newlib/libm/ld128/e_lgammal_r.c

   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 "../ld/fpmath.h"
  23 #include "math.h"
  24 #include "../ld/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 }