1 /* @(#)s_erf.c 5.1 93/09/24 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 */ 12 13 #include <sys/cdefs.h> 14 __FBSDID("$FreeBSD$"); 15 16 /* 17 * See s_erf.c for complete comments. 18 * 19 * Converted to long double by Steven G. Kargl. 20 */ 21 #include <float.h> 22 #ifdef __i386__ 23 #include <ieeefp.h> 24 #endif 25 26 #include "fpmath.h" 27 #include "math.h" 28 #include "math_private.h" 29 30 /* XXX Prevent compilers from erroneously constant folding: */ 31 static const volatile long double tiny = 0x1p-10000L; 32 33 static const double 34 half= 0.5, 35 one = 1, 36 two = 2; 37 /* 38 * In the domain [0, 2**-34], only the first term in the power series 39 * expansion of erf(x) is used. The magnitude of the first neglected 40 * terms is less than 2**-102. 41 */ 42 static const union IEEEl2bits 43 efxu = LD80C(0x8375d410a6db446c, -3, 1.28379167095512573902e-1L), 44 efx8u = LD80C(0x8375d410a6db446c, 0, 1.02703333676410059122e+0L), 45 /* 46 * Domain [0, 0.84375], range ~[-1.423e-22, 1.423e-22]: 47 * |(erf(x) - x)/x - pp(x)/qq(x)| < 2**-72.573 48 */ 49 pp0u = LD80C(0x8375d410a6db446c, -3, 1.28379167095512573902e-1L), 50 pp1u = LD80C(0xa46c7d09ec3d0cec, -2, -3.21140201054840180596e-1L), 51 pp2u = LD80C(0x9b31e66325576f86, -5, -3.78893851760347812082e-2L), 52 pp3u = LD80C(0x804ac72c9a0b97dd, -7, -7.83032847030604679616e-3L), 53 pp4u = LD80C(0x9f42bcbc3d5a601d, -12, -3.03765663857082048459e-4L), 54 pp5u = LD80C(0x9ec4ad6193470693, -16, -1.89266527398167917502e-5L), 55 qq1u = LD80C(0xdb4b8eb713188d6b, -2, 4.28310832832310510579e-1L), 56 qq2u = LD80C(0xa5750835b2459bd1, -4, 8.07896272074540216658e-2L), 57 qq3u = LD80C(0x8b85d6bd6a90b51c, -7, 8.51579638189385354266e-3L), 58 qq4u = LD80C(0x87332f82cff4ff96, -11, 5.15746855583604912827e-4L), 59 qq5u = LD80C(0x83466cb6bf9dca00, -16, 1.56492109706256700009e-5L), 60 qq6u = LD80C(0xf5bf98c2f996bf63, -24, 1.14435527803073879724e-7L); 61 #define efx (efxu.e) 62 #define efx8 (efx8u.e) 63 #define pp0 (pp0u.e) 64 #define pp1 (pp1u.e) 65 #define pp2 (pp2u.e) 66 #define pp3 (pp3u.e) 67 #define pp4 (pp4u.e) 68 #define pp5 (pp5u.e) 69 #define qq1 (qq1u.e) 70 #define qq2 (qq2u.e) 71 #define qq3 (qq3u.e) 72 #define qq4 (qq4u.e) 73 #define qq5 (qq5u.e) 74 #define qq6 (qq6u.e) 75 static const union IEEEl2bits 76 erxu = LD80C(0xd7bb3d0000000000, -1, 8.42700779438018798828e-1L), 77 /* 78 * Domain [0.84375, 1.25], range ~[-8.132e-22, 8.113e-22]: 79 * |(erf(x) - erx) - pa(x)/qa(x)| < 2**-71.762 80 */ 81 pa0u = LD80C(0xe8211158da02c692, -27, 1.35116960705131296711e-8L), 82 pa1u = LD80C(0xd488f89f36988618, -2, 4.15107507167065612570e-1L), 83 pa2u = LD80C(0xece74f8c63fa3942, -4, -1.15675565215949226989e-1L), 84 pa3u = LD80C(0xc8d31e020727c006, -4, 9.80589241379624665791e-2L), 85 pa4u = LD80C(0x985d5d5fafb0551f, -5, 3.71984145558422368847e-2L), 86 pa5u = LD80C(0xa5b6c4854d2f5452, -8, -5.05718799340957673661e-3L), 87 pa6u = LD80C(0x85c8d58fe3993a47, -8, 4.08277919612202243721e-3L), 88 pa7u = LD80C(0xddbfbc23677b35cf, -13, 2.11476292145347530794e-4L), 89 qa1u = LD80C(0xb8a977896f5eff3f, -1, 7.21335860303380361298e-1L), 90 qa2u = LD80C(0x9fcd662c3d4eac86, -1, 6.24227891731886593333e-1L), 91 qa3u = LD80C(0x9d0b618eac67ba07, -2, 3.06727455774491855801e-1L), 92 qa4u = LD80C(0x881a4293f6d6c92d, -3, 1.32912674218195890535e-1L), 93 qa5u = LD80C(0xbab144f07dea45bf, -5, 4.55792134233613027584e-2L), 94 qa6u = LD80C(0xa6c34ba438bdc900, -7, 1.01783980070527682680e-2L), 95 qa7u = LD80C(0x8fa866dc20717a91, -9, 2.19204436518951438183e-3L); 96 #define erx (erxu.e) 97 #define pa0 (pa0u.e) 98 #define pa1 (pa1u.e) 99 #define pa2 (pa2u.e) 100 #define pa3 (pa3u.e) 101 #define pa4 (pa4u.e) 102 #define pa5 (pa5u.e) 103 #define pa6 (pa6u.e) 104 #define pa7 (pa7u.e) 105 #define qa1 (qa1u.e) 106 #define qa2 (qa2u.e) 107 #define qa3 (qa3u.e) 108 #define qa4 (qa4u.e) 109 #define qa5 (qa5u.e) 110 #define qa6 (qa6u.e) 111 #define qa7 (qa7u.e) 112 static const union IEEEl2bits 113 /* 114 * Domain [1.25,2.85715], range ~[-2.334e-22,2.334e-22]: 115 * |log(x*erfc(x)) + x**2 + 0.5625 - ra(x)/sa(x)| < 2**-71.860 116 */ 117 ra0u = LD80C(0xa1a091e0fb4f335a, -7, -9.86494298915814308249e-3L), 118 ra1u = LD80C(0xc2b0d045ae37df6b, -1, -7.60510460864878271275e-1L), 119 ra2u = LD80C(0xf2cec3ee7da636c5, 3, -1.51754798236892278250e+1L), 120 ra3u = LD80C(0x813cc205395adc7d, 7, -1.29237335516455333420e+2L), 121 ra4u = LD80C(0x8737c8b7b4062c2f, 9, -5.40871625829510494776e+2L), 122 ra5u = LD80C(0x8ffe5383c08d4943, 10, -1.15194769466026108551e+3L), 123 ra6u = LD80C(0x983573e64d5015a9, 10, -1.21767039790249025544e+3L), 124 ra7u = LD80C(0x92a794e763a6d4db, 9, -5.86618463370624636688e+2L), 125 ra8u = LD80C(0xd5ad1fae77c3d9a3, 6, -1.06838132335777049840e+2L), 126 ra9u = LD80C(0x934c1a247807bb9c, 2, -4.60303980944467334806e+0L), 127 sa1u = LD80C(0xd342f90012bb1189, 4, 2.64077014928547064865e+1L), 128 sa2u = LD80C(0x839be13d9d5da883, 8, 2.63217811300123973067e+2L), 129 sa3u = LD80C(0x9f8cba6d1ae1b24b, 10, 1.27639775710344617587e+3L), 130 sa4u = LD80C(0xcaa83f403713e33e, 11, 3.24251544209971162003e+3L), 131 sa5u = LD80C(0x8796aff2f3c47968, 12, 4.33883591261332837874e+3L), 132 sa6u = LD80C(0xb6ef97f9c753157b, 11, 2.92697460344182158454e+3L), 133 sa7u = LD80C(0xe02aee5f83773d1c, 9, 8.96670799139389559818e+2L), 134 sa8u = LD80C(0xc82b83855b88e07e, 6, 1.00084987800048510018e+2L), 135 sa9u = LD80C(0x92f030aefadf28ad, 1, 2.29591004455459083843e+0L); 136 #define ra0 (ra0u.e) 137 #define ra1 (ra1u.e) 138 #define ra2 (ra2u.e) 139 #define ra3 (ra3u.e) 140 #define ra4 (ra4u.e) 141 #define ra5 (ra5u.e) 142 #define ra6 (ra6u.e) 143 #define ra7 (ra7u.e) 144 #define ra8 (ra8u.e) 145 #define ra9 (ra9u.e) 146 #define sa1 (sa1u.e) 147 #define sa2 (sa2u.e) 148 #define sa3 (sa3u.e) 149 #define sa4 (sa4u.e) 150 #define sa5 (sa5u.e) 151 #define sa6 (sa6u.e) 152 #define sa7 (sa7u.e) 153 #define sa8 (sa8u.e) 154 #define sa9 (sa9u.e) 155 /* 156 * Domain [2.85715,7], range ~[-8.323e-22,8.390e-22]: 157 * |log(x*erfc(x)) + x**2 + 0.5625 - rb(x)/sb(x)| < 2**-70.326 158 */ 159 static const union IEEEl2bits 160 rb0u = LD80C(0xa1a091cf43abcd26, -7, -9.86494292470284646962e-3L), 161 rb1u = LD80C(0xd19d2df1cbb8da0a, -1, -8.18804618389296662837e-1L), 162 rb2u = LD80C(0x9a4dd1383e5daf5b, 4, -1.92879967111618594779e+1L), 163 rb3u = LD80C(0xbff0ae9fc0751de6, 7, -1.91940164551245394969e+2L), 164 rb4u = LD80C(0xdde08465310b472b, 9, -8.87508080766577324539e+2L), 165 rb5u = LD80C(0xe796e1d38c8c70a9, 10, -1.85271506669474503781e+3L), 166 rb6u = LD80C(0xbaf655a76e0ab3b5, 10, -1.49569795581333675349e+3L), 167 rb7u = LD80C(0x95d21e3e75503c21, 8, -2.99641547972948019157e+2L), 168 sb1u = LD80C(0x814487ed823c8cbd, 5, 3.23169247732868256569e+1L), 169 sb2u = LD80C(0xbe4bfbb1301304be, 8, 3.80593618534539961773e+2L), 170 sb3u = LD80C(0x809c4ade46b927c7, 11, 2.05776827838541292848e+3L), 171 sb4u = LD80C(0xa55284359f3395a8, 12, 5.29031455540062116327e+3L), 172 sb5u = LD80C(0xbcfa72da9b820874, 12, 6.04730608102312640462e+3L), 173 sb6u = LD80C(0x9d09a35988934631, 11, 2.51260238030767176221e+3L), 174 sb7u = LD80C(0xd675bbe542c159fa, 7, 2.14459898308561015684e+2L); 175 #define rb0 (rb0u.e) 176 #define rb1 (rb1u.e) 177 #define rb2 (rb2u.e) 178 #define rb3 (rb3u.e) 179 #define rb4 (rb4u.e) 180 #define rb5 (rb5u.e) 181 #define rb6 (rb6u.e) 182 #define rb7 (rb7u.e) 183 #define sb1 (sb1u.e) 184 #define sb2 (sb2u.e) 185 #define sb3 (sb3u.e) 186 #define sb4 (sb4u.e) 187 #define sb5 (sb5u.e) 188 #define sb6 (sb6u.e) 189 #define sb7 (sb7u.e) 190 /* 191 * Domain [7,108], range ~[-4.422e-22,4.422e-22]: 192 * |log(x*erfc(x)) + x**2 + 0.5625 - rc(x)/sc(x)| < 2**-70.938 193 */ 194 static const union IEEEl2bits 195 /* err = -4.422092275318925082e-22 -70.937689 */ 196 rc0u = LD80C(0xa1a091cf437a17ad, -7, -9.86494292470008707260e-3L), 197 rc1u = LD80C(0xbe79c5a978122b00, -1, -7.44045595049165939261e-1L), 198 rc2u = LD80C(0xdb26f9bbe31a2794, 3, -1.36970155085888424425e+1L), 199 rc3u = LD80C(0xb5f69a38f5747ac8, 6, -9.09816453742625888546e+1L), 200 rc4u = LD80C(0xd79676d970d0a21a, 7, -2.15587750997584074147e+2L), 201 rc5u = LD80C(0xfe528153c45ec97c, 6, -1.27161142938347796666e+2L), 202 sc1u = LD80C(0xc5e8cd46d5604a96, 4, 2.47386727842204312937e+1L), 203 sc2u = LD80C(0xc5f0f5a5484520eb, 7, 1.97941248254913378865e+2L), 204 sc3u = LD80C(0x964e3c7b34db9170, 9, 6.01222441484087787522e+2L), 205 sc4u = LD80C(0x99be1b89faa0596a, 9, 6.14970430845978077827e+2L), 206 sc5u = LD80C(0xf80dfcbf37ffc5ea, 6, 1.24027318931184605891e+2L); 207 #define rc0 (rc0u.e) 208 #define rc1 (rc1u.e) 209 #define rc2 (rc2u.e) 210 #define rc3 (rc3u.e) 211 #define rc4 (rc4u.e) 212 #define rc5 (rc5u.e) 213 #define sc1 (sc1u.e) 214 #define sc2 (sc2u.e) 215 #define sc3 (sc3u.e) 216 #define sc4 (sc4u.e) 217 #define sc5 (sc5u.e) 218 219 long double 220 erfl(long double x) 221 { 222 long double ax,R,S,P,Q,s,y,z,r; 223 uint64_t lx; 224 int32_t i; 225 uint16_t hx; 226 227 EXTRACT_LDBL80_WORDS(hx, lx, x); 228 229 if((hx & 0x7fff) == 0x7fff) { /* erfl(nan)=nan */ 230 i = (hx>>15)<<1; 231 return (1-i)+one/x; /* erfl(+-inf)=+-1 */ 232 } 233 234 ENTERI(); 235 236 ax = fabsl(x); 237 if(ax < 0.84375) { 238 if(ax < 0x1p-34L) { 239 if(ax < 0x1p-16373L) 240 RETURNI((8*x+efx8*x)/8); /* avoid spurious underflow */ 241 RETURNI(x + efx*x); 242 } 243 z = x*x; 244 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*(pp4+z*pp5)))); 245 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*(qq5+z*qq6))))); 246 y = r/s; 247 RETURNI(x + x*y); 248 } 249 if(ax < 1.25) { 250 s = ax-one; 251 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*(pa6+s*pa7)))))); 252 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*(qa6+s*qa7)))))); 253 if(x>=0) RETURNI(erx + P/Q); else RETURNI(-erx - P/Q); 254 } 255 if(ax >= 7) { /* inf>|x|>= 7 */ 256 if(x>=0) RETURNI(one-tiny); else RETURNI(tiny-one); 257 } 258 s = one/(ax*ax); 259 if(ax < 2.85715) { /* |x| < 2.85715 */ 260 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*(ra7+ 261 s*(ra8+s*ra9)))))))); 262 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ 263 s*(sa8+s*sa9)))))))); 264 } else { /* |x| >= 2.85715 */ 265 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*(rb6+s*rb7)))))); 266 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); 267 } 268 z=(float)ax; 269 r=expl(-z*z-0.5625)*expl((z-ax)*(z+ax)+R/S); 270 if(x>=0) RETURNI(one-r/ax); else RETURNI(r/ax-one); 271 } 272 273 long double 274 erfcl(long double x) 275 { 276 long double ax,R,S,P,Q,s,y,z,r; 277 uint64_t lx; 278 uint16_t hx; 279 280 EXTRACT_LDBL80_WORDS(hx, lx, x); 281 282 if((hx & 0x7fff) == 0x7fff) { /* erfcl(nan)=nan */ 283 /* erfcl(+-inf)=0,2 */ 284 return ((hx>>15)<<1)+one/x; 285 } 286 287 ENTERI(); 288 289 ax = fabsl(x); 290 if(ax < 0.84375L) { 291 if(ax < 0x1p-34L) 292 RETURNI(one-x); 293 z = x*x; 294 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*(pp4+z*pp5)))); 295 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*(qq5+z*qq6))))); 296 y = r/s; 297 if(ax < 0.25L) { /* x<1/4 */ 298 RETURNI(one-(x+x*y)); 299 } else { 300 r = x*y; 301 r += (x-half); 302 RETURNI(half - r); 303 } 304 } 305 if(ax < 1.25L) { 306 s = ax-one; 307 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*(pa6+s*pa7)))))); 308 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*(qa6+s*qa7)))))); 309 if(x>=0) { 310 z = one-erx; RETURNI(z - P/Q); 311 } else { 312 z = (erx+P/Q); RETURNI(one+z); 313 } 314 } 315 316 if(ax < 108) { /* |x| < 108 */ 317 s = one/(ax*ax); 318 if(ax < 2.85715) { /* |x| < 2.85715 */ 319 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*(ra7+ 320 s*(ra8+s*ra9)))))))); 321 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ 322 s*(sa8+s*sa9)))))))); 323 } else if(ax < 7) { /* | |x| < 7 */ 324 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*(rb6+s*rb7)))))); 325 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); 326 } else { 327 if(x < -7) RETURNI(two-tiny);/* x < -7 */ 328 R=rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*rc5)))); 329 S=one+s*(sc1+s*(sc2+s*(sc3+s*(sc4+s*sc5)))); 330 } 331 z = (float)ax; 332 r = expl(-z*z-0.5625)*expl((z-ax)*(z+ax)+R/S); 333 if(x>0) RETURNI(r/ax); else RETURNI(two-r/ax); 334 } else { 335 if(x>0) RETURNI(tiny*tiny); else RETURNI(two-tiny); 336 } 337 } 338