1 /* $OpenBSD: s_log1pl.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 /* log1pl.c 20 * 21 * Relative error logarithm 22 * Natural logarithm of 1+x, 128-bit long double precision 23 * 24 * 25 * 26 * SYNOPSIS: 27 * 28 * long double x, y, log1pl(); 29 * 30 * y = log1pl( x ); 31 * 32 * 33 * 34 * DESCRIPTION: 35 * 36 * Returns the base e (2.718...) logarithm of 1+x. 37 * 38 * The argument 1+x is separated into its exponent and fractional 39 * parts. If the exponent is between -1 and +1, the logarithm 40 * of the fraction is approximated by 41 * 42 * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x). 43 * 44 * Otherwise, setting z = 2(w-1)/(w+1), 45 * 46 * log(w) = z + z^3 P(z)/Q(z). 47 * 48 * 49 * 50 * ACCURACY: 51 * 52 * Relative error: 53 * arithmetic domain # trials peak rms 54 * IEEE -1, 8 100000 1.9e-34 4.3e-35 55 */ 56 57 #include <openlibm.h> 58 59 #include "math_private.h" 60 61 /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x) 62 * 1/sqrt(2) <= 1+x < sqrt(2) 63 * Theoretical peak relative error = 5.3e-37, 64 * relative peak error spread = 2.3e-14 65 */ 66 static const long double 67 P12 = 1.538612243596254322971797716843006400388E-6L, 68 P11 = 4.998469661968096229986658302195402690910E-1L, 69 P10 = 2.321125933898420063925789532045674660756E1L, 70 P9 = 4.114517881637811823002128927449878962058E2L, 71 P8 = 3.824952356185897735160588078446136783779E3L, 72 P7 = 2.128857716871515081352991964243375186031E4L, 73 P6 = 7.594356839258970405033155585486712125861E4L, 74 P5 = 1.797628303815655343403735250238293741397E5L, 75 P4 = 2.854829159639697837788887080758954924001E5L, 76 P3 = 3.007007295140399532324943111654767187848E5L, 77 P2 = 2.014652742082537582487669938141683759923E5L, 78 P1 = 7.771154681358524243729929227226708890930E4L, 79 P0 = 1.313572404063446165910279910527789794488E4L, 80 /* Q12 = 1.000000000000000000000000000000000000000E0L, */ 81 Q11 = 4.839208193348159620282142911143429644326E1L, 82 Q10 = 9.104928120962988414618126155557301584078E2L, 83 Q9 = 9.147150349299596453976674231612674085381E3L, 84 Q8 = 5.605842085972455027590989944010492125825E4L, 85 Q7 = 2.248234257620569139969141618556349415120E5L, 86 Q6 = 6.132189329546557743179177159925690841200E5L, 87 Q5 = 1.158019977462989115839826904108208787040E6L, 88 Q4 = 1.514882452993549494932585972882995548426E6L, 89 Q3 = 1.347518538384329112529391120390701166528E6L, 90 Q2 = 7.777690340007566932935753241556479363645E5L, 91 Q1 = 2.626900195321832660448791748036714883242E5L, 92 Q0 = 3.940717212190338497730839731583397586124E4L; 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 = 1.1e-35, 98 * relative peak error spread 1.1e-9 99 */ 100 static const long double 101 R5 = -8.828896441624934385266096344596648080902E-1L, 102 R4 = 8.057002716646055371965756206836056074715E1L, 103 R3 = -2.024301798136027039250415126250455056397E3L, 104 R2 = 2.048819892795278657810231591630928516206E4L, 105 R1 = -8.977257995689735303686582344659576526998E4L, 106 R0 = 1.418134209872192732479751274970992665513E5L, 107 /* S6 = 1.000000000000000000000000000000000000000E0L, */ 108 S5 = -1.186359407982897997337150403816839480438E2L, 109 S4 = 3.998526750980007367835804959888064681098E3L, 110 S3 = -5.748542087379434595104154610899551484314E4L, 111 S2 = 4.001557694070773974936904547424676279307E5L, 112 S1 = -1.332535117259762928288745111081235577029E6L, 113 S0 = 1.701761051846631278975701529965589676574E6L; 114 115 /* C1 + C2 = ln 2 */ 116 static const long double C1 = 6.93145751953125E-1L; 117 static const long double C2 = 1.428606820309417232121458176568075500134E-6L; 118 119 static const long double sqrth = 0.7071067811865475244008443621048490392848L; 120 /* ln (2^16384 * (1 - 2^-113)) */ 121 static const long double zero = 0.0L; 122 123 long double 124 log1pl(long double xm1) 125 { 126 long double x, y, z, r, s; 127 ieee_quad_shape_type u; 128 int32_t hx; 129 int e; 130 131 /* Test for NaN or infinity input. */ 132 u.value = xm1; 133 hx = u.parts32.mswhi; 134 if (hx >= 0x7fff0000) 135 return xm1; 136 137 /* log1p(+- 0) = +- 0. */ 138 if (((hx & 0x7fffffff) == 0) 139 && (u.parts32.mswlo | u.parts32.lswhi | u.parts32.lswlo) == 0) 140 return xm1; 141 142 x = xm1 + 1.0L; 143 144 /* log1p(-1) = -inf */ 145 if (x <= 0.0L) 146 { 147 if (x == 0.0L) 148 return (-1.0L / (x - x)); 149 else 150 return (zero / (x - x)); 151 } 152 153 /* Separate mantissa from exponent. */ 154 155 /* Use frexp used so that denormal numbers will be handled properly. */ 156 x = frexpl (x, &e); 157 158 /* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2), 159 where z = 2(x-1)/x+1). */ 160 if ((e > 2) || (e < -2)) 161 { 162 if (x < sqrth) 163 { /* 2( 2x-1 )/( 2x+1 ) */ 164 e -= 1; 165 z = x - 0.5L; 166 y = 0.5L * z + 0.5L; 167 } 168 else 169 { /* 2 (x-1)/(x+1) */ 170 z = x - 0.5L; 171 z -= 0.5L; 172 y = 0.5L * x + 0.5L; 173 } 174 x = z / y; 175 z = x * x; 176 r = ((((R5 * z 177 + R4) * z 178 + R3) * z 179 + R2) * z 180 + R1) * z 181 + R0; 182 s = (((((z 183 + S5) * z 184 + S4) * z 185 + S3) * z 186 + S2) * z 187 + S1) * z 188 + S0; 189 z = x * (z * r / s); 190 z = z + e * C2; 191 z = z + x; 192 z = z + e * C1; 193 return (z); 194 } 195 196 197 /* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */ 198 199 if (x < sqrth) 200 { 201 e -= 1; 202 if (e != 0) 203 x = 2.0L * x - 1.0L; /* 2x - 1 */ 204 else 205 x = xm1; 206 } 207 else 208 { 209 if (e != 0) 210 x = x - 1.0L; 211 else 212 x = xm1; 213 } 214 z = x * x; 215 r = (((((((((((P12 * x 216 + P11) * x 217 + P10) * x 218 + P9) * x 219 + P8) * x 220 + P7) * x 221 + P6) * x 222 + P5) * x 223 + P4) * x 224 + P3) * x 225 + P2) * x 226 + P1) * x 227 + P0; 228 s = (((((((((((x 229 + Q11) * x 230 + Q10) * x 231 + Q9) * x 232 + Q8) * x 233 + Q7) * x 234 + Q6) * x 235 + Q5) * x 236 + Q4) * x 237 + Q3) * x 238 + Q2) * x 239 + Q1) * x 240 + Q0; 241 y = x * (z * r / s); 242 y = y + e * C2; 243 z = y - 0.5L * z; 244 z = z + x; 245 z = z + e * C1; 246 return (z); 247 } 248