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 #include <sys/cdefs.h> 17 #if defined(LIBM_SCCS) && !defined(lint) 18 __RCSID("$NetBSD: e_j0f.c,v 1.5 1997/10/09 11:29:11 lukem Exp $"); 19 #endif 20 21 #include "math.h" 22 #include "math_private.h" 23 24 #ifdef __STDC__ 25 static float pzerof(float), qzerof(float); 26 #else 27 static float pzerof(), qzerof(); 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, 2.00] */ 40 R02 = 1.5625000000e-02, /* 0x3c800000 */ 41 R03 = -1.8997929874e-04, /* 0xb947352e */ 42 R04 = 1.8295404516e-06, /* 0x35f58e88 */ 43 R05 = -4.6183270541e-09, /* 0xb19eaf3c */ 44 S01 = 1.5619102865e-02, /* 0x3c7fe744 */ 45 S02 = 1.1692678527e-04, /* 0x38f53697 */ 46 S03 = 5.1354652442e-07, /* 0x3509daa6 */ 47 S04 = 1.1661400734e-09; /* 0x30a045e8 */ 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_j0f(float x) 57 #else 58 float __ieee754_j0f(x) 59 float x; 60 #endif 61 { 62 float z, s,c,ss,cc,r,u,v; 63 int32_t hx,ix; 64 65 GET_FLOAT_WORD(hx,x); 66 ix = hx&0x7fffffff; 67 if(ix>=0x7f800000) return one/(x*x); 68 x = fabsf(x); 69 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 70 s = sinf(x); 71 c = cosf(x); 72 ss = s-c; 73 cc = s+c; 74 if(ix<0x7f000000) { /* make sure x+x not overflow */ 75 z = -cosf(x+x); 76 if ((s*c)<zero) cc = z/ss; 77 else ss = z/cc; 78 } 79 /* 80 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 81 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 82 */ 83 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(x); 84 else { 85 u = pzerof(x); v = qzerof(x); 86 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x); 87 } 88 return z; 89 } 90 if(ix<0x39000000) { /* |x| < 2**-13 */ 91 if(huge+x>one) { /* raise inexact if x != 0 */ 92 if(ix<0x32000000) return one; /* |x|<2**-27 */ 93 else return one - (float)0.25*x*x; 94 } 95 } 96 z = x*x; 97 r = z*(R02+z*(R03+z*(R04+z*R05))); 98 s = one+z*(S01+z*(S02+z*(S03+z*S04))); 99 if(ix < 0x3F800000) { /* |x| < 1.00 */ 100 return one + z*((float)-0.25+(r/s)); 101 } else { 102 u = (float)0.5*x; 103 return((one+u)*(one-u)+z*(r/s)); 104 } 105 } 106 107 #ifdef __STDC__ 108 static const float 109 #else 110 static float 111 #endif 112 u00 = -7.3804296553e-02, /* 0xbd9726b5 */ 113 u01 = 1.7666645348e-01, /* 0x3e34e80d */ 114 u02 = -1.3818567619e-02, /* 0xbc626746 */ 115 u03 = 3.4745343146e-04, /* 0x39b62a69 */ 116 u04 = -3.8140706238e-06, /* 0xb67ff53c */ 117 u05 = 1.9559013964e-08, /* 0x32a802ba */ 118 u06 = -3.9820518410e-11, /* 0xae2f21eb */ 119 v01 = 1.2730483897e-02, /* 0x3c509385 */ 120 v02 = 7.6006865129e-05, /* 0x389f65e0 */ 121 v03 = 2.5915085189e-07, /* 0x348b216c */ 122 v04 = 4.4111031494e-10; /* 0x2ff280c2 */ 123 124 #ifdef __STDC__ 125 float __ieee754_y0f(float x) 126 #else 127 float __ieee754_y0f(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 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(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 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) 142 * where x0 = x-pi/4 143 * Better formula: 144 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 145 * = 1/sqrt(2) * (sin(x) + cos(x)) 146 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 147 * = 1/sqrt(2) * (sin(x) - cos(x)) 148 * To avoid cancellation, use 149 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 150 * to compute the worse one. 151 */ 152 s = sinf(x); 153 c = cosf(x); 154 ss = s-c; 155 cc = s+c; 156 /* 157 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 158 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 159 */ 160 if(ix<0x7f000000) { /* make sure x+x not overflow */ 161 z = -cosf(x+x); 162 if ((s*c)<zero) cc = z/ss; 163 else ss = z/cc; 164 } 165 if(ix>0x80000000) z = (invsqrtpi*ss)/sqrtf(x); 166 else { 167 u = pzerof(x); v = qzerof(x); 168 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 169 } 170 return z; 171 } 172 if(ix<=0x32000000) { /* x < 2**-27 */ 173 return(u00 + tpi*__ieee754_logf(x)); 174 } 175 z = x*x; 176 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 177 v = one+z*(v01+z*(v02+z*(v03+z*v04))); 178 return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x))); 179 } 180 181 /* The asymptotic expansions of pzero is 182 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 183 * For x >= 2, We approximate pzero by 184 * pzero(x) = 1 + (R/S) 185 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 186 * S = 1 + pS0*s^2 + ... + pS4*s^10 187 * and 188 * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 189 */ 190 #ifdef __STDC__ 191 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 192 #else 193 static float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 194 #endif 195 0.0000000000e+00, /* 0x00000000 */ 196 -7.0312500000e-02, /* 0xbd900000 */ 197 -8.0816707611e+00, /* 0xc1014e86 */ 198 -2.5706311035e+02, /* 0xc3808814 */ 199 -2.4852163086e+03, /* 0xc51b5376 */ 200 -5.2530439453e+03, /* 0xc5a4285a */ 201 }; 202 #ifdef __STDC__ 203 static const float pS8[5] = { 204 #else 205 static float pS8[5] = { 206 #endif 207 1.1653436279e+02, /* 0x42e91198 */ 208 3.8337448730e+03, /* 0x456f9beb */ 209 4.0597855469e+04, /* 0x471e95db */ 210 1.1675296875e+05, /* 0x47e4087c */ 211 4.7627726562e+04, /* 0x473a0bba */ 212 }; 213 #ifdef __STDC__ 214 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 215 #else 216 static float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 217 #endif 218 -1.1412546255e-11, /* 0xad48c58a */ 219 -7.0312492549e-02, /* 0xbd8fffff */ 220 -4.1596107483e+00, /* 0xc0851b88 */ 221 -6.7674766541e+01, /* 0xc287597b */ 222 -3.3123129272e+02, /* 0xc3a59d9b */ 223 -3.4643338013e+02, /* 0xc3ad3779 */ 224 }; 225 #ifdef __STDC__ 226 static const float pS5[5] = { 227 #else 228 static float pS5[5] = { 229 #endif 230 6.0753936768e+01, /* 0x42730408 */ 231 1.0512523193e+03, /* 0x44836813 */ 232 5.9789707031e+03, /* 0x45bad7c4 */ 233 9.6254453125e+03, /* 0x461665c8 */ 234 2.4060581055e+03, /* 0x451660ee */ 235 }; 236 237 #ifdef __STDC__ 238 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 239 #else 240 static float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 241 #endif 242 -2.5470459075e-09, /* 0xb12f081b */ 243 -7.0311963558e-02, /* 0xbd8fffb8 */ 244 -2.4090321064e+00, /* 0xc01a2d95 */ 245 -2.1965976715e+01, /* 0xc1afba52 */ 246 -5.8079170227e+01, /* 0xc2685112 */ 247 -3.1447946548e+01, /* 0xc1fb9565 */ 248 }; 249 #ifdef __STDC__ 250 static const float pS3[5] = { 251 #else 252 static float pS3[5] = { 253 #endif 254 3.5856033325e+01, /* 0x420f6c94 */ 255 3.6151397705e+02, /* 0x43b4c1ca */ 256 1.1936077881e+03, /* 0x44953373 */ 257 1.1279968262e+03, /* 0x448cffe6 */ 258 1.7358093262e+02, /* 0x432d94b8 */ 259 }; 260 261 #ifdef __STDC__ 262 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 263 #else 264 static float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 265 #endif 266 -8.8753431271e-08, /* 0xb3be98b7 */ 267 -7.0303097367e-02, /* 0xbd8ffb12 */ 268 -1.4507384300e+00, /* 0xbfb9b1cc */ 269 -7.6356959343e+00, /* 0xc0f4579f */ 270 -1.1193166733e+01, /* 0xc1331736 */ 271 -3.2336456776e+00, /* 0xc04ef40d */ 272 }; 273 #ifdef __STDC__ 274 static const float pS2[5] = { 275 #else 276 static float pS2[5] = { 277 #endif 278 2.2220300674e+01, /* 0x41b1c32d */ 279 1.3620678711e+02, /* 0x430834f0 */ 280 2.7047027588e+02, /* 0x43873c32 */ 281 1.5387539673e+02, /* 0x4319e01a */ 282 1.4657617569e+01, /* 0x416a859a */ 283 }; 284 285 #ifdef __STDC__ 286 static float pzerof(float x) 287 #else 288 static float pzerof(x) 289 float x; 290 #endif 291 { 292 #ifdef __STDC__ 293 const float *p,*q; 294 #else 295 float *p,*q; 296 #endif 297 float z,r,s; 298 int32_t ix; 299 300 p = q = 0; 301 GET_FLOAT_WORD(ix,x); 302 ix &= 0x7fffffff; 303 if(ix>=0x41000000) {p = pR8; q= pS8;} 304 else if(ix>=0x40f71c58){p = pR5; q= pS5;} 305 else if(ix>=0x4036db68){p = pR3; q= pS3;} 306 else if(ix>=0x40000000){p = pR2; q= pS2;} 307 z = one/(x*x); 308 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 309 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 310 return one+ r/s; 311 } 312 313 314 /* For x >= 8, the asymptotic expansions of qzero is 315 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 316 * We approximate pzero by 317 * qzero(x) = s*(-1.25 + (R/S)) 318 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 319 * S = 1 + qS0*s^2 + ... + qS5*s^12 320 * and 321 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 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 7.3242187500e-02, /* 0x3d960000 */ 330 1.1768206596e+01, /* 0x413c4a93 */ 331 5.5767340088e+02, /* 0x440b6b19 */ 332 8.8591972656e+03, /* 0x460a6cca */ 333 3.7014625000e+04, /* 0x471096a0 */ 334 }; 335 #ifdef __STDC__ 336 static const float qS8[6] = { 337 #else 338 static float qS8[6] = { 339 #endif 340 1.6377603149e+02, /* 0x4323c6aa */ 341 8.0983447266e+03, /* 0x45fd12c2 */ 342 1.4253829688e+05, /* 0x480b3293 */ 343 8.0330925000e+05, /* 0x49441ed4 */ 344 8.4050156250e+05, /* 0x494d3359 */ 345 -3.4389928125e+05, /* 0xc8a7eb69 */ 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 1.8408595828e-11, /* 0x2da1ec79 */ 354 7.3242180049e-02, /* 0x3d95ffff */ 355 5.8356351852e+00, /* 0x40babd86 */ 356 1.3511157227e+02, /* 0x43071c90 */ 357 1.0272437744e+03, /* 0x448067cd */ 358 1.9899779053e+03, /* 0x44f8bf4b */ 359 }; 360 #ifdef __STDC__ 361 static const float qS5[6] = { 362 #else 363 static float qS5[6] = { 364 #endif 365 8.2776611328e+01, /* 0x42a58da0 */ 366 2.0778142090e+03, /* 0x4501dd07 */ 367 1.8847289062e+04, /* 0x46933e94 */ 368 5.6751113281e+04, /* 0x475daf1d */ 369 3.5976753906e+04, /* 0x470c88c1 */ 370 -5.3543427734e+03, /* 0xc5a752be */ 371 }; 372 373 #ifdef __STDC__ 374 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 375 #else 376 static float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 377 #endif 378 4.3774099900e-09, /* 0x3196681b */ 379 7.3241114616e-02, /* 0x3d95ff70 */ 380 3.3442313671e+00, /* 0x405607e3 */ 381 4.2621845245e+01, /* 0x422a7cc5 */ 382 1.7080809021e+02, /* 0x432acedf */ 383 1.6673394775e+02, /* 0x4326bbe4 */ 384 }; 385 #ifdef __STDC__ 386 static const float qS3[6] = { 387 #else 388 static float qS3[6] = { 389 #endif 390 4.8758872986e+01, /* 0x42430916 */ 391 7.0968920898e+02, /* 0x44316c1c */ 392 3.7041481934e+03, /* 0x4567825f */ 393 6.4604252930e+03, /* 0x45c9e367 */ 394 2.5163337402e+03, /* 0x451d4557 */ 395 -1.4924745178e+02, /* 0xc3153f59 */ 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 1.5044444979e-07, /* 0x342189db */ 404 7.3223426938e-02, /* 0x3d95f62a */ 405 1.9981917143e+00, /* 0x3fffc4bf */ 406 1.4495602608e+01, /* 0x4167edfd */ 407 3.1666231155e+01, /* 0x41fd5471 */ 408 1.6252708435e+01, /* 0x4182058c */ 409 }; 410 #ifdef __STDC__ 411 static const float qS2[6] = { 412 #else 413 static float qS2[6] = { 414 #endif 415 3.0365585327e+01, /* 0x41f2ecb8 */ 416 2.6934811401e+02, /* 0x4386ac8f */ 417 8.4478375244e+02, /* 0x44533229 */ 418 8.8293585205e+02, /* 0x445cbbe5 */ 419 2.1266638184e+02, /* 0x4354aa98 */ 420 -5.3109550476e+00, /* 0xc0a9f358 */ 421 }; 422 423 #ifdef __STDC__ 424 static float qzerof(float x) 425 #else 426 static float qzerof(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>=0x41000000) {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).125 + r/s)/x; 449 } 450