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 #include <sys/cdefs.h> 17 #if defined(LIBM_SCCS) && !defined(lint) 18 __RCSID("$NetBSD: e_j1f.c,v 1.5 1997/10/09 11:29:16 lukem Exp $"); 19 #endif 20 21 #include "math.h" 22 #include "math_private.h" 23 24 #ifdef __STDC__ 25 static float ponef(float), qonef(float); 26 #else 27 static float ponef(), qonef(); 28 #endif 29 30 #ifdef __STDC__ 31 static const float 32 #else 33 static float 34 #endif 35 huge = 1e30, 36 one = 1.0, 37 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 38 tpi = 6.3661974669e-01, /* 0x3f22f983 */ 39 /* R0/S0 on [0,2] */ 40 r00 = -6.2500000000e-02, /* 0xbd800000 */ 41 r01 = 1.4070566976e-03, /* 0x3ab86cfd */ 42 r02 = -1.5995563444e-05, /* 0xb7862e36 */ 43 r03 = 4.9672799207e-08, /* 0x335557d2 */ 44 s01 = 1.9153760746e-02, /* 0x3c9ce859 */ 45 s02 = 1.8594678841e-04, /* 0x3942fab6 */ 46 s03 = 1.1771846857e-06, /* 0x359dffc2 */ 47 s04 = 5.0463624390e-09, /* 0x31ad6446 */ 48 s05 = 1.2354227016e-11; /* 0x2d59567e */ 49 50 #ifdef __STDC__ 51 static const float zero = 0.0; 52 #else 53 static float zero = 0.0; 54 #endif 55 56 #ifdef __STDC__ 57 float __ieee754_j1f(float x) 58 #else 59 float __ieee754_j1f(x) 60 float x; 61 #endif 62 { 63 float z, s,c,ss,cc,r,u,v,y; 64 int32_t hx,ix; 65 66 GET_FLOAT_WORD(hx,x); 67 ix = hx&0x7fffffff; 68 if(ix>=0x7f800000) return one/x; 69 y = fabsf(x); 70 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 71 s = sinf(y); 72 c = cosf(y); 73 ss = -s-c; 74 cc = s-c; 75 if(ix<0x7f000000) { /* make sure y+y not overflow */ 76 z = cosf(y+y); 77 if ((s*c)>zero) cc = z/ss; 78 else ss = z/cc; 79 } 80 /* 81 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) 82 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) 83 */ 84 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y); 85 else { 86 u = ponef(y); v = qonef(y); 87 z = invsqrtpi*(u*cc-v*ss)/sqrtf(y); 88 } 89 if(hx<0) return -z; 90 else return z; 91 } 92 if(ix<0x32000000) { /* |x|<2**-27 */ 93 if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */ 94 } 95 z = x*x; 96 r = z*(r00+z*(r01+z*(r02+z*r03))); 97 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); 98 r *= x; 99 return(x*(float)0.5+r/s); 100 } 101 102 #ifdef __STDC__ 103 static const float U0[5] = { 104 #else 105 static float U0[5] = { 106 #endif 107 -1.9605709612e-01, /* 0xbe48c331 */ 108 5.0443872809e-02, /* 0x3d4e9e3c */ 109 -1.9125689287e-03, /* 0xbafaaf2a */ 110 2.3525259166e-05, /* 0x37c5581c */ 111 -9.1909917899e-08, /* 0xb3c56003 */ 112 }; 113 #ifdef __STDC__ 114 static const float V0[5] = { 115 #else 116 static float V0[5] = { 117 #endif 118 1.9916731864e-02, /* 0x3ca3286a */ 119 2.0255257550e-04, /* 0x3954644b */ 120 1.3560879779e-06, /* 0x35b602d4 */ 121 6.2274145840e-09, /* 0x31d5f8eb */ 122 1.6655924903e-11, /* 0x2d9281cf */ 123 }; 124 125 #ifdef __STDC__ 126 float __ieee754_y1f(float x) 127 #else 128 float __ieee754_y1f(x) 129 float x; 130 #endif 131 { 132 float z, s,c,ss,cc,u,v; 133 int32_t hx,ix; 134 135 GET_FLOAT_WORD(hx,x); 136 ix = 0x7fffffff&hx; 137 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ 138 if(ix>=0x7f800000) return one/(x+x*x); 139 if(ix==0) return -one/zero; 140 if(hx<0) return zero/zero; 141 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 142 s = sinf(x); 143 c = cosf(x); 144 ss = -s-c; 145 cc = s-c; 146 if(ix<0x7f000000) { /* make sure x+x not overflow */ 147 z = cosf(x+x); 148 if ((s*c)>zero) cc = z/ss; 149 else ss = z/cc; 150 } 151 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) 152 * where x0 = x-3pi/4 153 * Better formula: 154 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 155 * = 1/sqrt(2) * (sin(x) - cos(x)) 156 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 157 * = -1/sqrt(2) * (cos(x) + sin(x)) 158 * To avoid cancellation, use 159 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 160 * to compute the worse one. 161 */ 162 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x); 163 else { 164 u = ponef(x); v = qonef(x); 165 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 166 } 167 return z; 168 } 169 if(ix<=0x24800000) { /* x < 2**-54 */ 170 return(-tpi/x); 171 } 172 z = x*x; 173 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); 174 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); 175 return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x)); 176 } 177 178 /* For x >= 8, the asymptotic expansions of pone is 179 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. 180 * We approximate pone by 181 * pone(x) = 1 + (R/S) 182 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 183 * S = 1 + ps0*s^2 + ... + ps4*s^10 184 * and 185 * | pone(x)-1-R/S | <= 2 ** ( -60.06) 186 */ 187 188 #ifdef __STDC__ 189 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 190 #else 191 static float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 192 #endif 193 0.0000000000e+00, /* 0x00000000 */ 194 1.1718750000e-01, /* 0x3df00000 */ 195 1.3239480972e+01, /* 0x4153d4ea */ 196 4.1205184937e+02, /* 0x43ce06a3 */ 197 3.8747453613e+03, /* 0x45722bed */ 198 7.9144794922e+03, /* 0x45f753d6 */ 199 }; 200 #ifdef __STDC__ 201 static const float ps8[5] = { 202 #else 203 static float ps8[5] = { 204 #endif 205 1.1420736694e+02, /* 0x42e46a2c */ 206 3.6509309082e+03, /* 0x45642ee5 */ 207 3.6956207031e+04, /* 0x47105c35 */ 208 9.7602796875e+04, /* 0x47bea166 */ 209 3.0804271484e+04, /* 0x46f0a88b */ 210 }; 211 212 #ifdef __STDC__ 213 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 214 #else 215 static float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 216 #endif 217 1.3199052094e-11, /* 0x2d68333f */ 218 1.1718749255e-01, /* 0x3defffff */ 219 6.8027510643e+00, /* 0x40d9b023 */ 220 1.0830818176e+02, /* 0x42d89dca */ 221 5.1763616943e+02, /* 0x440168b7 */ 222 5.2871520996e+02, /* 0x44042dc6 */ 223 }; 224 #ifdef __STDC__ 225 static const float ps5[5] = { 226 #else 227 static float ps5[5] = { 228 #endif 229 5.9280597687e+01, /* 0x426d1f55 */ 230 9.9140142822e+02, /* 0x4477d9b1 */ 231 5.3532670898e+03, /* 0x45a74a23 */ 232 7.8446904297e+03, /* 0x45f52586 */ 233 1.5040468750e+03, /* 0x44bc0180 */ 234 }; 235 236 #ifdef __STDC__ 237 static const float pr3[6] = { 238 #else 239 static float pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 240 #endif 241 3.0250391081e-09, /* 0x314fe10d */ 242 1.1718686670e-01, /* 0x3defffab */ 243 3.9329774380e+00, /* 0x407bb5e7 */ 244 3.5119403839e+01, /* 0x420c7a45 */ 245 9.1055007935e+01, /* 0x42b61c2a */ 246 4.8559066772e+01, /* 0x42423c7c */ 247 }; 248 #ifdef __STDC__ 249 static const float ps3[5] = { 250 #else 251 static float ps3[5] = { 252 #endif 253 3.4791309357e+01, /* 0x420b2a4d */ 254 3.3676245117e+02, /* 0x43a86198 */ 255 1.0468714600e+03, /* 0x4482dbe3 */ 256 8.9081134033e+02, /* 0x445eb3ed */ 257 1.0378793335e+02, /* 0x42cf936c */ 258 }; 259 260 #ifdef __STDC__ 261 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 262 #else 263 static float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 264 #endif 265 1.0771083225e-07, /* 0x33e74ea8 */ 266 1.1717621982e-01, /* 0x3deffa16 */ 267 2.3685150146e+00, /* 0x401795c0 */ 268 1.2242610931e+01, /* 0x4143e1bc */ 269 1.7693971634e+01, /* 0x418d8d41 */ 270 5.0735230446e+00, /* 0x40a25a4d */ 271 }; 272 #ifdef __STDC__ 273 static const float ps2[5] = { 274 #else 275 static float ps2[5] = { 276 #endif 277 2.1436485291e+01, /* 0x41ab7dec */ 278 1.2529022980e+02, /* 0x42fa9499 */ 279 2.3227647400e+02, /* 0x436846c7 */ 280 1.1767937469e+02, /* 0x42eb5bd7 */ 281 8.3646392822e+00, /* 0x4105d590 */ 282 }; 283 284 #ifdef __STDC__ 285 static float ponef(float x) 286 #else 287 static float ponef(x) 288 float x; 289 #endif 290 { 291 #ifdef __STDC__ 292 const float *p,*q; 293 #else 294 float *p,*q; 295 #endif 296 float z,r,s; 297 int32_t ix; 298 299 p = q = 0; 300 GET_FLOAT_WORD(ix,x); 301 ix &= 0x7fffffff; 302 if(ix>=0x41000000) {p = pr8; q= ps8;} 303 else if(ix>=0x40f71c58){p = pr5; q= ps5;} 304 else if(ix>=0x4036db68){p = pr3; q= ps3;} 305 else if(ix>=0x40000000){p = pr2; q= ps2;} 306 z = one/(x*x); 307 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 308 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 309 return one+ r/s; 310 } 311 312 313 /* For x >= 8, the asymptotic expansions of qone is 314 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. 315 * We approximate pone by 316 * qone(x) = s*(0.375 + (R/S)) 317 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 318 * S = 1 + qs1*s^2 + ... + qs6*s^12 319 * and 320 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) 321 */ 322 323 #ifdef __STDC__ 324 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 325 #else 326 static float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 327 #endif 328 0.0000000000e+00, /* 0x00000000 */ 329 -1.0253906250e-01, /* 0xbdd20000 */ 330 -1.6271753311e+01, /* 0xc1822c8d */ 331 -7.5960174561e+02, /* 0xc43de683 */ 332 -1.1849806641e+04, /* 0xc639273a */ 333 -4.8438511719e+04, /* 0xc73d3683 */ 334 }; 335 #ifdef __STDC__ 336 static const float qs8[6] = { 337 #else 338 static float qs8[6] = { 339 #endif 340 1.6139537048e+02, /* 0x43216537 */ 341 7.8253862305e+03, /* 0x45f48b17 */ 342 1.3387534375e+05, /* 0x4802bcd6 */ 343 7.1965775000e+05, /* 0x492fb29c */ 344 6.6660125000e+05, /* 0x4922be94 */ 345 -2.9449025000e+05, /* 0xc88fcb48 */ 346 }; 347 348 #ifdef __STDC__ 349 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 350 #else 351 static float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 352 #endif 353 -2.0897993405e-11, /* 0xadb7d219 */ 354 -1.0253904760e-01, /* 0xbdd1fffe */ 355 -8.0564479828e+00, /* 0xc100e736 */ 356 -1.8366960144e+02, /* 0xc337ab6b */ 357 -1.3731937256e+03, /* 0xc4aba633 */ 358 -2.6124443359e+03, /* 0xc523471c */ 359 }; 360 #ifdef __STDC__ 361 static const float qs5[6] = { 362 #else 363 static float qs5[6] = { 364 #endif 365 8.1276550293e+01, /* 0x42a28d98 */ 366 1.9917987061e+03, /* 0x44f8f98f */ 367 1.7468484375e+04, /* 0x468878f8 */ 368 4.9851425781e+04, /* 0x4742bb6d */ 369 2.7948074219e+04, /* 0x46da5826 */ 370 -4.7191835938e+03, /* 0xc5937978 */ 371 }; 372 373 #ifdef __STDC__ 374 static const float qr3[6] = { 375 #else 376 static float qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 377 #endif 378 -5.0783124372e-09, /* 0xb1ae7d4f */ 379 -1.0253783315e-01, /* 0xbdd1ff5b */ 380 -4.6101160049e+00, /* 0xc0938612 */ 381 -5.7847221375e+01, /* 0xc267638e */ 382 -2.2824453735e+02, /* 0xc3643e9a */ 383 -2.1921012878e+02, /* 0xc35b35cb */ 384 }; 385 #ifdef __STDC__ 386 static const float qs3[6] = { 387 #else 388 static float qs3[6] = { 389 #endif 390 4.7665153503e+01, /* 0x423ea91e */ 391 6.7386511230e+02, /* 0x4428775e */ 392 3.3801528320e+03, /* 0x45534272 */ 393 5.5477290039e+03, /* 0x45ad5dd5 */ 394 1.9031191406e+03, /* 0x44ede3d0 */ 395 -1.3520118713e+02, /* 0xc3073381 */ 396 }; 397 398 #ifdef __STDC__ 399 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 400 #else 401 static float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 402 #endif 403 -1.7838172539e-07, /* 0xb43f8932 */ 404 -1.0251704603e-01, /* 0xbdd1f475 */ 405 -2.7522056103e+00, /* 0xc0302423 */ 406 -1.9663616180e+01, /* 0xc19d4f16 */ 407 -4.2325313568e+01, /* 0xc2294d1f */ 408 -2.1371921539e+01, /* 0xc1aaf9b2 */ 409 }; 410 #ifdef __STDC__ 411 static const float qs2[6] = { 412 #else 413 static float qs2[6] = { 414 #endif 415 2.9533363342e+01, /* 0x41ec4454 */ 416 2.5298155212e+02, /* 0x437cfb47 */ 417 7.5750280762e+02, /* 0x443d602e */ 418 7.3939318848e+02, /* 0x4438d92a */ 419 1.5594900513e+02, /* 0x431bf2f2 */ 420 -4.9594988823e+00, /* 0xc09eb437 */ 421 }; 422 423 #ifdef __STDC__ 424 static float qonef(float x) 425 #else 426 static float qonef(x) 427 float x; 428 #endif 429 { 430 #ifdef __STDC__ 431 const float *p,*q; 432 #else 433 float *p,*q; 434 #endif 435 float s,r,z; 436 int32_t ix; 437 438 p = q = 0; 439 GET_FLOAT_WORD(ix,x); 440 ix &= 0x7fffffff; 441 if(ix>=0x40200000) {p = qr8; q= qs8;} 442 else if(ix>=0x40f71c58){p = qr5; q= qs5;} 443 else if(ix>=0x4036db68){p = qr3; q= qs3;} 444 else if(ix>=0x40000000){p = qr2; q= qs2;} 445 z = one/(x*x); 446 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 447 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 448 return ((float).375 + r/s)/x; 449 } 450