| OLD | NEW |
| 1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_powl.c */ | 1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_powl.c */ |
| 2 /* | 2 /* |
| 3 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> | 3 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> |
| 4 * | 4 * |
| 5 * Permission to use, copy, modify, and distribute this software for any | 5 * Permission to use, copy, modify, and distribute this software for any |
| 6 * purpose with or without fee is hereby granted, provided that the above | 6 * purpose with or without fee is hereby granted, provided that the above |
| 7 * copyright notice and this permission notice appear in all copies. | 7 * copyright notice and this permission notice appear in all copies. |
| 8 * | 8 * |
| 9 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES | 9 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES |
| 10 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF | 10 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF |
| (...skipping 52 matching lines...) Expand 10 before | Expand all | Expand 10 after Loading... |
| 63 * message condition value returned | 63 * message condition value returned |
| 64 * pow overflow x**y > MAXNUM INFINITY | 64 * pow overflow x**y > MAXNUM INFINITY |
| 65 * pow underflow x**y < 1/MAXNUM 0.0 | 65 * pow underflow x**y < 1/MAXNUM 0.0 |
| 66 * pow domain x<0 and y noninteger 0.0 | 66 * pow domain x<0 and y noninteger 0.0 |
| 67 * | 67 * |
| 68 */ | 68 */ |
| 69 | 69 |
| 70 #include "libm.h" | 70 #include "libm.h" |
| 71 | 71 |
| 72 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 | 72 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 |
| 73 long double powl(long double x, long double y) | 73 long double powl(long double x, long double y) { |
| 74 { | 74 return pow(x, y); |
| 75 » return pow(x, y); | |
| 76 } | 75 } |
| 77 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 | 76 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 |
| 78 | 77 |
| 79 /* Table size */ | 78 /* Table size */ |
| 80 #define NXT 32 | 79 #define NXT 32 |
| 81 | 80 |
| 82 /* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z) | 81 /* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z) |
| 83 * on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1 | 82 * on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1 |
| 84 */ | 83 */ |
| 85 static const long double P[] = { | 84 static const long double P[] = { |
| 86 8.3319510773868690346226E-4L, | 85 8.3319510773868690346226E-4L, 4.9000050881978028599627E-1L, |
| 87 4.9000050881978028599627E-1L, | 86 1.7500123722550302671919E0L, 1.4000100839971580279335E0L, |
| 88 1.7500123722550302671919E0L, | |
| 89 1.4000100839971580279335E0L, | |
| 90 }; | 87 }; |
| 91 static const long double Q[] = { | 88 static const long double Q[] = { |
| 92 /* 1.0000000000000000000000E0L,*/ | 89 /* 1.0000000000000000000000E0L,*/ |
| 93 5.2500282295834889175431E0L, | 90 5.2500282295834889175431E0L, 8.4000598057587009834666E0L, |
| 94 8.4000598057587009834666E0L, | 91 4.2000302519914740834728E0L, |
| 95 4.2000302519914740834728E0L, | |
| 96 }; | 92 }; |
| 97 /* A[i] = 2^(-i/32), rounded to IEEE long double precision. | 93 /* A[i] = 2^(-i/32), rounded to IEEE long double precision. |
| 98 * If i is even, A[i] + B[i/2] gives additional accuracy. | 94 * If i is even, A[i] + B[i/2] gives additional accuracy. |
| 99 */ | 95 */ |
| 100 static const long double A[33] = { | 96 static const long double A[33] = { |
| 101 1.0000000000000000000000E0L, | 97 1.0000000000000000000000E0L, 9.7857206208770013448287E-1L, |
| 102 9.7857206208770013448287E-1L, | 98 9.5760328069857364691013E-1L, 9.3708381705514995065011E-1L, |
| 103 9.5760328069857364691013E-1L, | 99 9.1700404320467123175367E-1L, 8.9735453750155359320742E-1L, |
| 104 9.3708381705514995065011E-1L, | 100 8.7812608018664974155474E-1L, 8.5930964906123895780165E-1L, |
| 105 9.1700404320467123175367E-1L, | 101 8.4089641525371454301892E-1L, 8.2287773907698242225554E-1L, |
| 106 8.9735453750155359320742E-1L, | 102 8.0524516597462715409607E-1L, 7.8799042255394324325455E-1L, |
| 107 8.7812608018664974155474E-1L, | 103 7.7110541270397041179298E-1L, 7.5458221379671136985669E-1L, |
| 108 8.5930964906123895780165E-1L, | 104 7.3841307296974965571198E-1L, 7.2259040348852331001267E-1L, |
| 109 8.4089641525371454301892E-1L, | 105 7.0710678118654752438189E-1L, 6.9195494098191597746178E-1L, |
| 110 8.2287773907698242225554E-1L, | 106 6.7712777346844636413344E-1L, 6.6261832157987064729696E-1L, |
| 111 8.0524516597462715409607E-1L, | 107 6.4841977732550483296079E-1L, 6.3452547859586661129850E-1L, |
| 112 7.8799042255394324325455E-1L, | 108 6.2092890603674202431705E-1L, 6.0762367999023443907803E-1L, |
| 113 7.7110541270397041179298E-1L, | 109 5.9460355750136053334378E-1L, 5.8186242938878875689693E-1L, |
| 114 7.5458221379671136985669E-1L, | 110 5.6939431737834582684856E-1L, 5.5719337129794626814472E-1L, |
| 115 7.3841307296974965571198E-1L, | 111 5.4525386633262882960438E-1L, 5.3357020033841180906486E-1L, |
| 116 7.2259040348852331001267E-1L, | 112 5.2213689121370692017331E-1L, 5.1094857432705833910408E-1L, |
| 117 7.0710678118654752438189E-1L, | 113 5.0000000000000000000000E-1L, |
| 118 6.9195494098191597746178E-1L, | |
| 119 6.7712777346844636413344E-1L, | |
| 120 6.6261832157987064729696E-1L, | |
| 121 6.4841977732550483296079E-1L, | |
| 122 6.3452547859586661129850E-1L, | |
| 123 6.2092890603674202431705E-1L, | |
| 124 6.0762367999023443907803E-1L, | |
| 125 5.9460355750136053334378E-1L, | |
| 126 5.8186242938878875689693E-1L, | |
| 127 5.6939431737834582684856E-1L, | |
| 128 5.5719337129794626814472E-1L, | |
| 129 5.4525386633262882960438E-1L, | |
| 130 5.3357020033841180906486E-1L, | |
| 131 5.2213689121370692017331E-1L, | |
| 132 5.1094857432705833910408E-1L, | |
| 133 5.0000000000000000000000E-1L, | |
| 134 }; | 114 }; |
| 135 static const long double B[17] = { | 115 static const long double B[17] = { |
| 136 0.0000000000000000000000E0L, | 116 0.0000000000000000000000E0L, 2.6176170809902549338711E-20L, |
| 137 2.6176170809902549338711E-20L, | 117 -1.0126791927256478897086E-20L, 1.3438228172316276937655E-21L, |
| 138 -1.0126791927256478897086E-20L, | 118 1.2207982955417546912101E-20L, -6.3084814358060867200133E-21L, |
| 139 1.3438228172316276937655E-21L, | 119 1.3164426894366316434230E-20L, -1.8527916071632873716786E-20L, |
| 140 1.2207982955417546912101E-20L, | 120 1.8950325588932570796551E-20L, 1.5564775779538780478155E-20L, |
| 141 -6.3084814358060867200133E-21L, | 121 6.0859793637556860974380E-21L, -2.0208749253662532228949E-20L, |
| 142 1.3164426894366316434230E-20L, | 122 1.4966292219224761844552E-20L, 3.3540909728056476875639E-21L, |
| 143 -1.8527916071632873716786E-20L, | 123 -8.6987564101742849540743E-22L, -1.2327176863327626135542E-20L, |
| 144 1.8950325588932570796551E-20L, | 124 0.0000000000000000000000E0L, |
| 145 1.5564775779538780478155E-20L, | |
| 146 6.0859793637556860974380E-21L, | |
| 147 -2.0208749253662532228949E-20L, | |
| 148 1.4966292219224761844552E-20L, | |
| 149 3.3540909728056476875639E-21L, | |
| 150 -8.6987564101742849540743E-22L, | |
| 151 -1.2327176863327626135542E-20L, | |
| 152 0.0000000000000000000000E0L, | |
| 153 }; | 125 }; |
| 154 | 126 |
| 155 /* 2^x = 1 + x P(x), | 127 /* 2^x = 1 + x P(x), |
| 156 * on the interval -1/32 <= x <= 0 | 128 * on the interval -1/32 <= x <= 0 |
| 157 */ | 129 */ |
| 158 static const long double R[] = { | 130 static const long double R[] = { |
| 159 1.5089970579127659901157E-5L, | 131 1.5089970579127659901157E-5L, 1.5402715328927013076125E-4L, |
| 160 1.5402715328927013076125E-4L, | 132 1.3333556028915671091390E-3L, 9.6181291046036762031786E-3L, |
| 161 1.3333556028915671091390E-3L, | 133 5.5504108664798463044015E-2L, 2.4022650695910062854352E-1L, |
| 162 9.6181291046036762031786E-3L, | 134 6.9314718055994530931447E-1L, |
| 163 5.5504108664798463044015E-2L, | |
| 164 2.4022650695910062854352E-1L, | |
| 165 6.9314718055994530931447E-1L, | |
| 166 }; | 135 }; |
| 167 | 136 |
| 168 #define MEXP (NXT*16384.0L) | 137 #define MEXP (NXT * 16384.0L) |
| 169 /* The following if denormal numbers are supported, else -MEXP: */ | 138 /* The following if denormal numbers are supported, else -MEXP: */ |
| 170 #define MNEXP (-NXT*(16384.0L+64.0L)) | 139 #define MNEXP (-NXT * (16384.0L + 64.0L)) |
| 171 /* log2(e) - 1 */ | 140 /* log2(e) - 1 */ |
| 172 #define LOG2EA 0.44269504088896340735992L | 141 #define LOG2EA 0.44269504088896340735992L |
| 173 | 142 |
| 174 #define F W | 143 #define F W |
| 175 #define Fa Wa | 144 #define Fa Wa |
| 176 #define Fb Wb | 145 #define Fb Wb |
| 177 #define G W | 146 #define G W |
| 178 #define Ga Wa | 147 #define Ga Wa |
| 179 #define Gb u | 148 #define Gb u |
| 180 #define H W | 149 #define H W |
| 181 #define Ha Wb | 150 #define Ha Wb |
| 182 #define Hb Wb | 151 #define Hb Wb |
| 183 | 152 |
| 184 static const long double MAXLOGL = 1.1356523406294143949492E4L; | 153 static const long double MAXLOGL = 1.1356523406294143949492E4L; |
| 185 static const long double MINLOGL = -1.13994985314888605586758E4L; | 154 static const long double MINLOGL = -1.13994985314888605586758E4L; |
| 186 static const long double LOGE2L = 6.9314718055994530941723E-1L; | 155 static const long double LOGE2L = 6.9314718055994530941723E-1L; |
| 187 static const long double huge = 0x1p10000L; | 156 static const long double huge = 0x1p10000L; |
| 188 /* XXX Prevent gcc from erroneously constant folding this. */ | 157 /* XXX Prevent gcc from erroneously constant folding this. */ |
| 189 static const volatile long double twom10000 = 0x1p-10000L; | 158 static const volatile long double twom10000 = 0x1p-10000L; |
| 190 | 159 |
| 191 static long double reducl(long double); | 160 static long double reducl(long double); |
| 192 static long double powil(long double, int); | 161 static long double powil(long double, int); |
| 193 | 162 |
| 194 long double powl(long double x, long double y) | 163 long double powl(long double x, long double y) { |
| 195 { | 164 /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */ |
| 196 » /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */ | 165 int i, nflg, iyflg, yoddint; |
| 197 » int i, nflg, iyflg, yoddint; | 166 long e; |
| 198 » long e; | 167 volatile long double z = 0; |
| 199 » volatile long double z=0; | 168 long double w = 0, W = 0, Wa = 0, Wb = 0, ya = 0, yb = 0, u = 0; |
| 200 » long double w=0, W=0, Wa=0, Wb=0, ya=0, yb=0, u=0; | 169 |
| 201 | 170 /* make sure no invalid exception is raised by nan comparision */ |
| 202 » /* make sure no invalid exception is raised by nan comparision */ | 171 if (isnan(x)) { |
| 203 » if (isnan(x)) { | 172 if (!isnan(y) && y == 0.0) |
| 204 » » if (!isnan(y) && y == 0.0) | 173 return 1.0; |
| 205 » » » return 1.0; | 174 return x; |
| 206 » » return x; | 175 } |
| 207 » } | 176 if (isnan(y)) { |
| 208 » if (isnan(y)) { | 177 if (x == 1.0) |
| 209 » » if (x == 1.0) | 178 return 1.0; |
| 210 » » » return 1.0; | 179 return y; |
| 211 » » return y; | 180 } |
| 212 » } | 181 if (x == 1.0) |
| 213 » if (x == 1.0) | 182 return 1.0; /* 1**y = 1, even if y is nan */ |
| 214 » » return 1.0; /* 1**y = 1, even if y is nan */ | 183 if (x == -1.0 && !isfinite(y)) |
| 215 » if (x == -1.0 && !isfinite(y)) | 184 return 1.0; /* -1**inf = 1 */ |
| 216 » » return 1.0; /* -1**inf = 1 */ | 185 if (y == 0.0) |
| 217 » if (y == 0.0) | 186 return 1.0; /* x**0 = 1, even if x is nan */ |
| 218 » » return 1.0; /* x**0 = 1, even if x is nan */ | 187 if (y == 1.0) |
| 219 » if (y == 1.0) | 188 return x; |
| 220 » » return x; | 189 if (y >= LDBL_MAX) { |
| 221 » if (y >= LDBL_MAX) { | 190 if (x > 1.0 || x < -1.0) |
| 222 » » if (x > 1.0 || x < -1.0) | 191 return INFINITY; |
| 223 » » » return INFINITY; | 192 if (x != 0.0) |
| 224 » » if (x != 0.0) | 193 return 0.0; |
| 225 » » » return 0.0; | 194 } |
| 226 » } | 195 if (y <= -LDBL_MAX) { |
| 227 » if (y <= -LDBL_MAX) { | 196 if (x > 1.0 || x < -1.0) |
| 228 » » if (x > 1.0 || x < -1.0) | 197 return 0.0; |
| 229 » » » return 0.0; | 198 if (x != 0.0 || y == -INFINITY) |
| 230 » » if (x != 0.0 || y == -INFINITY) | 199 return INFINITY; |
| 231 » » » return INFINITY; | 200 } |
| 232 » } | 201 if (x >= LDBL_MAX) { |
| 233 » if (x >= LDBL_MAX) { | 202 if (y > 0.0) |
| 234 » » if (y > 0.0) | 203 return INFINITY; |
| 235 » » » return INFINITY; | 204 return 0.0; |
| 236 » » return 0.0; | 205 } |
| 237 » } | 206 |
| 238 | 207 w = floorl(y); |
| 239 » w = floorl(y); | 208 |
| 240 | 209 /* Set iyflg to 1 if y is an integer. */ |
| 241 » /* Set iyflg to 1 if y is an integer. */ | 210 iyflg = 0; |
| 242 » iyflg = 0; | 211 if (w == y) |
| 243 » if (w == y) | 212 iyflg = 1; |
| 244 » » iyflg = 1; | 213 |
| 245 | 214 /* Test for odd integer y. */ |
| 246 » /* Test for odd integer y. */ | 215 yoddint = 0; |
| 247 » yoddint = 0; | 216 if (iyflg) { |
| 248 » if (iyflg) { | 217 ya = fabsl(y); |
| 249 » » ya = fabsl(y); | 218 ya = floorl(0.5 * ya); |
| 250 » » ya = floorl(0.5 * ya); | 219 yb = 0.5 * fabsl(w); |
| 251 » » yb = 0.5 * fabsl(w); | 220 if (ya != yb) |
| 252 » » if( ya != yb ) | 221 yoddint = 1; |
| 253 » » » yoddint = 1; | 222 } |
| 254 » } | 223 |
| 255 | 224 if (x <= -LDBL_MAX) { |
| 256 » if (x <= -LDBL_MAX) { | 225 if (y > 0.0) { |
| 257 » » if (y > 0.0) { | 226 if (yoddint) |
| 258 » » » if (yoddint) | 227 return -INFINITY; |
| 259 » » » » return -INFINITY; | 228 return INFINITY; |
| 260 » » » return INFINITY; | 229 } |
| 261 » » } | 230 if (y < 0.0) { |
| 262 » » if (y < 0.0) { | 231 if (yoddint) |
| 263 » » » if (yoddint) | 232 return -0.0; |
| 264 » » » » return -0.0; | 233 return 0.0; |
| 265 » » » return 0.0; | 234 } |
| 266 » » } | 235 } |
| 267 » } | 236 nflg = 0; /* (x<0)**(odd int) */ |
| 268 » nflg = 0; /* (x<0)**(odd int) */ | 237 if (x <= 0.0) { |
| 269 » if (x <= 0.0) { | 238 if (x == 0.0) { |
| 270 » » if (x == 0.0) { | 239 if (y < 0.0) { |
| 271 » » » if (y < 0.0) { | 240 if (signbit(x) && yoddint) |
| 272 » » » » if (signbit(x) && yoddint) | 241 /* (-0.0)**(-odd int) = -inf, divbyzero */ |
| 273 » » » » » /* (-0.0)**(-odd int) = -inf, divbyzero
*/ | 242 return -1.0 / 0.0; |
| 274 » » » » » return -1.0/0.0; | 243 /* (+-0.0)**(negative) = inf, divbyzero */ |
| 275 » » » » /* (+-0.0)**(negative) = inf, divbyzero */ | 244 return 1.0 / 0.0; |
| 276 » » » » return 1.0/0.0; | 245 } |
| 277 » » » } | 246 if (signbit(x) && yoddint) |
| 278 » » » if (signbit(x) && yoddint) | 247 return -0.0; |
| 279 » » » » return -0.0; | 248 return 0.0; |
| 280 » » » return 0.0; | 249 } |
| 281 » » } | 250 if (iyflg == 0) |
| 282 » » if (iyflg == 0) | 251 return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */ |
| 283 » » » return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */ | 252 /* (x<0)**(integer) */ |
| 284 » » /* (x<0)**(integer) */ | 253 if (yoddint) |
| 285 » » if (yoddint) | 254 nflg = 1; /* negate result */ |
| 286 » » » nflg = 1; /* negate result */ | 255 x = -x; |
| 287 » » x = -x; | 256 } |
| 288 » } | 257 /* (+integer)**(integer) */ |
| 289 » /* (+integer)**(integer) */ | 258 if (iyflg && floorl(x) == x && fabsl(y) < 32768.0) { |
| 290 » if (iyflg && floorl(x) == x && fabsl(y) < 32768.0) { | 259 w = powil(x, (int)y); |
| 291 » » w = powil(x, (int)y); | 260 return nflg ? -w : w; |
| 292 » » return nflg ? -w : w; | 261 } |
| 293 » } | 262 |
| 294 | 263 /* separate significand from exponent */ |
| 295 » /* separate significand from exponent */ | 264 x = frexpl(x, &i); |
| 296 » x = frexpl(x, &i); | 265 e = i; |
| 297 » e = i; | 266 |
| 298 | 267 /* find significand in antilog table A[] */ |
| 299 » /* find significand in antilog table A[] */ | 268 i = 1; |
| 300 » i = 1; | 269 if (x <= A[17]) |
| 301 » if (x <= A[17]) | 270 i = 17; |
| 302 » » i = 17; | 271 if (x <= A[i + 8]) |
| 303 » if (x <= A[i+8]) | 272 i += 8; |
| 304 » » i += 8; | 273 if (x <= A[i + 4]) |
| 305 » if (x <= A[i+4]) | 274 i += 4; |
| 306 » » i += 4; | 275 if (x <= A[i + 2]) |
| 307 » if (x <= A[i+2]) | 276 i += 2; |
| 308 » » i += 2; | 277 if (x >= A[1]) |
| 309 » if (x >= A[1]) | 278 i = -1; |
| 310 » » i = -1; | 279 i += 1; |
| 311 » i += 1; | 280 |
| 312 | 281 /* Find (x - A[i])/A[i] |
| 313 » /* Find (x - A[i])/A[i] | 282 * in order to compute log(x/A[i]): |
| 314 » * in order to compute log(x/A[i]): | 283 * |
| 315 » * | 284 * log(x) = log( a x/a ) = log(a) + log(x/a) |
| 316 » * log(x) = log( a x/a ) = log(a) + log(x/a) | 285 * |
| 317 » * | 286 * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a |
| 318 » * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a | 287 */ |
| 319 » */ | 288 x -= A[i]; |
| 320 » x -= A[i]; | 289 x -= B[i / 2]; |
| 321 » x -= B[i/2]; | 290 x /= A[i]; |
| 322 » x /= A[i]; | 291 |
| 323 | 292 /* rational approximation for log(1+v): |
| 324 » /* rational approximation for log(1+v): | 293 * |
| 325 » * | 294 * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v) |
| 326 » * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v) | 295 */ |
| 327 » */ | 296 z = x * x; |
| 328 » z = x*x; | 297 w = x * (z * __polevll(x, P, 3) / __p1evll(x, Q, 3)); |
| 329 » w = x * (z * __polevll(x, P, 3) / __p1evll(x, Q, 3)); | 298 w = w - 0.5 * z; |
| 330 » w = w - 0.5*z; | 299 |
| 331 | 300 /* Convert to base 2 logarithm: |
| 332 » /* Convert to base 2 logarithm: | 301 * multiply by log2(e) = 1 + LOG2EA |
| 333 » * multiply by log2(e) = 1 + LOG2EA | 302 */ |
| 334 » */ | 303 z = LOG2EA * w; |
| 335 » z = LOG2EA * w; | 304 z += w; |
| 336 » z += w; | 305 z += LOG2EA * x; |
| 337 » z += LOG2EA * x; | 306 z += x; |
| 338 » z += x; | 307 |
| 339 | 308 /* Compute exponent term of the base 2 logarithm. */ |
| 340 » /* Compute exponent term of the base 2 logarithm. */ | 309 w = -i; |
| 341 » w = -i; | 310 w /= NXT; |
| 342 » w /= NXT; | 311 w += e; |
| 343 » w += e; | 312 /* Now base 2 log of x is w + z. */ |
| 344 » /* Now base 2 log of x is w + z. */ | 313 |
| 345 | 314 /* Multiply base 2 log by y, in extended precision. */ |
| 346 » /* Multiply base 2 log by y, in extended precision. */ | 315 |
| 347 | 316 /* separate y into large part ya |
| 348 » /* separate y into large part ya | 317 * and small part yb less than 1/NXT |
| 349 » * and small part yb less than 1/NXT | 318 */ |
| 350 » */ | 319 ya = reducl(y); |
| 351 » ya = reducl(y); | 320 yb = y - ya; |
| 352 » yb = y - ya; | 321 |
| 353 | 322 /* (w+z)(ya+yb) |
| 354 » /* (w+z)(ya+yb) | 323 * = w*ya + w*yb + z*y |
| 355 » * = w*ya + w*yb + z*y | 324 */ |
| 356 » */ | 325 F = z * y + w * yb; |
| 357 » F = z * y + w * yb; | 326 Fa = reducl(F); |
| 358 » Fa = reducl(F); | 327 Fb = F - Fa; |
| 359 » Fb = F - Fa; | 328 |
| 360 | 329 G = Fa + w * ya; |
| 361 » G = Fa + w * ya; | 330 Ga = reducl(G); |
| 362 » Ga = reducl(G); | 331 Gb = G - Ga; |
| 363 » Gb = G - Ga; | 332 |
| 364 | 333 H = Fb + Gb; |
| 365 » H = Fb + Gb; | 334 Ha = reducl(H); |
| 366 » Ha = reducl(H); | 335 w = (Ga + Ha) * NXT; |
| 367 » w = (Ga + Ha) * NXT; | 336 |
| 368 | 337 /* Test the power of 2 for overflow */ |
| 369 » /* Test the power of 2 for overflow */ | 338 if (w > MEXP) |
| 370 » if (w > MEXP) | 339 return huge * huge; /* overflow */ |
| 371 » » return huge * huge; /* overflow */ | 340 if (w < MNEXP) |
| 372 » if (w < MNEXP) | 341 return twom10000 * twom10000; /* underflow */ |
| 373 » » return twom10000 * twom10000; /* underflow */ | 342 |
| 374 | 343 e = w; |
| 375 » e = w; | 344 Hb = H - Ha; |
| 376 » Hb = H - Ha; | 345 |
| 377 | 346 if (Hb > 0.0) { |
| 378 » if (Hb > 0.0) { | 347 e += 1; |
| 379 » » e += 1; | 348 Hb -= 1.0 / NXT; /*0.0625L;*/ |
| 380 » » Hb -= 1.0/NXT; /*0.0625L;*/ | 349 } |
| 381 » } | 350 |
| 382 | 351 /* Now the product y * log2(x) = Hb + e/NXT. |
| 383 » /* Now the product y * log2(x) = Hb + e/NXT. | 352 * |
| 384 » * | 353 * Compute base 2 exponential of Hb, |
| 385 » * Compute base 2 exponential of Hb, | 354 * where -0.0625 <= Hb <= 0. |
| 386 » * where -0.0625 <= Hb <= 0. | 355 */ |
| 387 » */ | 356 z = Hb * __polevll(Hb, R, 6); /* z = 2**Hb - 1 */ |
| 388 » z = Hb * __polevll(Hb, R, 6); /* z = 2**Hb - 1 */ | 357 |
| 389 | 358 /* Express e/NXT as an integer plus a negative number of (1/NXT)ths. |
| 390 » /* Express e/NXT as an integer plus a negative number of (1/NXT)ths. | 359 * Find lookup table entry for the fractional power of 2. |
| 391 » * Find lookup table entry for the fractional power of 2. | 360 */ |
| 392 » */ | 361 if (e < 0) |
| 393 » if (e < 0) | 362 i = 0; |
| 394 » » i = 0; | 363 else |
| 395 » else | 364 i = 1; |
| 396 » » i = 1; | 365 i = e / NXT + i; |
| 397 » i = e/NXT + i; | 366 e = NXT * i - e; |
| 398 » e = NXT*i - e; | 367 w = A[e]; |
| 399 » w = A[e]; | 368 z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */ |
| 400 » z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */ | 369 z = z + w; |
| 401 » z = z + w; | 370 z = scalbnl(z, i); /* multiply by integer power of 2 */ |
| 402 » z = scalbnl(z, i); /* multiply by integer power of 2 */ | 371 |
| 403 | 372 if (nflg) |
| 404 » if (nflg) | 373 z = -z; |
| 405 » » z = -z; | 374 return z; |
| 406 » return z; | |
| 407 } | 375 } |
| 408 | 376 |
| 409 | |
| 410 /* Find a multiple of 1/NXT that is within 1/NXT of x. */ | 377 /* Find a multiple of 1/NXT that is within 1/NXT of x. */ |
| 411 static long double reducl(long double x) | 378 static long double reducl(long double x) { |
| 412 { | 379 long double t; |
| 413 » long double t; | 380 |
| 414 | 381 t = x * NXT; |
| 415 » t = x * NXT; | 382 t = floorl(t); |
| 416 » t = floorl(t); | 383 t = t / NXT; |
| 417 » t = t / NXT; | 384 return t; |
| 418 » return t; | |
| 419 } | 385 } |
| 420 | 386 |
| 421 /* | 387 /* |
| 422 * Positive real raised to integer power, long double precision | 388 * Positive real raised to integer power, long double precision |
| 423 * | 389 * |
| 424 * | 390 * |
| 425 * SYNOPSIS: | 391 * SYNOPSIS: |
| 426 * | 392 * |
| 427 * long double x, y, powil(); | 393 * long double x, y, powil(); |
| 428 * int n; | 394 * int n; |
| (...skipping 14 matching lines...) Expand all Loading... |
| 443 * | 409 * |
| 444 * Relative error: | 410 * Relative error: |
| 445 * arithmetic x domain n domain # trials peak rms | 411 * arithmetic x domain n domain # trials peak rms |
| 446 * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18 | 412 * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18 |
| 447 * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18 | 413 * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18 |
| 448 * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17 | 414 * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17 |
| 449 * | 415 * |
| 450 * Returns MAXNUM on overflow, zero on underflow. | 416 * Returns MAXNUM on overflow, zero on underflow. |
| 451 */ | 417 */ |
| 452 | 418 |
| 453 static long double powil(long double x, int nn) | 419 static long double powil(long double x, int nn) { |
| 454 { | 420 long double ww, y; |
| 455 » long double ww, y; | 421 long double s; |
| 456 » long double s; | 422 int n, e, sign, lx; |
| 457 » int n, e, sign, lx; | |
| 458 | 423 |
| 459 » if (nn == 0) | 424 if (nn == 0) |
| 460 » » return 1.0; | 425 return 1.0; |
| 461 | 426 |
| 462 » if (nn < 0) { | 427 if (nn < 0) { |
| 463 » » sign = -1; | 428 sign = -1; |
| 464 » » n = -nn; | 429 n = -nn; |
| 465 » } else { | 430 } else { |
| 466 » » sign = 1; | 431 sign = 1; |
| 467 » » n = nn; | 432 n = nn; |
| 468 » } | 433 } |
| 469 | 434 |
| 470 » /* Overflow detection */ | 435 /* Overflow detection */ |
| 471 | 436 |
| 472 » /* Calculate approximate logarithm of answer */ | 437 /* Calculate approximate logarithm of answer */ |
| 473 » s = x; | 438 s = x; |
| 474 » s = frexpl( s, &lx); | 439 s = frexpl(s, &lx); |
| 475 » e = (lx - 1)*n; | 440 e = (lx - 1) * n; |
| 476 » if ((e == 0) || (e > 64) || (e < -64)) { | 441 if ((e == 0) || (e > 64) || (e < -64)) { |
| 477 » » s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L)
; | 442 s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L); |
| 478 » » s = (2.9142135623730950L * s - 0.5 + lx) * nn * LOGE2L; | 443 s = (2.9142135623730950L * s - 0.5 + lx) * nn * LOGE2L; |
| 479 » } else { | 444 } else { |
| 480 » » s = LOGE2L * e; | 445 s = LOGE2L * e; |
| 481 » } | 446 } |
| 482 | 447 |
| 483 » if (s > MAXLOGL) | 448 if (s > MAXLOGL) |
| 484 » » return huge * huge; /* overflow */ | 449 return huge * huge; /* overflow */ |
| 485 | 450 |
| 486 » if (s < MINLOGL) | 451 if (s < MINLOGL) |
| 487 » » return twom10000 * twom10000; /* underflow */ | 452 return twom10000 * twom10000; /* underflow */ |
| 488 » /* Handle tiny denormal answer, but with less accuracy | 453 /* Handle tiny denormal answer, but with less accuracy |
| 489 » * since roundoff error in 1.0/x will be amplified. | 454 * since roundoff error in 1.0/x will be amplified. |
| 490 » * The precise demarcation should be the gradual underflow threshold. | 455 * The precise demarcation should be the gradual underflow threshold. |
| 491 » */ | 456 */ |
| 492 » if (s < -MAXLOGL+2.0) { | 457 if (s < -MAXLOGL + 2.0) { |
| 493 » » x = 1.0/x; | 458 x = 1.0 / x; |
| 494 » » sign = -sign; | 459 sign = -sign; |
| 495 » } | 460 } |
| 496 | 461 |
| 497 » /* First bit of the power */ | 462 /* First bit of the power */ |
| 498 » if (n & 1) | 463 if (n & 1) |
| 499 » » y = x; | 464 y = x; |
| 500 » else | 465 else |
| 501 » » y = 1.0; | 466 y = 1.0; |
| 502 | 467 |
| 503 » ww = x; | 468 ww = x; |
| 504 » n >>= 1; | 469 n >>= 1; |
| 505 » while (n) { | 470 while (n) { |
| 506 » » ww = ww * ww; /* arg to the 2-to-the-kth power */ | 471 ww = ww * ww; /* arg to the 2-to-the-kth power */ |
| 507 » » if (n & 1) /* if that bit is set, then include in product */ | 472 if (n & 1) /* if that bit is set, then include in product */ |
| 508 » » » y *= ww; | 473 y *= ww; |
| 509 » » n >>= 1; | 474 n >>= 1; |
| 510 » } | 475 } |
| 511 | 476 |
| 512 » if (sign < 0) | 477 if (sign < 0) |
| 513 » » y = 1.0/y; | 478 y = 1.0 / y; |
| 514 » return y; | 479 return y; |
| 515 } | 480 } |
| 516 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384 | 481 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384 |
| 517 // TODO: broken implementation to make things compile | 482 // TODO: broken implementation to make things compile |
| 518 long double powl(long double x, long double y) | 483 long double powl(long double x, long double y) { |
| 519 { | 484 return pow(x, y); |
| 520 » return pow(x, y); | |
| 521 } | 485 } |
| 522 #endif | 486 #endif |
| OLD | NEW |