1 /* e_j0f.c -- float version of e_j0.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: e_j0f.c,v 1.1.1.1 1994/08/19 09:39:55 jkh Exp $"; 18 #endif 19 20 #include "math.h" 21 #include "math_private.h" 22 23 #ifdef __STDC__ 24 static float pzerof(float), qzerof(float); 25 #else 26 static float pzerof(), qzerof(); 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, 2.00] */ 39 R02 = 1.5625000000e-02, /* 0x3c800000 */ 40 R03 = -1.8997929874e-04, /* 0xb947352e */ 41 R04 = 1.8295404516e-06, /* 0x35f58e88 */ 42 R05 = -4.6183270541e-09, /* 0xb19eaf3c */ 43 S01 = 1.5619102865e-02, /* 0x3c7fe744 */ 44 S02 = 1.1692678527e-04, /* 0x38f53697 */ 45 S03 = 5.1354652442e-07, /* 0x3509daa6 */ 46 S04 = 1.1661400734e-09; /* 0x30a045e8 */ 47 48 #ifdef __STDC__ 49 static const float zero = 0.0; 50 #else 51 static float zero = 0.0; 52 #endif 53 54 #ifdef __STDC__ 55 float __ieee754_j0f(float x) 56 #else 57 float __ieee754_j0f(x) 58 float x; 59 #endif 60 { 61 float z, s,c,ss,cc,r,u,v; 62 int32_t hx,ix; 63 64 GET_FLOAT_WORD(hx,x); 65 ix = hx&0x7fffffff; 66 if(ix>=0x7f800000) return one/(x*x); 67 x = fabsf(x); 68 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 69 s = sinf(x); 70 c = cosf(x); 71 ss = s-c; 72 cc = s+c; 73 if(ix<0x7f000000) { /* make sure x+x not overflow */ 74 z = -cosf(x+x); 75 if ((s*c)<zero) cc = z/ss; 76 else ss = z/cc; 77 } 78 /* 79 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 80 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 81 */ 82 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(x); 83 else { 84 u = pzerof(x); v = qzerof(x); 85 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x); 86 } 87 return z; 88 } 89 if(ix<0x39000000) { /* |x| < 2**-13 */ 90 if(huge+x>one) { /* raise inexact if x != 0 */ 91 if(ix<0x32000000) return one; /* |x|<2**-27 */ 92 else return one - (float)0.25*x*x; 93 } 94 } 95 z = x*x; 96 r = z*(R02+z*(R03+z*(R04+z*R05))); 97 s = one+z*(S01+z*(S02+z*(S03+z*S04))); 98 if(ix < 0x3F800000) { /* |x| < 1.00 */ 99 return one + z*((float)-0.25+(r/s)); 100 } else { 101 u = (float)0.5*x; 102 return((one+u)*(one-u)+z*(r/s)); 103 } 104 } 105 106 #ifdef __STDC__ 107 static const float 108 #else 109 static float 110 #endif 111 u00 = -7.3804296553e-02, /* 0xbd9726b5 */ 112 u01 = 1.7666645348e-01, /* 0x3e34e80d */ 113 u02 = -1.3818567619e-02, /* 0xbc626746 */ 114 u03 = 3.4745343146e-04, /* 0x39b62a69 */ 115 u04 = -3.8140706238e-06, /* 0xb67ff53c */ 116 u05 = 1.9559013964e-08, /* 0x32a802ba */ 117 u06 = -3.9820518410e-11, /* 0xae2f21eb */ 118 v01 = 1.2730483897e-02, /* 0x3c509385 */ 119 v02 = 7.6006865129e-05, /* 0x389f65e0 */ 120 v03 = 2.5915085189e-07, /* 0x348b216c */ 121 v04 = 4.4111031494e-10; /* 0x2ff280c2 */ 122 123 #ifdef __STDC__ 124 float __ieee754_y0f(float x) 125 #else 126 float __ieee754_y0f(x) 127 float x; 128 #endif 129 { 130 float z, s,c,ss,cc,u,v; 131 int32_t hx,ix; 132 133 GET_FLOAT_WORD(hx,x); 134 ix = 0x7fffffff&hx; 135 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */ 136 if(ix>=0x7f800000) return one/(x+x*x); 137 if(ix==0) return -one/zero; 138 if(hx<0) return zero/zero; 139 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 140 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) 141 * where x0 = x-pi/4 142 * Better formula: 143 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 144 * = 1/sqrt(2) * (sin(x) + cos(x)) 145 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 146 * = 1/sqrt(2) * (sin(x) - cos(x)) 147 * To avoid cancellation, use 148 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 149 * to compute the worse one. 150 */ 151 s = sinf(x); 152 c = cosf(x); 153 ss = s-c; 154 cc = s+c; 155 /* 156 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 157 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 158 */ 159 if(ix<0x7f000000) { /* make sure x+x not overflow */ 160 z = -cosf(x+x); 161 if ((s*c)<zero) cc = z/ss; 162 else ss = z/cc; 163 } 164 if(ix>0x80000000) z = (invsqrtpi*ss)/sqrtf(x); 165 else { 166 u = pzerof(x); v = qzerof(x); 167 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 168 } 169 return z; 170 } 171 if(ix<=0x32000000) { /* x < 2**-27 */ 172 return(u00 + tpi*__ieee754_logf(x)); 173 } 174 z = x*x; 175 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 176 v = one+z*(v01+z*(v02+z*(v03+z*v04))); 177 return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x))); 178 } 179 180 /* The asymptotic expansions of pzero is 181 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 182 * For x >= 2, We approximate pzero by 183 * pzero(x) = 1 + (R/S) 184 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 185 * S = 1 + pS0*s^2 + ... + pS4*s^10 186 * and 187 * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 188 */ 189 #ifdef __STDC__ 190 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 191 #else 192 static float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 193 #endif 194 0.0000000000e+00, /* 0x00000000 */ 195 -7.0312500000e-02, /* 0xbd900000 */ 196 -8.0816707611e+00, /* 0xc1014e86 */ 197 -2.5706311035e+02, /* 0xc3808814 */ 198 -2.4852163086e+03, /* 0xc51b5376 */ 199 -5.2530439453e+03, /* 0xc5a4285a */ 200 }; 201 #ifdef __STDC__ 202 static const float pS8[5] = { 203 #else 204 static float pS8[5] = { 205 #endif 206 1.1653436279e+02, /* 0x42e91198 */ 207 3.8337448730e+03, /* 0x456f9beb */ 208 4.0597855469e+04, /* 0x471e95db */ 209 1.1675296875e+05, /* 0x47e4087c */ 210 4.7627726562e+04, /* 0x473a0bba */ 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 -1.1412546255e-11, /* 0xad48c58a */ 218 -7.0312492549e-02, /* 0xbd8fffff */ 219 -4.1596107483e+00, /* 0xc0851b88 */ 220 -6.7674766541e+01, /* 0xc287597b */ 221 -3.3123129272e+02, /* 0xc3a59d9b */ 222 -3.4643338013e+02, /* 0xc3ad3779 */ 223 }; 224 #ifdef __STDC__ 225 static const float pS5[5] = { 226 #else 227 static float pS5[5] = { 228 #endif 229 6.0753936768e+01, /* 0x42730408 */ 230 1.0512523193e+03, /* 0x44836813 */ 231 5.9789707031e+03, /* 0x45bad7c4 */ 232 9.6254453125e+03, /* 0x461665c8 */ 233 2.4060581055e+03, /* 0x451660ee */ 234 }; 235 236 #ifdef __STDC__ 237 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 238 #else 239 static float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 240 #endif 241 -2.5470459075e-09, /* 0xb12f081b */ 242 -7.0311963558e-02, /* 0xbd8fffb8 */ 243 -2.4090321064e+00, /* 0xc01a2d95 */ 244 -2.1965976715e+01, /* 0xc1afba52 */ 245 -5.8079170227e+01, /* 0xc2685112 */ 246 -3.1447946548e+01, /* 0xc1fb9565 */ 247 }; 248 #ifdef __STDC__ 249 static const float pS3[5] = { 250 #else 251 static float pS3[5] = { 252 #endif 253 3.5856033325e+01, /* 0x420f6c94 */ 254 3.6151397705e+02, /* 0x43b4c1ca */ 255 1.1936077881e+03, /* 0x44953373 */ 256 1.1279968262e+03, /* 0x448cffe6 */ 257 1.7358093262e+02, /* 0x432d94b8 */ 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 -8.8753431271e-08, /* 0xb3be98b7 */ 266 -7.0303097367e-02, /* 0xbd8ffb12 */ 267 -1.4507384300e+00, /* 0xbfb9b1cc */ 268 -7.6356959343e+00, /* 0xc0f4579f */ 269 -1.1193166733e+01, /* 0xc1331736 */ 270 -3.2336456776e+00, /* 0xc04ef40d */ 271 }; 272 #ifdef __STDC__ 273 static const float pS2[5] = { 274 #else 275 static float pS2[5] = { 276 #endif 277 2.2220300674e+01, /* 0x41b1c32d */ 278 1.3620678711e+02, /* 0x430834f0 */ 279 2.7047027588e+02, /* 0x43873c32 */ 280 1.5387539673e+02, /* 0x4319e01a */ 281 1.4657617569e+01, /* 0x416a859a */ 282 }; 283 284 #ifdef __STDC__ 285 static float pzerof(float x) 286 #else 287 static float pzerof(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 GET_FLOAT_WORD(ix,x); 299 ix &= 0x7fffffff; 300 if(ix>=0x41000000) {p = pR8; q= pS8;} 301 else if(ix>=0x40f71c58){p = pR5; q= pS5;} 302 else if(ix>=0x4036db68){p = pR3; q= pS3;} 303 else if(ix>=0x40000000){p = pR2; q= pS2;} 304 z = one/(x*x); 305 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 306 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 307 return one+ r/s; 308 } 309 310 311 /* For x >= 8, the asymptotic expansions of qzero is 312 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 313 * We approximate pzero by 314 * qzero(x) = s*(-1.25 + (R/S)) 315 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 316 * S = 1 + qS0*s^2 + ... + qS5*s^12 317 * and 318 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 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 7.3242187500e-02, /* 0x3d960000 */ 327 1.1768206596e+01, /* 0x413c4a93 */ 328 5.5767340088e+02, /* 0x440b6b19 */ 329 8.8591972656e+03, /* 0x460a6cca */ 330 3.7014625000e+04, /* 0x471096a0 */ 331 }; 332 #ifdef __STDC__ 333 static const float qS8[6] = { 334 #else 335 static float qS8[6] = { 336 #endif 337 1.6377603149e+02, /* 0x4323c6aa */ 338 8.0983447266e+03, /* 0x45fd12c2 */ 339 1.4253829688e+05, /* 0x480b3293 */ 340 8.0330925000e+05, /* 0x49441ed4 */ 341 8.4050156250e+05, /* 0x494d3359 */ 342 -3.4389928125e+05, /* 0xc8a7eb69 */ 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 1.8408595828e-11, /* 0x2da1ec79 */ 351 7.3242180049e-02, /* 0x3d95ffff */ 352 5.8356351852e+00, /* 0x40babd86 */ 353 1.3511157227e+02, /* 0x43071c90 */ 354 1.0272437744e+03, /* 0x448067cd */ 355 1.9899779053e+03, /* 0x44f8bf4b */ 356 }; 357 #ifdef __STDC__ 358 static const float qS5[6] = { 359 #else 360 static float qS5[6] = { 361 #endif 362 8.2776611328e+01, /* 0x42a58da0 */ 363 2.0778142090e+03, /* 0x4501dd07 */ 364 1.8847289062e+04, /* 0x46933e94 */ 365 5.6751113281e+04, /* 0x475daf1d */ 366 3.5976753906e+04, /* 0x470c88c1 */ 367 -5.3543427734e+03, /* 0xc5a752be */ 368 }; 369 370 #ifdef __STDC__ 371 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 372 #else 373 static float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 374 #endif 375 4.3774099900e-09, /* 0x3196681b */ 376 7.3241114616e-02, /* 0x3d95ff70 */ 377 3.3442313671e+00, /* 0x405607e3 */ 378 4.2621845245e+01, /* 0x422a7cc5 */ 379 1.7080809021e+02, /* 0x432acedf */ 380 1.6673394775e+02, /* 0x4326bbe4 */ 381 }; 382 #ifdef __STDC__ 383 static const float qS3[6] = { 384 #else 385 static float qS3[6] = { 386 #endif 387 4.8758872986e+01, /* 0x42430916 */ 388 7.0968920898e+02, /* 0x44316c1c */ 389 3.7041481934e+03, /* 0x4567825f */ 390 6.4604252930e+03, /* 0x45c9e367 */ 391 2.5163337402e+03, /* 0x451d4557 */ 392 -1.4924745178e+02, /* 0xc3153f59 */ 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 1.5044444979e-07, /* 0x342189db */ 401 7.3223426938e-02, /* 0x3d95f62a */ 402 1.9981917143e+00, /* 0x3fffc4bf */ 403 1.4495602608e+01, /* 0x4167edfd */ 404 3.1666231155e+01, /* 0x41fd5471 */ 405 1.6252708435e+01, /* 0x4182058c */ 406 }; 407 #ifdef __STDC__ 408 static const float qS2[6] = { 409 #else 410 static float qS2[6] = { 411 #endif 412 3.0365585327e+01, /* 0x41f2ecb8 */ 413 2.6934811401e+02, /* 0x4386ac8f */ 414 8.4478375244e+02, /* 0x44533229 */ 415 8.8293585205e+02, /* 0x445cbbe5 */ 416 2.1266638184e+02, /* 0x4354aa98 */ 417 -5.3109550476e+00, /* 0xc0a9f358 */ 418 }; 419 420 #ifdef __STDC__ 421 static float qzerof(float x) 422 #else 423 static float qzerof(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>=0x41000000) {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).125 + r/s)/x; 444 } 445