1 /* $OpenBSD: e_log2l.c,v 1.1 2011/07/06 00:02:42 martynas 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 * Base 2 logarithm, 128-bit long double precision 21 * 22 * 23 * 24 * SYNOPSIS: 25 * 26 * long double x, y, log2l(); 27 * 28 * y = log2l( x ); 29 * 30 * 31 * 32 * DESCRIPTION: 33 * 34 * Returns the base 2 logarithm of x. 35 * 36 * The argument is separated into its exponent and fractional 37 * parts. If the exponent is between -1 and +1, the (natural) 38 * logarithm of the fraction is approximated by 39 * 40 * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x). 41 * 42 * Otherwise, setting z = 2(x-1)/x+1), 43 * 44 * log(x) = z + z^3 P(z)/Q(z). 45 * 46 * 47 * 48 * ACCURACY: 49 * 50 * Relative error: 51 * arithmetic domain # trials peak rms 52 * IEEE 0.5, 2.0 100,000 2.6e-34 4.9e-35 53 * IEEE exp(+-10000) 100,000 9.6e-35 4.0e-35 54 * 55 * In the tests over the interval exp(+-10000), the logarithms 56 * of the random arguments were uniformly distributed over 57 * [-10000, +10000]. 58 * 59 */ 60 61 #include <openlibm_math.h> 62 63 #include "math_private.h" 64 65 /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x) 66 * 1/sqrt(2) <= x < sqrt(2) 67 * Theoretical peak relative error = 5.3e-37, 68 * relative peak error spread = 2.3e-14 69 */ 70 static const long double P[13] = 71 { 72 1.313572404063446165910279910527789794488E4L, 73 7.771154681358524243729929227226708890930E4L, 74 2.014652742082537582487669938141683759923E5L, 75 3.007007295140399532324943111654767187848E5L, 76 2.854829159639697837788887080758954924001E5L, 77 1.797628303815655343403735250238293741397E5L, 78 7.594356839258970405033155585486712125861E4L, 79 2.128857716871515081352991964243375186031E4L, 80 3.824952356185897735160588078446136783779E3L, 81 4.114517881637811823002128927449878962058E2L, 82 2.321125933898420063925789532045674660756E1L, 83 4.998469661968096229986658302195402690910E-1L, 84 1.538612243596254322971797716843006400388E-6L 85 }; 86 static const long double Q[12] = 87 { 88 3.940717212190338497730839731583397586124E4L, 89 2.626900195321832660448791748036714883242E5L, 90 7.777690340007566932935753241556479363645E5L, 91 1.347518538384329112529391120390701166528E6L, 92 1.514882452993549494932585972882995548426E6L, 93 1.158019977462989115839826904108208787040E6L, 94 6.132189329546557743179177159925690841200E5L, 95 2.248234257620569139969141618556349415120E5L, 96 5.605842085972455027590989944010492125825E4L, 97 9.147150349299596453976674231612674085381E3L, 98 9.104928120962988414618126155557301584078E2L, 99 4.839208193348159620282142911143429644326E1L 100 /* 1.000000000000000000000000000000000000000E0L, */ 101 }; 102 103 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), 104 * where z = 2(x-1)/(x+1) 105 * 1/sqrt(2) <= x < sqrt(2) 106 * Theoretical peak relative error = 1.1e-35, 107 * relative peak error spread 1.1e-9 108 */ 109 static const long double R[6] = 110 { 111 1.418134209872192732479751274970992665513E5L, 112 -8.977257995689735303686582344659576526998E4L, 113 2.048819892795278657810231591630928516206E4L, 114 -2.024301798136027039250415126250455056397E3L, 115 8.057002716646055371965756206836056074715E1L, 116 -8.828896441624934385266096344596648080902E-1L 117 }; 118 static const long double S[6] = 119 { 120 1.701761051846631278975701529965589676574E6L, 121 -1.332535117259762928288745111081235577029E6L, 122 4.001557694070773974936904547424676279307E5L, 123 -5.748542087379434595104154610899551484314E4L, 124 3.998526750980007367835804959888064681098E3L, 125 -1.186359407982897997337150403816839480438E2L 126 /* 1.000000000000000000000000000000000000000E0L, */ 127 }; 128 129 static const long double 130 /* log2(e) - 1 */ 131 LOG2EA = 4.4269504088896340735992468100189213742664595E-1L, 132 /* sqrt(2)/2 */ 133 SQRTH = 7.071067811865475244008443621048490392848359E-1L; 134 135 136 /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ 137 138 static long double 139 neval (long double x, const long double *p, int n) 140 { 141 long double y; 142 143 p += n; 144 y = *p--; 145 do 146 { 147 y = y * x + *p--; 148 } 149 while (--n > 0); 150 return y; 151 } 152 153 154 /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ 155 156 static long double 157 deval (long double x, const long double *p, int n) 158 { 159 long double y; 160 161 p += n; 162 y = x + *p--; 163 do 164 { 165 y = y * x + *p--; 166 } 167 while (--n > 0); 168 return y; 169 } 170 171 172 173 long double 174 log2l(long double x) 175 { 176 long double z; 177 long double y; 178 int e; 179 int64_t hx, lx; 180 181 /* Test for domain */ 182 GET_LDOUBLE_WORDS64 (hx, lx, x); 183 if (((hx & 0x7fffffffffffffffLL) | lx) == 0) 184 return (-1.0L / (x - x)); 185 if (hx < 0) 186 return (x - x) / (x - x); 187 if (hx >= 0x7fff000000000000LL) 188 return (x + x); 189 190 /* separate mantissa from exponent */ 191 192 /* Note, frexp is used so that denormal numbers 193 * will be handled properly. 194 */ 195 x = frexpl (x, &e); 196 197 198 /* logarithm using log(x) = z + z**3 P(z)/Q(z), 199 * where z = 2(x-1)/x+1) 200 */ 201 if ((e > 2) || (e < -2)) 202 { 203 if (x < SQRTH) 204 { /* 2( 2x-1 )/( 2x+1 ) */ 205 e -= 1; 206 z = x - 0.5L; 207 y = 0.5L * z + 0.5L; 208 } 209 else 210 { /* 2 (x-1)/(x+1) */ 211 z = x - 0.5L; 212 z -= 0.5L; 213 y = 0.5L * x + 0.5L; 214 } 215 x = z / y; 216 z = x * x; 217 y = x * (z * neval (z, R, 5) / deval (z, S, 5)); 218 goto done; 219 } 220 221 222 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ 223 224 if (x < SQRTH) 225 { 226 e -= 1; 227 x = 2.0 * x - 1.0L; /* 2x - 1 */ 228 } 229 else 230 { 231 x = x - 1.0L; 232 } 233 z = x * x; 234 y = x * (z * neval (x, P, 12) / deval (x, Q, 11)); 235 y = y - 0.5 * z; 236 237 done: 238 239 /* Multiply log of fraction by log2(e) 240 * and base 2 exponent by 1 241 */ 242 z = y * LOG2EA; 243 z += x * LOG2EA; 244 z += y; 245 z += x; 246 z += e; 247 return (z); 248 } 249