1 /*- 2 * Copyright (c) 2017 Steven G. Kargl 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 1. Redistributions of source code must retain the above copyright 9 * notice unmodified, this list of conditions, and the following 10 * disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 16 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 17 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 18 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 19 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 20 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 21 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 22 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 23 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 24 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 25 */ 26 27 /** 28 * cospi(x) computes cos(pi*x) without multiplication by pi (almost). First, 29 * note that cospi(-x) = cospi(x), so the algorithm considers only |x|. The 30 * method used depends on the magnitude of x. 31 * 32 * 1. For small |x|, cospi(x) = 1 with FE_INEXACT raised where a sloppy 33 * threshold is used. The threshold is |x| < 0x1pN with N = -(P/2+M). 34 * P is the precision of the floating-point type and M = 2 to 4. 35 * 36 * 2. For |x| < 1, argument reduction is not required and sinpi(x) is 37 * computed by calling a kernel that leverages the kernels for sin(x) 38 * ans cos(x). See k_sinpi.c and k_cospi.c for details. 39 * 40 * 3. For 1 <= |x| < 0x1p(P-1), argument reduction is required where 41 * |x| = j0 + r with j0 an integer and the remainder r satisfies 42 * 0 <= r < 1. With the given domain, a simplified inline floor(x) 43 * is used. Also, note the following identity 44 * 45 * cospi(x) = cos(pi*(j0+r)) 46 * = cos(pi*j0) * cos(pi*r) - sin(pi*j0) * sin(pi*r) 47 * = cos(pi*j0) * cos(pi*r) 48 * = +-cospi(r) 49 * 50 * If j0 is even, then cos(pi*j0) = 1. If j0 is odd, then cos(pi*j0) = -1. 51 * cospi(r) is then computed via an appropriate kernel. 52 * 53 * 4. For |x| >= 0x1p(P-1), |x| is integral and cospi(x) = 1. 54 * 55 * 5. Special cases: 56 * 57 * cospi(+-0) = 1. 58 * cospi(n.5) = 0 for n an integer. 59 * cospi(+-inf) = nan. Raises the "invalid" floating-point exception. 60 * cospi(nan) = nan. Raises the "invalid" floating-point exception. 61 */ 62 63 #include <float.h> 64 #include "math.h" 65 #include "math_private.h" 66 67 static const double 68 pi_hi = 3.1415926814079285e+00, /* 0x400921fb 0x58000000 */ 69 pi_lo =-2.7818135228334233e-08; /* 0xbe5dde97 0x3dcb3b3a */ 70 71 #include "k_cospi.h" 72 #include "k_sinpi.h" 73 74 volatile static const double vzero = 0; 75 76 double 77 cospi(double x) 78 { 79 double ax, c; 80 uint32_t hx, ix, j0, lx; 81 82 EXTRACT_WORDS(hx, lx, x); 83 ix = hx & 0x7fffffff; 84 INSERT_WORDS(ax, ix, lx); 85 86 if (ix < 0x3ff00000) { /* |x| < 1 */ 87 if (ix < 0x3fd00000) { /* |x| < 0.25 */ 88 if (ix < 0x3e200000) { /* |x| < 0x1p-29 */ 89 if ((int)ax == 0) 90 return (1); 91 } 92 return (__kernel_cospi(ax)); 93 } 94 95 if (ix < 0x3fe00000) /* |x| < 0.5 */ 96 c = __kernel_sinpi(0.5 - ax); 97 else if (ix < 0x3fe80000){ /* |x| < 0.75 */ 98 if (ax == 0.5) 99 return (0); 100 c = -__kernel_sinpi(ax - 0.5); 101 } else 102 c = -__kernel_cospi(1 - ax); 103 return (c); 104 } 105 106 if (ix < 0x43300000) { /* 1 <= |x| < 0x1p52 */ 107 /* Determine integer part of ax. */ 108 j0 = ((ix >> 20) & 0x7ff) - 0x3ff; 109 if (j0 < 20) { 110 ix &= ~(0x000fffff >> j0); 111 lx = 0; 112 } else { 113 lx &= ~((uint32_t)0xffffffff >> (j0 - 20)); 114 } 115 INSERT_WORDS(x, ix, lx); 116 117 ax -= x; 118 EXTRACT_WORDS(ix, lx, ax); 119 120 121 if (ix < 0x3fe00000) { /* |x| < 0.5 */ 122 if (ix < 0x3fd00000) /* |x| < 0.25 */ 123 c = ix == 0 ? 1 : __kernel_cospi(ax); 124 else 125 c = __kernel_sinpi(0.5 - ax); 126 } else { 127 if (ix < 0x3fe80000) { /* |x| < 0.75 */ 128 if (ax == 0.5) 129 return (0); 130 c = -__kernel_sinpi(ax - 0.5); 131 } else 132 c = -__kernel_cospi(1 - ax); 133 } 134 135 if (j0 > 30) 136 x -= 0x1p30; 137 j0 = (uint32_t)x; 138 return (j0 & 1 ? -c : c); 139 } 140 141 if (ix >= 0x7f800000) 142 return (vzero / vzero); 143 144 /* 145 * |x| >= 0x1p52 is always an even integer, so return 1. 146 */ 147 return (1); 148 } 149 150 #if LDBL_MANT_DIG == 53 151 __weak_reference(cospi, cospil); 152 #endif 153