1 /* @(#)e_lgamma_r.c 1.3 95/01/18 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunSoft, 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 e_lgamma_r.c for complete comments. 18 * 19 * Converted to long double by Steven G. Kargl. 20 */ 21 22 #include "fpmath.h" 23 #include "math.h" 24 #include "math_private.h" 25 26 static const volatile double vzero = 0; 27 28 static const double 29 zero= 0, 30 half= 0.5, 31 one = 1; 32 33 static const long double 34 pi = 3.14159265358979323846264338327950288e+00L; 35 /* 36 * Domain y in [0x1p-119, 0.28], range ~[-1.4065e-36, 1.4065e-36]: 37 * |(lgamma(2 - y) + y / 2) / y - a(y)| < 2**-119.1 38 */ 39 static const long double 40 a0 = 7.72156649015328606065120900824024296e-02L, 41 a1 = 3.22467033424113218236207583323018498e-01L, 42 a2 = 6.73523010531980951332460538330282217e-02L, 43 a3 = 2.05808084277845478790009252803463129e-02L, 44 a4 = 7.38555102867398526627292839296001626e-03L, 45 a5 = 2.89051033074152328576829509522483468e-03L, 46 a6 = 1.19275391170326097618357349881842913e-03L, 47 a7 = 5.09669524743042462515256340206203019e-04L, 48 a8 = 2.23154758453578096143609255559576017e-04L, 49 a9 = 9.94575127818397632126978731542755129e-05L, 50 a10 = 4.49262367375420471287545895027098145e-05L, 51 a11 = 2.05072127845117995426519671481628849e-05L, 52 a12 = 9.43948816959096748454087141447939513e-06L, 53 a13 = 4.37486780697359330303852050718287419e-06L, 54 a14 = 2.03920783892362558276037363847651809e-06L, 55 a15 = 9.55191070057967287877923073200324649e-07L, 56 a16 = 4.48993286185740853170657139487620560e-07L, 57 a17 = 2.13107543597620911675316728179563522e-07L, 58 a18 = 9.70745379855304499867546549551023473e-08L, 59 a19 = 5.61889970390290257926487734695402075e-08L, 60 a20 = 6.42739653024130071866684358960960951e-09L, 61 a21 = 3.34491062143649291746195612991870119e-08L, 62 a22 = -1.57068547394315223934653011440641472e-08L, 63 a23 = 1.30812825422415841213733487745200632e-08L; 64 /* 65 * Domain x in [tc-0.24, tc+0.28], range ~[-6.3201e-37, 6.3201e-37]: 66 * |(lgamma(x) - tf) - t(x - tc)| < 2**-120.3. 67 */ 68 static const long double 69 tc = 1.46163214496836234126265954232572133e+00L, 70 tf = -1.21486290535849608095514557177691584e-01L, 71 tt = 1.57061739945077675484237837992951704e-36L, 72 t0 = -1.99238329499314692728655623767019240e-36L, 73 t1 = -6.08453430711711404116887457663281416e-35L, 74 t2 = 4.83836122723810585213722380854828904e-01L, 75 t3 = -1.47587722994530702030955093950668275e-01L, 76 t4 = 6.46249402389127526561003464202671923e-02L, 77 t5 = -3.27885410884813055008502586863748063e-02L, 78 t6 = 1.79706751152103942928638276067164935e-02L, 79 t7 = -1.03142230366363872751602029672767978e-02L, 80 t8 = 6.10053602051788840313573150785080958e-03L, 81 t9 = -3.68456960831637325470641021892968954e-03L, 82 t10 = 2.25976482322181046611440855340968560e-03L, 83 t11 = -1.40225144590445082933490395950664961e-03L, 84 t12 = 8.78232634717681264035014878172485575e-04L, 85 t13 = -5.54194952796682301220684760591403899e-04L, 86 t14 = 3.51912956837848209220421213975000298e-04L, 87 t15 = -2.24653443695947456542669289367055542e-04L, 88 t16 = 1.44070395420840737695611929680511823e-04L, 89 t17 = -9.27609865550394140067059487518862512e-05L, 90 t18 = 5.99347334438437081412945428365433073e-05L, 91 t19 = -3.88458388854572825603964274134801009e-05L, 92 t20 = 2.52476631610328129217896436186551043e-05L, 93 t21 = -1.64508584981658692556994212457518536e-05L, 94 t22 = 1.07434583475987007495523340296173839e-05L, 95 t23 = -7.03070407519397260929482550448878399e-06L, 96 t24 = 4.60968590693753579648385629003100469e-06L, 97 t25 = -3.02765473778832036018438676945512661e-06L, 98 t26 = 1.99238771545503819972741288511303401e-06L, 99 t27 = -1.31281299822614084861868817951788579e-06L, 100 t28 = 8.60844432267399655055574642052370223e-07L, 101 t29 = -5.64535486432397413273248363550536374e-07L, 102 t30 = 3.99357783676275660934903139592727737e-07L, 103 t31 = -2.95849029193433121795495215869311610e-07L, 104 t32 = 1.37790144435073124976696250804940384e-07L; 105 /* 106 * Domain y in [-0.1, 0.232], range ~[-1.4046e-37, 1.4181e-37]: 107 * |(lgamma(1 + y) + 0.5 * y) / y - u(y) / v(y)| < 2**-122.8 108 */ 109 static const long double 110 u0 = -7.72156649015328606065120900824024311e-02L, 111 u1 = 4.24082772271938167430983113242482656e-01L, 112 u2 = 2.96194003481457101058321977413332171e+00L, 113 u3 = 6.49503267711258043997790983071543710e+00L, 114 u4 = 7.40090051288150177152835698948644483e+00L, 115 u5 = 4.94698036296756044610805900340723464e+00L, 116 u6 = 2.00194224610796294762469550684947768e+00L, 117 u7 = 4.82073087750608895996915051568834949e-01L, 118 u8 = 6.46694052280506568192333848437585427e-02L, 119 u9 = 4.17685526755100259316625348933108810e-03L, 120 u10 = 9.06361003550314327144119307810053410e-05L, 121 v1 = 5.15937098592887275994320496999951947e+00L, 122 v2 = 1.14068418766251486777604403304717558e+01L, 123 v3 = 1.41164839437524744055723871839748489e+01L, 124 v4 = 1.07170702656179582805791063277960532e+01L, 125 v5 = 5.14448694179047879915042998453632434e+00L, 126 v6 = 1.55210088094585540637493826431170289e+00L, 127 v7 = 2.82975732849424562719893657416365673e-01L, 128 v8 = 2.86424622754753198010525786005443539e-02L, 129 v9 = 1.35364253570403771005922441442688978e-03L, 130 v10 = 1.91514173702398375346658943749580666e-05L, 131 v11 = -3.25364686890242327944584691466034268e-08L; 132 /* 133 * Domain x in (2, 3], range ~[-1.3341e-36, 1.3536e-36]: 134 * |(lgamma(y+2) - 0.5 * y) / y - s(y)/r(y)| < 2**-120.1 135 * with y = x - 2. 136 */ 137 static const long double 138 s0 = -7.72156649015328606065120900824024297e-02L, 139 s1 = 1.23221687850916448903914170805852253e-01L, 140 s2 = 5.43673188699937239808255378293820020e-01L, 141 s3 = 6.31998137119005233383666791176301800e-01L, 142 s4 = 3.75885340179479850993811501596213763e-01L, 143 s5 = 1.31572908743275052623410195011261575e-01L, 144 s6 = 2.82528453299138685507186287149699749e-02L, 145 s7 = 3.70262021550340817867688714880797019e-03L, 146 s8 = 2.83374000312371199625774129290973648e-04L, 147 s9 = 1.15091830239148290758883505582343691e-05L, 148 s10 = 2.04203474281493971326506384646692446e-07L, 149 s11 = 9.79544198078992058548607407635645763e-10L, 150 r1 = 2.58037466655605285937112832039537492e+00L, 151 r2 = 2.86289413392776399262513849911531180e+00L, 152 r3 = 1.78691044735267497452847829579514367e+00L, 153 r4 = 6.89400381446725342846854215600008055e-01L, 154 r5 = 1.70135865462567955867134197595365343e-01L, 155 r6 = 2.68794816183964420375498986152766763e-02L, 156 r7 = 2.64617234244861832870088893332006679e-03L, 157 r8 = 1.52881761239180800640068128681725702e-04L, 158 r9 = 4.63264813762296029824851351257638558e-06L, 159 r10 = 5.89461519146957343083848967333671142e-08L, 160 r11 = 1.79027678176582527798327441636552968e-10L; 161 /* 162 * Domain z in [8, 0x1p70], range ~[-9.8214e-35, 9.8214e-35]: 163 * |lgamma(x) - (x - 0.5) * (log(x) - 1) - w(1/x)| < 2**-113.0 164 */ 165 static const long double 166 w0 = 4.18938533204672741780329736405617738e-01L, 167 w1 = 8.33333333333333333333333333332852026e-02L, 168 w2 = -2.77777777777777777777777727810123528e-03L, 169 w3 = 7.93650793650793650791708939493907380e-04L, 170 w4 = -5.95238095238095234390450004444370959e-04L, 171 w5 = 8.41750841750837633887817658848845695e-04L, 172 w6 = -1.91752691752396849943172337347259743e-03L, 173 w7 = 6.41025640880333069429106541459015557e-03L, 174 w8 = -2.95506530801732133437990433080327074e-02L, 175 w9 = 1.79644237328444101596766586979576927e-01L, 176 w10 = -1.39240539108367641920172649259736394e+00L, 177 w11 = 1.33987701479007233325288857758641761e+01L, 178 w12 = -1.56363596431084279780966590116006255e+02L, 179 w13 = 2.14830978044410267201172332952040777e+03L, 180 w14 = -3.28636067474227378352761516589092334e+04L, 181 w15 = 5.06201257747865138432663574251462485e+05L, 182 w16 = -6.79720123352023636706247599728048344e+06L, 183 w17 = 6.57556601705472106989497289465949255e+07L, 184 w18 = -3.26229058141181783534257632389415580e+08L; 185 186 static long double 187 sin_pil(long double x) 188 { 189 volatile long double vz; 190 long double y,z; 191 uint64_t lx, n; 192 uint16_t hx; 193 194 y = -x; 195 196 vz = y+0x1.p112; 197 z = vz-0x1.p112; 198 if (z == y) 199 return zero; 200 201 vz = y+0x1.p110; 202 EXTRACT_LDBL128_WORDS(hx,lx,n,vz); 203 z = vz-0x1.p110; 204 if (z > y) { 205 z -= 0.25; 206 n--; 207 } 208 n &= 7; 209 y = y - z + n * 0.25; 210 211 switch (n) { 212 case 0: y = __kernel_sinl(pi*y,zero,0); break; 213 case 1: 214 case 2: y = __kernel_cosl(pi*(0.5-y),zero); break; 215 case 3: 216 case 4: y = __kernel_sinl(pi*(one-y),zero,0); break; 217 case 5: 218 case 6: y = -__kernel_cosl(pi*(y-1.5),zero); break; 219 default: y = __kernel_sinl(pi*(y-2.0),zero,0); break; 220 } 221 return -y; 222 } 223 224 long double 225 lgammal_r(long double x, int *signgamp) 226 { 227 long double nadj,p,p1,p2,p3,q,r,t,w,y,z; 228 uint64_t llx,lx; 229 int i; 230 uint16_t hx,ix; 231 232 EXTRACT_LDBL128_WORDS(hx,lx,llx,x); 233 234 /* purge +-Inf and NaNs */ 235 *signgamp = 1; 236 ix = hx&0x7fff; 237 if(ix==0x7fff) return x*x; 238 239 /* purge +-0 and tiny arguments */ 240 *signgamp = 1-2*(hx>>15); 241 if(ix<0x3fff-116) { /* |x|<2**-(p+3), return -log(|x|) */ 242 if((ix|lx|llx)==0) 243 return one/vzero; 244 return -logl(fabsl(x)); 245 } 246 247 /* purge negative integers and start evaluation for other x < 0 */ 248 if(hx&0x8000) { 249 *signgamp = 1; 250 if(ix>=0x3fff+112) /* |x|>=2**(p-1), must be -integer */ 251 return one/vzero; 252 t = sin_pil(x); 253 if(t==zero) return one/vzero; 254 nadj = logl(pi/fabsl(t*x)); 255 if(t<zero) *signgamp = -1; 256 x = -x; 257 } 258 259 /* purge 1 and 2 */ 260 if((ix==0x3fff || ix==0x4000) && (lx|llx)==0) r = 0; 261 /* for x < 2.0 */ 262 else if(ix<0x4000) { 263 if(x<=8.9999961853027344e-01) { 264 r = -logl(x); 265 if(x>=7.3159980773925781e-01) {y = 1-x; i= 0;} 266 else if(x>=2.3163998126983643e-01) {y= x-(tc-1); i=1;} 267 else {y = x; i=2;} 268 } else { 269 r = 0; 270 if(x>=1.7316312789916992e+00) {y=2-x;i=0;} 271 else if(x>=1.2316322326660156e+00) {y=x-tc;i=1;} 272 else {y=x-1;i=2;} 273 } 274 switch(i) { 275 case 0: 276 z = y*y; 277 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*(a10+z*(a12+z*(a14+z*(a16+ 278 z*(a18+z*(a20+z*a22)))))))))); 279 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*(a11+z*(a13+z*(a15+ 280 z*(a17+z*(a19+z*(a21+z*a23))))))))))); 281 p = y*p1+p2; 282 r += p-y/2; break; 283 case 1: 284 p = t0+y*t1+tt+y*y*(t2+y*(t3+y*(t4+y*(t5+y*(t6+y*(t7+y*(t8+ 285 y*(t9+y*(t10+y*(t11+y*(t12+y*(t13+y*(t14+y*(t15+y*(t16+ 286 y*(t17+y*(t18+y*(t19+y*(t20+y*(t21+y*(t22+y*(t23+ 287 y*(t24+y*(t25+y*(t26+y*(t27+y*(t28+y*(t29+y*(t30+ 288 y*(t31+y*t32)))))))))))))))))))))))))))))); 289 r += tf + p; break; 290 case 2: 291 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*(u5+y*(u6+y*(u7+ 292 y*(u8+y*(u9+y*u10)))))))))); 293 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*(v5+y*(v6+y*(v7+ 294 y*(v8+y*(v9+y*(v10+y*v11)))))))))); 295 r += p1/p2-y/2; 296 } 297 } 298 /* x < 8.0 */ 299 else if(ix<0x4002) { 300 i = x; 301 y = x-i; 302 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*(s6+y*(s7+y*(s8+ 303 y*(s9+y*(s10+y*s11))))))))))); 304 q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*(r6+y*(r7+y*(r8+ 305 y*(r9+y*(r10+y*r11)))))))))); 306 r = y/2+p/q; 307 z = 1; /* lgamma(1+s) = log(s) + lgamma(s) */ 308 switch(i) { 309 case 7: z *= (y+6); /* FALLTHRU */ 310 case 6: z *= (y+5); /* FALLTHRU */ 311 case 5: z *= (y+4); /* FALLTHRU */ 312 case 4: z *= (y+3); /* FALLTHRU */ 313 case 3: z *= (y+2); /* FALLTHRU */ 314 r += logl(z); break; 315 } 316 /* 8.0 <= x < 2**(p+3) */ 317 } else if (ix<0x3fff+116) { 318 t = logl(x); 319 z = one/x; 320 y = z*z; 321 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*(w6+y*(w7+y*(w8+ 322 y*(w9+y*(w10+y*(w11+y*(w12+y*(w13+y*(w14+y*(w15+y*(w16+ 323 y*(w17+y*w18))))))))))))))))); 324 r = (x-half)*(t-one)+w; 325 /* 2**(p+3) <= x <= inf */ 326 } else 327 r = x*(logl(x)-1); 328 if(hx&0x8000) r = nadj - r; 329 return r; 330 } 331