1 /* $OpenBSD: e_log2l.c,v 1.3 2017/01/21 08:29:13 krw Exp $ */ 2 3 /* 4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> 5 * 6 * Permission to use, copy, modify, and distribute this software for any 7 * purpose with or without fee is hereby granted, provided that the above 8 * copyright notice and this permission notice appear in all copies. 9 * 10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN 15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF 16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 17 */ 18 19 /* log2l.c 20 * 21 * Base 2 logarithm, long double precision 22 * 23 * 24 * 25 * SYNOPSIS: 26 * 27 * long double x, y, log2l(); 28 * 29 * y = log2l( x ); 30 * 31 * 32 * 33 * DESCRIPTION: 34 * 35 * Returns the base 2 logarithm of x. 36 * 37 * The argument is separated into its exponent and fractional 38 * parts. If the exponent is between -1 and +1, the (natural) 39 * logarithm of the fraction is approximated by 40 * 41 * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x). 42 * 43 * Otherwise, setting z = 2(x-1)/x+1), 44 * 45 * log(x) = z + z**3 P(z)/Q(z). 46 * 47 * 48 * 49 * ACCURACY: 50 * 51 * Relative error: 52 * arithmetic domain # trials peak rms 53 * IEEE 0.5, 2.0 30000 9.8e-20 2.7e-20 54 * IEEE exp(+-10000) 70000 5.4e-20 2.3e-20 55 * 56 * In the tests over the interval exp(+-10000), the logarithms 57 * of the random arguments were uniformly distributed over 58 * [-10000, +10000]. 59 * 60 * ERROR MESSAGES: 61 * 62 * log singularity: x = 0; returns -INFINITY 63 * log domain: x < 0; returns NAN 64 */ 65 66 #include <math.h> 67 68 #include "math_private.h" 69 70 /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x) 71 * 1/sqrt(2) <= x < sqrt(2) 72 * Theoretical peak relative error = 6.2e-22 73 */ 74 static long double P[] = { 75 4.9962495940332550844739E-1L, 76 1.0767376367209449010438E1L, 77 7.7671073698359539859595E1L, 78 2.5620629828144409632571E2L, 79 4.2401812743503691187826E2L, 80 3.4258224542413922935104E2L, 81 1.0747524399916215149070E2L, 82 }; 83 static long double Q[] = { 84 /* 1.0000000000000000000000E0,*/ 85 2.3479774160285863271658E1L, 86 1.9444210022760132894510E2L, 87 7.7952888181207260646090E2L, 88 1.6911722418503949084863E3L, 89 2.0307734695595183428202E3L, 90 1.2695660352705325274404E3L, 91 3.2242573199748645407652E2L, 92 }; 93 94 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), 95 * where z = 2(x-1)/(x+1) 96 * 1/sqrt(2) <= x < sqrt(2) 97 * Theoretical peak relative error = 6.16e-22 98 */ 99 static long double R[4] = { 100 1.9757429581415468984296E-3L, 101 -7.1990767473014147232598E-1L, 102 1.0777257190312272158094E1L, 103 -3.5717684488096787370998E1L, 104 }; 105 static long double S[4] = { 106 /* 1.00000000000000000000E0L,*/ 107 -2.6201045551331104417768E1L, 108 1.9361891836232102174846E2L, 109 -4.2861221385716144629696E2L, 110 }; 111 /* log2(e) - 1 */ 112 #define LOG2EA 4.4269504088896340735992e-1L 113 114 #define SQRTH 0.70710678118654752440L 115 116 long double 117 log2l(long double x) 118 { 119 volatile long double z; 120 long double y; 121 int e; 122 123 if( isnan(x) ) 124 return(x); 125 if( x == INFINITY ) 126 return(x); 127 /* Test for domain */ 128 if( x <= 0.0L ) 129 { 130 if( x == 0.0L ) 131 return( -INFINITY ); 132 else 133 return( NAN ); 134 } 135 136 /* separate mantissa from exponent */ 137 138 /* Note, frexp is used so that denormal numbers 139 * will be handled properly. 140 */ 141 x = frexpl( x, &e ); 142 143 144 /* logarithm using log(x) = z + z**3 P(z)/Q(z), 145 * where z = 2(x-1)/x+1) 146 */ 147 if( (e > 2) || (e < -2) ) 148 { 149 if( x < SQRTH ) 150 { /* 2( 2x-1 )/( 2x+1 ) */ 151 e -= 1; 152 z = x - 0.5L; 153 y = 0.5L * z + 0.5L; 154 } 155 else 156 { /* 2 (x-1)/(x+1) */ 157 z = x - 0.5L; 158 z -= 0.5L; 159 y = 0.5L * x + 0.5L; 160 } 161 x = z / y; 162 z = x*x; 163 y = x * ( z * __polevll( z, R, 3 ) / __p1evll( z, S, 3 ) ); 164 goto done; 165 } 166 167 168 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ 169 170 if( x < SQRTH ) 171 { 172 e -= 1; 173 x = ldexpl( x, 1 ) - 1.0L; /* 2x - 1 */ 174 } 175 else 176 { 177 x = x - 1.0L; 178 } 179 z = x*x; 180 y = x * ( z * __polevll( x, P, 6 ) / __p1evll( x, Q, 7 ) ); 181 y = y - ldexpl( z, -1 ); /* -0.5x^2 + ... */ 182 183 done: 184 185 /* Multiply log of fraction by log2(e) 186 * and base 2 exponent by 1 187 * 188 * ***CAUTION*** 189 * 190 * This sequence of operations is critical and it may 191 * be horribly defeated by some compiler optimizers. 192 */ 193 z = y * LOG2EA; 194 z += x * LOG2EA; 195 z += y; 196 z += x; 197 z += e; 198 return( z ); 199 } 200