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 /* @(#)k_cos.c 5.1 93/09/24 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #if defined(LIBM_SCCS) && !defined(lint) static const char rcsid[] = "\$NetBSD: k_cos.c,v 1.8 1995/05/10 20:46:22 jtc Exp \$"; #endif /* * __kernel_cos( x, y ) * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 * Input x is assumed to be bounded by ~pi/4 in magnitude. * Input y is the tail of x. * * Algorithm * 1. Since cos(-x) = cos(x), we need only to consider positive x. * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. * 3. cos(x) is approximated by a polynomial of degree 14 on * [0,pi/4] * 4 14 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x * where the remez error is * * | 2 4 6 8 10 12 14 | -58 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 * | | * * 4 6 8 10 12 14 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then * cos(x) = 1 - x*x/2 + r * since cos(x+y) ~ cos(x) - sin(x)*y * ~ cos(x) - x*y, * a correction term is necessary in cos(x) and hence * cos(x+y) = 1 - (x*x/2 - (r - x*y)) * For better accuracy when x > 0.3, let qx = |x|/4 with * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. * Then * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). * Note that 1-qx and (x*x/2-qx) is EXACT here, and the * magnitude of the latter is at least a quarter of x*x/2, * thus, reducing the rounding error in the subtraction. */ #include "math_libm.h" #include "math_private.h" #ifdef __STDC__ static const double #else static double #endif one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ #ifdef __STDC__ double attribute_hidden __kernel_cos(double x, double y) #else double attribute_hidden __kernel_cos(x, y) double x, y; #endif { double a, hz, z, r, qx; int32_t ix; GET_HIGH_WORD(ix, x); ix &= 0x7fffffff; /* ix = |x|'s high word */ if (ix < 0x3e400000) { /* if x < 2**27 */ if (((int) x) == 0) return one; /* generate inexact */ } z = x * x; r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6))))); if (ix < 0x3FD33333) /* if |x| < 0.3 */ return one - (0.5 * z - (z * r - x * y)); else { if (ix > 0x3fe90000) { /* x > 0.78125 */ qx = 0.28125; } else { INSERT_WORDS(qx, ix - 0x00200000, 0); /* x/4 */ } hz = 0.5 * z - qx; a = one - qx; return a - (hz - (z * r - x * y)); } }