static const volatile double C1hi = 0.041666666666666664, /* 0x15555555555555.0p-57 */ C1lo = 2.2598839032744733e-18; /* 0x14d80000000000.0p-111 */ #define C1 ((long double)C1hi + C1lo) #else static const long double C1 = 0.0416666666666666666136L; /* 0xaaaaaaaaaaaaaa9b.0p-68 */ #endif static const double C2 = -0.0013888888888888874, /* -0x16c16c16c16c10.0p-62 */ C3 = 0.000024801587301571716, /* 0x1a01a01a018e22.0p-68 */ C4 = -0.00000027557319215507120, /* -0x127e4fb7602f22.0p-74 */ C5 = 0.0000000020876754400407278, /* 0x11eed8caaeccf1.0p-81 */ C6 = -1.1470297442401303e-11, /* -0x19393412bd1529.0p-89 */ C7 = 4.7383039476436467e-14; /* 0x1aac9d9af5c43e.0p-97 */ long double __kernel_cosl(long double x, long double y) { long double hz,z,r,w; z = x*x; r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*C7)))))); hz = 0.5*z; w = one-hz; return w + (((one-w)-hz) + (z*r-x*y)); }