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 1.9 2006/03/19 20:42:44 christos Exp $"); 19 #endif 20 21 #include "math.h" 22 #include "math_private.h" 23 24 static float pzerof(float), qzerof(float); 25 26 static const float 27 huge = 1e30, 28 one = 1.0, 29 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 30 tpi = 6.3661974669e-01, /* 0x3f22f983 */ 31 /* R0/S0 on [0, 2.00] */ 32 R02 = 1.5625000000e-02, /* 0x3c800000 */ 33 R03 = -1.8997929874e-04, /* 0xb947352e */ 34 R04 = 1.8295404516e-06, /* 0x35f58e88 */ 35 R05 = -4.6183270541e-09, /* 0xb19eaf3c */ 36 S01 = 1.5619102865e-02, /* 0x3c7fe744 */ 37 S02 = 1.1692678527e-04, /* 0x38f53697 */ 38 S03 = 5.1354652442e-07, /* 0x3509daa6 */ 39 S04 = 1.1661400734e-09; /* 0x30a045e8 */ 40 41 static const float zero = 0.0; 42 43 float 44 __ieee754_j0f(float x) 45 { 46 float z, s,c,ss,cc,r,u,v; 47 int32_t hx,ix; 48 49 GET_FLOAT_WORD(hx,x); 50 ix = hx&0x7fffffff; 51 if(ix>=0x7f800000) return one/(x*x); 52 x = fabsf(x); 53 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 54 s = sinf(x); 55 c = cosf(x); 56 ss = s-c; 57 cc = s+c; 58 if(ix<0x7f000000) { /* make sure x+x not overflow */ 59 z = -cosf(x+x); 60 if ((s*c)<zero) cc = z/ss; 61 else ss = z/cc; 62 } 63 /* 64 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 65 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 66 */ 67 #ifdef DEAD_CODE 68 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(x); 69 else 70 #endif 71 { 72 u = pzerof(x); v = qzerof(x); 73 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x); 74 } 75 return z; 76 } 77 if(ix<0x39000000) { /* |x| < 2**-13 */ 78 if(huge+x>one) { /* raise inexact if x != 0 */ 79 if(ix<0x32000000) return one; /* |x|<2**-27 */ 80 else return one - (float)0.25*x*x; 81 } 82 } 83 z = x*x; 84 r = z*(R02+z*(R03+z*(R04+z*R05))); 85 s = one+z*(S01+z*(S02+z*(S03+z*S04))); 86 if(ix < 0x3F800000) { /* |x| < 1.00 */ 87 return one + z*((float)-0.25+(r/s)); 88 } else { 89 u = (float)0.5*x; 90 return((one+u)*(one-u)+z*(r/s)); 91 } 92 } 93 94 static const float 95 u00 = -7.3804296553e-02, /* 0xbd9726b5 */ 96 u01 = 1.7666645348e-01, /* 0x3e34e80d */ 97 u02 = -1.3818567619e-02, /* 0xbc626746 */ 98 u03 = 3.4745343146e-04, /* 0x39b62a69 */ 99 u04 = -3.8140706238e-06, /* 0xb67ff53c */ 100 u05 = 1.9559013964e-08, /* 0x32a802ba */ 101 u06 = -3.9820518410e-11, /* 0xae2f21eb */ 102 v01 = 1.2730483897e-02, /* 0x3c509385 */ 103 v02 = 7.6006865129e-05, /* 0x389f65e0 */ 104 v03 = 2.5915085189e-07, /* 0x348b216c */ 105 v04 = 4.4111031494e-10; /* 0x2ff280c2 */ 106 107 float 108 __ieee754_y0f(float x) 109 { 110 float z, s,c,ss,cc,u,v; 111 int32_t hx,ix; 112 113 GET_FLOAT_WORD(hx,x); 114 ix = 0x7fffffff&hx; 115 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */ 116 if(ix>=0x7f800000) return one/(x+x*x); 117 if(ix==0) return -one/zero; 118 if(hx<0) return zero/zero; 119 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 120 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) 121 * where x0 = x-pi/4 122 * Better formula: 123 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 124 * = 1/sqrt(2) * (sin(x) + cos(x)) 125 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 126 * = 1/sqrt(2) * (sin(x) - cos(x)) 127 * To avoid cancellation, use 128 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 129 * to compute the worse one. 130 */ 131 s = sinf(x); 132 c = cosf(x); 133 ss = s-c; 134 cc = s+c; 135 /* 136 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 137 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 138 */ 139 if(ix<0x7f000000) { /* make sure x+x not overflow */ 140 z = -cosf(x+x); 141 if ((s*c)<zero) cc = z/ss; 142 else ss = z/cc; 143 } 144 #ifdef DEAD_CODE 145 if(ix>0x80000000) z = (invsqrtpi*ss)/sqrtf(x); 146 else 147 #endif 148 { 149 u = pzerof(x); v = qzerof(x); 150 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 151 } 152 return z; 153 } 154 if(ix<=0x32000000) { /* x < 2**-27 */ 155 return(u00 + tpi*__ieee754_logf(x)); 156 } 157 z = x*x; 158 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 159 v = one+z*(v01+z*(v02+z*(v03+z*v04))); 160 return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x))); 161 } 162 163 /* The asymptotic expansions of pzero is 164 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 165 * For x >= 2, We approximate pzero by 166 * pzero(x) = 1 + (R/S) 167 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 168 * S = 1 + pS0*s^2 + ... + pS4*s^10 169 * and 170 * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 171 */ 172 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 173 0.0000000000e+00, /* 0x00000000 */ 174 -7.0312500000e-02, /* 0xbd900000 */ 175 -8.0816707611e+00, /* 0xc1014e86 */ 176 -2.5706311035e+02, /* 0xc3808814 */ 177 -2.4852163086e+03, /* 0xc51b5376 */ 178 -5.2530439453e+03, /* 0xc5a4285a */ 179 }; 180 static const float pS8[5] = { 181 1.1653436279e+02, /* 0x42e91198 */ 182 3.8337448730e+03, /* 0x456f9beb */ 183 4.0597855469e+04, /* 0x471e95db */ 184 1.1675296875e+05, /* 0x47e4087c */ 185 4.7627726562e+04, /* 0x473a0bba */ 186 }; 187 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 188 -1.1412546255e-11, /* 0xad48c58a */ 189 -7.0312492549e-02, /* 0xbd8fffff */ 190 -4.1596107483e+00, /* 0xc0851b88 */ 191 -6.7674766541e+01, /* 0xc287597b */ 192 -3.3123129272e+02, /* 0xc3a59d9b */ 193 -3.4643338013e+02, /* 0xc3ad3779 */ 194 }; 195 static const float pS5[5] = { 196 6.0753936768e+01, /* 0x42730408 */ 197 1.0512523193e+03, /* 0x44836813 */ 198 5.9789707031e+03, /* 0x45bad7c4 */ 199 9.6254453125e+03, /* 0x461665c8 */ 200 2.4060581055e+03, /* 0x451660ee */ 201 }; 202 203 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 204 -2.5470459075e-09, /* 0xb12f081b */ 205 -7.0311963558e-02, /* 0xbd8fffb8 */ 206 -2.4090321064e+00, /* 0xc01a2d95 */ 207 -2.1965976715e+01, /* 0xc1afba52 */ 208 -5.8079170227e+01, /* 0xc2685112 */ 209 -3.1447946548e+01, /* 0xc1fb9565 */ 210 }; 211 static const float pS3[5] = { 212 3.5856033325e+01, /* 0x420f6c94 */ 213 3.6151397705e+02, /* 0x43b4c1ca */ 214 1.1936077881e+03, /* 0x44953373 */ 215 1.1279968262e+03, /* 0x448cffe6 */ 216 1.7358093262e+02, /* 0x432d94b8 */ 217 }; 218 219 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 220 -8.8753431271e-08, /* 0xb3be98b7 */ 221 -7.0303097367e-02, /* 0xbd8ffb12 */ 222 -1.4507384300e+00, /* 0xbfb9b1cc */ 223 -7.6356959343e+00, /* 0xc0f4579f */ 224 -1.1193166733e+01, /* 0xc1331736 */ 225 -3.2336456776e+00, /* 0xc04ef40d */ 226 }; 227 static const float pS2[5] = { 228 2.2220300674e+01, /* 0x41b1c32d */ 229 1.3620678711e+02, /* 0x430834f0 */ 230 2.7047027588e+02, /* 0x43873c32 */ 231 1.5387539673e+02, /* 0x4319e01a */ 232 1.4657617569e+01, /* 0x416a859a */ 233 }; 234 235 static float 236 pzerof(float x) 237 { 238 const float *p,*q; 239 float z,r,s; 240 int32_t ix; 241 242 p = q = 0; 243 GET_FLOAT_WORD(ix,x); 244 ix &= 0x7fffffff; 245 if(ix>=0x41000000) {p = pR8; q= pS8;} 246 else if(ix>=0x40f71c58){p = pR5; q= pS5;} 247 else if(ix>=0x4036db68){p = pR3; q= pS3;} 248 else if(ix>=0x40000000){p = pR2; q= pS2;} 249 z = one/(x*x); 250 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 251 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 252 return one+ r/s; 253 } 254 255 256 /* For x >= 8, the asymptotic expansions of qzero is 257 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 258 * We approximate pzero by 259 * qzero(x) = s*(-1.25 + (R/S)) 260 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 261 * S = 1 + qS0*s^2 + ... + qS5*s^12 262 * and 263 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 264 */ 265 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 266 0.0000000000e+00, /* 0x00000000 */ 267 7.3242187500e-02, /* 0x3d960000 */ 268 1.1768206596e+01, /* 0x413c4a93 */ 269 5.5767340088e+02, /* 0x440b6b19 */ 270 8.8591972656e+03, /* 0x460a6cca */ 271 3.7014625000e+04, /* 0x471096a0 */ 272 }; 273 static const float qS8[6] = { 274 1.6377603149e+02, /* 0x4323c6aa */ 275 8.0983447266e+03, /* 0x45fd12c2 */ 276 1.4253829688e+05, /* 0x480b3293 */ 277 8.0330925000e+05, /* 0x49441ed4 */ 278 8.4050156250e+05, /* 0x494d3359 */ 279 -3.4389928125e+05, /* 0xc8a7eb69 */ 280 }; 281 282 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 283 1.8408595828e-11, /* 0x2da1ec79 */ 284 7.3242180049e-02, /* 0x3d95ffff */ 285 5.8356351852e+00, /* 0x40babd86 */ 286 1.3511157227e+02, /* 0x43071c90 */ 287 1.0272437744e+03, /* 0x448067cd */ 288 1.9899779053e+03, /* 0x44f8bf4b */ 289 }; 290 static const float qS5[6] = { 291 8.2776611328e+01, /* 0x42a58da0 */ 292 2.0778142090e+03, /* 0x4501dd07 */ 293 1.8847289062e+04, /* 0x46933e94 */ 294 5.6751113281e+04, /* 0x475daf1d */ 295 3.5976753906e+04, /* 0x470c88c1 */ 296 -5.3543427734e+03, /* 0xc5a752be */ 297 }; 298 299 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 300 4.3774099900e-09, /* 0x3196681b */ 301 7.3241114616e-02, /* 0x3d95ff70 */ 302 3.3442313671e+00, /* 0x405607e3 */ 303 4.2621845245e+01, /* 0x422a7cc5 */ 304 1.7080809021e+02, /* 0x432acedf */ 305 1.6673394775e+02, /* 0x4326bbe4 */ 306 }; 307 static const float qS3[6] = { 308 4.8758872986e+01, /* 0x42430916 */ 309 7.0968920898e+02, /* 0x44316c1c */ 310 3.7041481934e+03, /* 0x4567825f */ 311 6.4604252930e+03, /* 0x45c9e367 */ 312 2.5163337402e+03, /* 0x451d4557 */ 313 -1.4924745178e+02, /* 0xc3153f59 */ 314 }; 315 316 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 317 1.5044444979e-07, /* 0x342189db */ 318 7.3223426938e-02, /* 0x3d95f62a */ 319 1.9981917143e+00, /* 0x3fffc4bf */ 320 1.4495602608e+01, /* 0x4167edfd */ 321 3.1666231155e+01, /* 0x41fd5471 */ 322 1.6252708435e+01, /* 0x4182058c */ 323 }; 324 static const float qS2[6] = { 325 3.0365585327e+01, /* 0x41f2ecb8 */ 326 2.6934811401e+02, /* 0x4386ac8f */ 327 8.4478375244e+02, /* 0x44533229 */ 328 8.8293585205e+02, /* 0x445cbbe5 */ 329 2.1266638184e+02, /* 0x4354aa98 */ 330 -5.3109550476e+00, /* 0xc0a9f358 */ 331 }; 332 333 static float 334 qzerof(float x) 335 { 336 const float *p,*q; 337 float s,r,z; 338 int32_t ix; 339 340 p = q = 0; 341 GET_FLOAT_WORD(ix,x); 342 ix &= 0x7fffffff; 343 if(ix>=0x41000000) {p = qR8; q= qS8;} 344 else if(ix>=0x40f71c58){p = qR5; q= qS5;} 345 else if(ix>=0x4036db68){p = qR3; q= qS3;} 346 else if(ix>=0x40000000){p = qR2; q= qS2;} 347 z = one/(x*x); 348 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 349 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 350 return (-(float).125 + r/s)/x; 351 } 352