1 /* e_j1f.c -- float version of e_j1.c. 2 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. 3 */ 4 5 /* 6 * ==================================================== 7 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 8 * 9 * Developed at SunPro, a Sun Microsystems, Inc. business. 10 * Permission to use, copy, modify, and distribute this 11 * software is freely granted, provided that this notice 12 * is preserved. 13 * ==================================================== 14 */ 15 16 #ifndef lint 17 static char rcsid[] = "$Id$"; 18 #endif 19 20 #include "math.h" 21 #include "math_private.h" 22 23 #ifdef __STDC__ 24 static float ponef(float), qonef(float); 25 #else 26 static float ponef(), qonef(); 27 #endif 28 29 #ifdef __STDC__ 30 static const float 31 #else 32 static float 33 #endif 34 huge = 1e30, 35 one = 1.0, 36 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 37 tpi = 6.3661974669e-01, /* 0x3f22f983 */ 38 /* R0/S0 on [0,2] */ 39 r00 = -6.2500000000e-02, /* 0xbd800000 */ 40 r01 = 1.4070566976e-03, /* 0x3ab86cfd */ 41 r02 = -1.5995563444e-05, /* 0xb7862e36 */ 42 r03 = 4.9672799207e-08, /* 0x335557d2 */ 43 s01 = 1.9153760746e-02, /* 0x3c9ce859 */ 44 s02 = 1.8594678841e-04, /* 0x3942fab6 */ 45 s03 = 1.1771846857e-06, /* 0x359dffc2 */ 46 s04 = 5.0463624390e-09, /* 0x31ad6446 */ 47 s05 = 1.2354227016e-11; /* 0x2d59567e */ 48 49 #ifdef __STDC__ 50 static const float zero = 0.0; 51 #else 52 static float zero = 0.0; 53 #endif 54 55 #ifdef __STDC__ 56 float __ieee754_j1f(float x) 57 #else 58 float __ieee754_j1f(x) 59 float x; 60 #endif 61 { 62 float z, s,c,ss,cc,r,u,v,y; 63 int32_t hx,ix; 64 65 GET_FLOAT_WORD(hx,x); 66 ix = hx&0x7fffffff; 67 if(ix>=0x7f800000) return one/x; 68 y = fabsf(x); 69 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 70 s = sinf(y); 71 c = cosf(y); 72 ss = -s-c; 73 cc = s-c; 74 if(ix<0x7f000000) { /* make sure y+y not overflow */ 75 z = cosf(y+y); 76 if ((s*c)>zero) cc = z/ss; 77 else ss = z/cc; 78 } 79 /* 80 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) 81 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) 82 */ 83 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y); 84 else { 85 u = ponef(y); v = qonef(y); 86 z = invsqrtpi*(u*cc-v*ss)/sqrtf(y); 87 } 88 if(hx<0) return -z; 89 else return z; 90 } 91 if(ix<0x32000000) { /* |x|<2**-27 */ 92 if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */ 93 } 94 z = x*x; 95 r = z*(r00+z*(r01+z*(r02+z*r03))); 96 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); 97 r *= x; 98 return(x*(float)0.5+r/s); 99 } 100 101 #ifdef __STDC__ 102 static const float U0[5] = { 103 #else 104 static float U0[5] = { 105 #endif 106 -1.9605709612e-01, /* 0xbe48c331 */ 107 5.0443872809e-02, /* 0x3d4e9e3c */ 108 -1.9125689287e-03, /* 0xbafaaf2a */ 109 2.3525259166e-05, /* 0x37c5581c */ 110 -9.1909917899e-08, /* 0xb3c56003 */ 111 }; 112 #ifdef __STDC__ 113 static const float V0[5] = { 114 #else 115 static float V0[5] = { 116 #endif 117 1.9916731864e-02, /* 0x3ca3286a */ 118 2.0255257550e-04, /* 0x3954644b */ 119 1.3560879779e-06, /* 0x35b602d4 */ 120 6.2274145840e-09, /* 0x31d5f8eb */ 121 1.6655924903e-11, /* 0x2d9281cf */ 122 }; 123 124 #ifdef __STDC__ 125 float __ieee754_y1f(float x) 126 #else 127 float __ieee754_y1f(x) 128 float x; 129 #endif 130 { 131 float z, s,c,ss,cc,u,v; 132 int32_t hx,ix; 133 134 GET_FLOAT_WORD(hx,x); 135 ix = 0x7fffffff&hx; 136 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ 137 if(ix>=0x7f800000) return one/(x+x*x); 138 if(ix==0) return -one/zero; 139 if(hx<0) return zero/zero; 140 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 141 s = sinf(x); 142 c = cosf(x); 143 ss = -s-c; 144 cc = s-c; 145 if(ix<0x7f000000) { /* make sure x+x not overflow */ 146 z = cosf(x+x); 147 if ((s*c)>zero) cc = z/ss; 148 else ss = z/cc; 149 } 150 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) 151 * where x0 = x-3pi/4 152 * Better formula: 153 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 154 * = 1/sqrt(2) * (sin(x) - cos(x)) 155 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 156 * = -1/sqrt(2) * (cos(x) + sin(x)) 157 * To avoid cancellation, use 158 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 159 * to compute the worse one. 160 */ 161 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x); 162 else { 163 u = ponef(x); v = qonef(x); 164 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 165 } 166 return z; 167 } 168 if(ix<=0x24800000) { /* x < 2**-54 */ 169 return(-tpi/x); 170 } 171 z = x*x; 172 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); 173 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); 174 return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x)); 175 } 176 177 /* For x >= 8, the asymptotic expansions of pone is 178 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. 179 * We approximate pone by 180 * pone(x) = 1 + (R/S) 181 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 182 * S = 1 + ps0*s^2 + ... + ps4*s^10 183 * and 184 * | pone(x)-1-R/S | <= 2 ** ( -60.06) 185 */ 186 187 #ifdef __STDC__ 188 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 189 #else 190 static float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 191 #endif 192 0.0000000000e+00, /* 0x00000000 */ 193 1.1718750000e-01, /* 0x3df00000 */ 194 1.3239480972e+01, /* 0x4153d4ea */ 195 4.1205184937e+02, /* 0x43ce06a3 */ 196 3.8747453613e+03, /* 0x45722bed */ 197 7.9144794922e+03, /* 0x45f753d6 */ 198 }; 199 #ifdef __STDC__ 200 static const float ps8[5] = { 201 #else 202 static float ps8[5] = { 203 #endif 204 1.1420736694e+02, /* 0x42e46a2c */ 205 3.6509309082e+03, /* 0x45642ee5 */ 206 3.6956207031e+04, /* 0x47105c35 */ 207 9.7602796875e+04, /* 0x47bea166 */ 208 3.0804271484e+04, /* 0x46f0a88b */ 209 }; 210 211 #ifdef __STDC__ 212 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 213 #else 214 static float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 215 #endif 216 1.3199052094e-11, /* 0x2d68333f */ 217 1.1718749255e-01, /* 0x3defffff */ 218 6.8027510643e+00, /* 0x40d9b023 */ 219 1.0830818176e+02, /* 0x42d89dca */ 220 5.1763616943e+02, /* 0x440168b7 */ 221 5.2871520996e+02, /* 0x44042dc6 */ 222 }; 223 #ifdef __STDC__ 224 static const float ps5[5] = { 225 #else 226 static float ps5[5] = { 227 #endif 228 5.9280597687e+01, /* 0x426d1f55 */ 229 9.9140142822e+02, /* 0x4477d9b1 */ 230 5.3532670898e+03, /* 0x45a74a23 */ 231 7.8446904297e+03, /* 0x45f52586 */ 232 1.5040468750e+03, /* 0x44bc0180 */ 233 }; 234 235 #ifdef __STDC__ 236 static const float pr3[6] = { 237 #else 238 static float pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 239 #endif 240 3.0250391081e-09, /* 0x314fe10d */ 241 1.1718686670e-01, /* 0x3defffab */ 242 3.9329774380e+00, /* 0x407bb5e7 */ 243 3.5119403839e+01, /* 0x420c7a45 */ 244 9.1055007935e+01, /* 0x42b61c2a */ 245 4.8559066772e+01, /* 0x42423c7c */ 246 }; 247 #ifdef __STDC__ 248 static const float ps3[5] = { 249 #else 250 static float ps3[5] = { 251 #endif 252 3.4791309357e+01, /* 0x420b2a4d */ 253 3.3676245117e+02, /* 0x43a86198 */ 254 1.0468714600e+03, /* 0x4482dbe3 */ 255 8.9081134033e+02, /* 0x445eb3ed */ 256 1.0378793335e+02, /* 0x42cf936c */ 257 }; 258 259 #ifdef __STDC__ 260 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 261 #else 262 static float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 263 #endif 264 1.0771083225e-07, /* 0x33e74ea8 */ 265 1.1717621982e-01, /* 0x3deffa16 */ 266 2.3685150146e+00, /* 0x401795c0 */ 267 1.2242610931e+01, /* 0x4143e1bc */ 268 1.7693971634e+01, /* 0x418d8d41 */ 269 5.0735230446e+00, /* 0x40a25a4d */ 270 }; 271 #ifdef __STDC__ 272 static const float ps2[5] = { 273 #else 274 static float ps2[5] = { 275 #endif 276 2.1436485291e+01, /* 0x41ab7dec */ 277 1.2529022980e+02, /* 0x42fa9499 */ 278 2.3227647400e+02, /* 0x436846c7 */ 279 1.1767937469e+02, /* 0x42eb5bd7 */ 280 8.3646392822e+00, /* 0x4105d590 */ 281 }; 282 283 #ifdef __STDC__ 284 static float ponef(float x) 285 #else 286 static float ponef(x) 287 float x; 288 #endif 289 { 290 #ifdef __STDC__ 291 const float *p,*q; 292 #else 293 float *p,*q; 294 #endif 295 float z,r,s; 296 int32_t ix; 297 GET_FLOAT_WORD(ix,x); 298 ix &= 0x7fffffff; 299 if(ix>=0x41000000) {p = pr8; q= ps8;} 300 else if(ix>=0x40f71c58){p = pr5; q= ps5;} 301 else if(ix>=0x4036db68){p = pr3; q= ps3;} 302 else if(ix>=0x40000000){p = pr2; q= ps2;} 303 z = one/(x*x); 304 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 305 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 306 return one+ r/s; 307 } 308 309 310 /* For x >= 8, the asymptotic expansions of qone is 311 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. 312 * We approximate pone by 313 * qone(x) = s*(0.375 + (R/S)) 314 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 315 * S = 1 + qs1*s^2 + ... + qs6*s^12 316 * and 317 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) 318 */ 319 320 #ifdef __STDC__ 321 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 322 #else 323 static float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 324 #endif 325 0.0000000000e+00, /* 0x00000000 */ 326 -1.0253906250e-01, /* 0xbdd20000 */ 327 -1.6271753311e+01, /* 0xc1822c8d */ 328 -7.5960174561e+02, /* 0xc43de683 */ 329 -1.1849806641e+04, /* 0xc639273a */ 330 -4.8438511719e+04, /* 0xc73d3683 */ 331 }; 332 #ifdef __STDC__ 333 static const float qs8[6] = { 334 #else 335 static float qs8[6] = { 336 #endif 337 1.6139537048e+02, /* 0x43216537 */ 338 7.8253862305e+03, /* 0x45f48b17 */ 339 1.3387534375e+05, /* 0x4802bcd6 */ 340 7.1965775000e+05, /* 0x492fb29c */ 341 6.6660125000e+05, /* 0x4922be94 */ 342 -2.9449025000e+05, /* 0xc88fcb48 */ 343 }; 344 345 #ifdef __STDC__ 346 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 347 #else 348 static float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 349 #endif 350 -2.0897993405e-11, /* 0xadb7d219 */ 351 -1.0253904760e-01, /* 0xbdd1fffe */ 352 -8.0564479828e+00, /* 0xc100e736 */ 353 -1.8366960144e+02, /* 0xc337ab6b */ 354 -1.3731937256e+03, /* 0xc4aba633 */ 355 -2.6124443359e+03, /* 0xc523471c */ 356 }; 357 #ifdef __STDC__ 358 static const float qs5[6] = { 359 #else 360 static float qs5[6] = { 361 #endif 362 8.1276550293e+01, /* 0x42a28d98 */ 363 1.9917987061e+03, /* 0x44f8f98f */ 364 1.7468484375e+04, /* 0x468878f8 */ 365 4.9851425781e+04, /* 0x4742bb6d */ 366 2.7948074219e+04, /* 0x46da5826 */ 367 -4.7191835938e+03, /* 0xc5937978 */ 368 }; 369 370 #ifdef __STDC__ 371 static const float qr3[6] = { 372 #else 373 static float qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 374 #endif 375 -5.0783124372e-09, /* 0xb1ae7d4f */ 376 -1.0253783315e-01, /* 0xbdd1ff5b */ 377 -4.6101160049e+00, /* 0xc0938612 */ 378 -5.7847221375e+01, /* 0xc267638e */ 379 -2.2824453735e+02, /* 0xc3643e9a */ 380 -2.1921012878e+02, /* 0xc35b35cb */ 381 }; 382 #ifdef __STDC__ 383 static const float qs3[6] = { 384 #else 385 static float qs3[6] = { 386 #endif 387 4.7665153503e+01, /* 0x423ea91e */ 388 6.7386511230e+02, /* 0x4428775e */ 389 3.3801528320e+03, /* 0x45534272 */ 390 5.5477290039e+03, /* 0x45ad5dd5 */ 391 1.9031191406e+03, /* 0x44ede3d0 */ 392 -1.3520118713e+02, /* 0xc3073381 */ 393 }; 394 395 #ifdef __STDC__ 396 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 397 #else 398 static float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 399 #endif 400 -1.7838172539e-07, /* 0xb43f8932 */ 401 -1.0251704603e-01, /* 0xbdd1f475 */ 402 -2.7522056103e+00, /* 0xc0302423 */ 403 -1.9663616180e+01, /* 0xc19d4f16 */ 404 -4.2325313568e+01, /* 0xc2294d1f */ 405 -2.1371921539e+01, /* 0xc1aaf9b2 */ 406 }; 407 #ifdef __STDC__ 408 static const float qs2[6] = { 409 #else 410 static float qs2[6] = { 411 #endif 412 2.9533363342e+01, /* 0x41ec4454 */ 413 2.5298155212e+02, /* 0x437cfb47 */ 414 7.5750280762e+02, /* 0x443d602e */ 415 7.3939318848e+02, /* 0x4438d92a */ 416 1.5594900513e+02, /* 0x431bf2f2 */ 417 -4.9594988823e+00, /* 0xc09eb437 */ 418 }; 419 420 #ifdef __STDC__ 421 static float qonef(float x) 422 #else 423 static float qonef(x) 424 float x; 425 #endif 426 { 427 #ifdef __STDC__ 428 const float *p,*q; 429 #else 430 float *p,*q; 431 #endif 432 float s,r,z; 433 int32_t ix; 434 GET_FLOAT_WORD(ix,x); 435 ix &= 0x7fffffff; 436 if(ix>=0x40200000) {p = qr8; q= qs8;} 437 else if(ix>=0x40f71c58){p = qr5; q= qs5;} 438 else if(ix>=0x4036db68){p = qr3; q= qs3;} 439 else if(ix>=0x40000000){p = qr2; q= qs2;} 440 z = one/(x*x); 441 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 442 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 443 return ((float).375 + r/s)/x; 444 } 445