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 * tanpi(x) computes tan(pi*x) without multiplication by pi (almost). First, 29 * note that tanpi(-x) = -tanpi(x), so the algorithm considers only |x| and 30 * includes reflection symmetry by considering the sign of x on output. The 31 * method used depends on the magnitude of x. 32 * 33 * 1. For small |x|, tanpi(x) = pi * x where a sloppy threshold is used. The 34 * threshold is |x| < 0x1pN with N = -(P/2+M). P is the precision of the 35 * floating-point type and M = 2 to 4. To achieve high accuracy, pi is 36 * decomposed into high and low parts with the high part containing a 37 * number of trailing zero bits. x is also split into high and low parts. 38 * 39 * 2. For |x| < 1, argument reduction is not required and tanpi(x) is 40 * computed by a direct call to a kernel, which uses the kernel for 41 * tan(x). See below. 42 * 43 * 3. For 1 <= |x| < 0x1p(P-1), argument reduction is required where 44 * |x| = j0 + r with j0 an integer and the remainder r satisfies 45 * 0 <= r < 1. With the given domain, a simplified inline floor(x) 46 * is used. Also, note the following identity 47 * 48 * tan(pi*j0) + tan(pi*r) 49 * tanpi(x) = tan(pi*(j0+r)) = ---------------------------- = tanpi(r) 50 * 1 - tan(pi*j0) * tan(pi*r) 51 * 52 * So, after argument reduction, the kernel is again invoked. 53 * 54 * 4. For |x| >= 0x1p(P-1), |x| is integral and tanpi(x) = copysign(0,x). 55 * 56 * 5. Special cases: 57 * 58 * tanpi(+-0) = +-0 59 * tanpi(+-n) = +-0, for positive integers n. 60 * tanpi(+-n+1/4) = +-1, for positive integers n. 61 * tanpi(+-n+1/2) = NaN, for positive integers n. 62 * tanpi(+-inf) = NaN. Raises the "invalid" floating-point exception. 63 * tanpi(nan) = NaN. Raises the "invalid" floating-point exception. 64 */ 65 66 #include <float.h> 67 #include "math.h" 68 #include "math_private.h" 69 70 static const double 71 pi_hi = 3.1415926814079285e+00, /* 0x400921fb 0x58000000 */ 72 pi_lo = -2.7818135228334233e-08; /* 0xbe5dde97 0x3dcb3b3a */ 73 74 /* 75 * The kernel for tanpi(x) multiplies x by an 80-bit approximation of 76 * pi, where the hi and lo parts are used with with kernel for tan(x). 77 */ 78 static inline double 79 __kernel_tanpi(double x) 80 { 81 double_t hi, lo, t; 82 83 if (x < 0.25) { 84 hi = (float)x; 85 lo = x - hi; 86 lo = lo * (pi_lo + pi_hi) + hi * pi_lo; 87 hi *= pi_hi; 88 _2sumF(hi, lo); 89 t = __kernel_tan(hi, lo, 1); 90 } else if (x > 0.25) { 91 x = 0.5 - x; 92 hi = (float)x; 93 lo = x - hi; 94 lo = lo * (pi_lo + pi_hi) + hi * pi_lo; 95 hi *= pi_hi; 96 _2sumF(hi, lo); 97 t = - __kernel_tan(hi, lo, -1); 98 } else 99 t = 1; 100 101 return (t); 102 } 103 104 volatile static const double vzero = 0; 105 106 double 107 tanpi(double x) 108 { 109 double ax, hi, lo, t; 110 uint32_t hx, ix, j0, lx; 111 112 EXTRACT_WORDS(hx, lx, x); 113 ix = hx & 0x7fffffff; 114 INSERT_WORDS(ax, ix, lx); 115 116 if (ix < 0x3ff00000) { /* |x| < 1 */ 117 if (ix < 0x3fe00000) { /* |x| < 0.5 */ 118 if (ix < 0x3e200000) { /* |x| < 0x1p-29 */ 119 if (x == 0) 120 return (x); 121 /* 122 * To avoid issues with subnormal values, 123 * scale the computation and rescale on 124 * return. 125 */ 126 INSERT_WORDS(hi, hx, 0); 127 hi *= 0x1p53; 128 lo = x * 0x1p53 - hi; 129 t = (pi_lo + pi_hi) * lo + pi_lo * hi + 130 pi_hi * hi; 131 return (t * 0x1p-53); 132 } 133 t = __kernel_tanpi(ax); 134 } else if (ax == 0.5) 135 return ((ax - ax) / (ax - ax)); 136 else 137 t = - __kernel_tanpi(1 - ax); 138 return ((hx & 0x80000000) ? -t : t); 139 } 140 141 if (ix < 0x43300000) { /* 1 <= |x| < 0x1p52 */ 142 /* Determine integer part of ax. */ 143 j0 = ((ix >> 20) & 0x7ff) - 0x3ff; 144 if (j0 < 20) { 145 ix &= ~(0x000fffff >> j0); 146 lx = 0; 147 } else { 148 lx &= ~(((uint32_t)(0xffffffff)) >> (j0 - 20)); 149 } 150 INSERT_WORDS(x,ix,lx); 151 152 ax -= x; 153 EXTRACT_WORDS(ix, lx, ax); 154 155 if (ix < 0x3fe00000) /* |x| < 0.5 */ 156 t = ax == 0 ? 0 : __kernel_tanpi(ax); 157 else if (ax == 0.5) 158 return ((ax - ax) / (ax - ax)); 159 else 160 t = - __kernel_tanpi(1 - ax); 161 162 return ((hx & 0x80000000) ? -t : t); 163 } 164 165 /* x = +-inf or nan. */ 166 if (ix >= 0x7ff00000) 167 return (vzero / vzero); 168 169 /* 170 * |x| >= 0x1p52 is always an integer, so return +-0. 171 */ 172 return (copysign(0, x)); 173 } 174 175 #if LDBL_MANT_DIG == 53 176 __weak_reference(tanpi, tanpil); 177 #endif 178