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 __FBSDID("$FreeBSD$"); 18 19 /* 20 * See e_j0.c for complete comments. 21 */ 22 23 #include "math.h" 24 #include "math_private.h" 25 26 static __inline float pzerof(float), qzerof(float); 27 28 static const volatile float vone = 1, vzero = 0; 29 30 static const float 31 huge = 1e30, 32 one = 1.0, 33 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 34 tpi = 6.3661974669e-01, /* 0x3f22f983 */ 35 /* R0/S0 on [0, 2.00] */ 36 R02 = 1.5625000000e-02, /* 0x3c800000 */ 37 R03 = -1.8997929874e-04, /* 0xb947352e */ 38 R04 = 1.8295404516e-06, /* 0x35f58e88 */ 39 R05 = -4.6183270541e-09, /* 0xb19eaf3c */ 40 S01 = 1.5619102865e-02, /* 0x3c7fe744 */ 41 S02 = 1.1692678527e-04, /* 0x38f53697 */ 42 S03 = 5.1354652442e-07, /* 0x3509daa6 */ 43 S04 = 1.1661400734e-09; /* 0x30a045e8 */ 44 45 static const float zero = 0.0; 46 47 float 48 __ieee754_j0f(float x) 49 { 50 float z, s,c,ss,cc,r,u,v; 51 int32_t hx,ix; 52 53 GET_FLOAT_WORD(hx,x); 54 ix = hx&0x7fffffff; 55 if(ix>=0x7f800000) return one/(x*x); 56 x = fabsf(x); 57 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 58 s = sinf(x); 59 c = cosf(x); 60 ss = s-c; 61 cc = s+c; 62 if(ix<0x7f000000) { /* make sure x+x not overflow */ 63 z = -cosf(x+x); 64 if ((s*c)<zero) cc = z/ss; 65 else ss = z/cc; 66 } 67 /* 68 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 69 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 70 */ 71 if(ix>0x58000000) z = (invsqrtpi*cc)/sqrtf(x); /* |x|>2**49 */ 72 else { 73 u = pzerof(x); v = qzerof(x); 74 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x); 75 } 76 return z; 77 } 78 if(ix<0x3b000000) { /* |x| < 2**-9 */ 79 if(huge+x>one) { /* raise inexact if x != 0 */ 80 if(ix<0x39800000) return one; /* |x|<2**-12 */ 81 else return one - x*x/4; 82 } 83 } 84 z = x*x; 85 r = z*(R02+z*(R03+z*(R04+z*R05))); 86 s = one+z*(S01+z*(S02+z*(S03+z*S04))); 87 if(ix < 0x3F800000) { /* |x| < 1.00 */ 88 return one + z*((float)-0.25+(r/s)); 89 } else { 90 u = (float)0.5*x; 91 return((one+u)*(one-u)+z*(r/s)); 92 } 93 } 94 95 static const float 96 u00 = -7.3804296553e-02, /* 0xbd9726b5 */ 97 u01 = 1.7666645348e-01, /* 0x3e34e80d */ 98 u02 = -1.3818567619e-02, /* 0xbc626746 */ 99 u03 = 3.4745343146e-04, /* 0x39b62a69 */ 100 u04 = -3.8140706238e-06, /* 0xb67ff53c */ 101 u05 = 1.9559013964e-08, /* 0x32a802ba */ 102 u06 = -3.9820518410e-11, /* 0xae2f21eb */ 103 v01 = 1.2730483897e-02, /* 0x3c509385 */ 104 v02 = 7.6006865129e-05, /* 0x389f65e0 */ 105 v03 = 2.5915085189e-07, /* 0x348b216c */ 106 v04 = 4.4111031494e-10; /* 0x2ff280c2 */ 107 108 float 109 __ieee754_y0f(float x) 110 { 111 float z, s,c,ss,cc,u,v; 112 int32_t hx,ix; 113 114 GET_FLOAT_WORD(hx,x); 115 ix = 0x7fffffff&hx; 116 if(ix>=0x7f800000) return vone/(x+x*x); 117 if(ix==0) return -one/vzero; 118 if(hx<0) return vzero/vzero; 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 if(ix>0x58000000) z = (invsqrtpi*ss)/sqrtf(x); /* |x|>2**49 */ 145 else { 146 u = pzerof(x); v = qzerof(x); 147 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 148 } 149 return z; 150 } 151 if(ix<=0x39000000) { /* x < 2**-13 */ 152 return(u00 + tpi*__ieee754_logf(x)); 153 } 154 z = x*x; 155 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 156 v = one+z*(v01+z*(v02+z*(v03+z*v04))); 157 return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x))); 158 } 159 160 /* The asymptotic expansions of pzero is 161 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 162 * For x >= 2, We approximate pzero by 163 * pzero(x) = 1 + (R/S) 164 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 165 * S = 1 + pS0*s^2 + ... + pS4*s^10 166 * and 167 * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 168 */ 169 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 170 0.0000000000e+00, /* 0x00000000 */ 171 -7.0312500000e-02, /* 0xbd900000 */ 172 -8.0816707611e+00, /* 0xc1014e86 */ 173 -2.5706311035e+02, /* 0xc3808814 */ 174 -2.4852163086e+03, /* 0xc51b5376 */ 175 -5.2530439453e+03, /* 0xc5a4285a */ 176 }; 177 static const float pS8[5] = { 178 1.1653436279e+02, /* 0x42e91198 */ 179 3.8337448730e+03, /* 0x456f9beb */ 180 4.0597855469e+04, /* 0x471e95db */ 181 1.1675296875e+05, /* 0x47e4087c */ 182 4.7627726562e+04, /* 0x473a0bba */ 183 }; 184 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 185 -1.1412546255e-11, /* 0xad48c58a */ 186 -7.0312492549e-02, /* 0xbd8fffff */ 187 -4.1596107483e+00, /* 0xc0851b88 */ 188 -6.7674766541e+01, /* 0xc287597b */ 189 -3.3123129272e+02, /* 0xc3a59d9b */ 190 -3.4643338013e+02, /* 0xc3ad3779 */ 191 }; 192 static const float pS5[5] = { 193 6.0753936768e+01, /* 0x42730408 */ 194 1.0512523193e+03, /* 0x44836813 */ 195 5.9789707031e+03, /* 0x45bad7c4 */ 196 9.6254453125e+03, /* 0x461665c8 */ 197 2.4060581055e+03, /* 0x451660ee */ 198 }; 199 200 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 201 -2.5470459075e-09, /* 0xb12f081b */ 202 -7.0311963558e-02, /* 0xbd8fffb8 */ 203 -2.4090321064e+00, /* 0xc01a2d95 */ 204 -2.1965976715e+01, /* 0xc1afba52 */ 205 -5.8079170227e+01, /* 0xc2685112 */ 206 -3.1447946548e+01, /* 0xc1fb9565 */ 207 }; 208 static const float pS3[5] = { 209 3.5856033325e+01, /* 0x420f6c94 */ 210 3.6151397705e+02, /* 0x43b4c1ca */ 211 1.1936077881e+03, /* 0x44953373 */ 212 1.1279968262e+03, /* 0x448cffe6 */ 213 1.7358093262e+02, /* 0x432d94b8 */ 214 }; 215 216 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 217 -8.8753431271e-08, /* 0xb3be98b7 */ 218 -7.0303097367e-02, /* 0xbd8ffb12 */ 219 -1.4507384300e+00, /* 0xbfb9b1cc */ 220 -7.6356959343e+00, /* 0xc0f4579f */ 221 -1.1193166733e+01, /* 0xc1331736 */ 222 -3.2336456776e+00, /* 0xc04ef40d */ 223 }; 224 static const float pS2[5] = { 225 2.2220300674e+01, /* 0x41b1c32d */ 226 1.3620678711e+02, /* 0x430834f0 */ 227 2.7047027588e+02, /* 0x43873c32 */ 228 1.5387539673e+02, /* 0x4319e01a */ 229 1.4657617569e+01, /* 0x416a859a */ 230 }; 231 232 static __inline float 233 pzerof(float x) 234 { 235 const float *p,*q; 236 float z,r,s; 237 int32_t ix; 238 GET_FLOAT_WORD(ix,x); 239 ix &= 0x7fffffff; 240 if(ix>=0x41000000) {p = pR8; q= pS8;} 241 else if(ix>=0x409173eb){p = pR5; q= pS5;} 242 else if(ix>=0x4036d917){p = pR3; q= pS3;} 243 else {p = pR2; q= pS2;} /* ix>=0x40000000 */ 244 z = one/(x*x); 245 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 246 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 247 return one+ r/s; 248 } 249 250 251 /* For x >= 8, the asymptotic expansions of qzero is 252 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 253 * We approximate pzero by 254 * qzero(x) = s*(-1.25 + (R/S)) 255 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 256 * S = 1 + qS0*s^2 + ... + qS5*s^12 257 * and 258 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 259 */ 260 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 261 0.0000000000e+00, /* 0x00000000 */ 262 7.3242187500e-02, /* 0x3d960000 */ 263 1.1768206596e+01, /* 0x413c4a93 */ 264 5.5767340088e+02, /* 0x440b6b19 */ 265 8.8591972656e+03, /* 0x460a6cca */ 266 3.7014625000e+04, /* 0x471096a0 */ 267 }; 268 static const float qS8[6] = { 269 1.6377603149e+02, /* 0x4323c6aa */ 270 8.0983447266e+03, /* 0x45fd12c2 */ 271 1.4253829688e+05, /* 0x480b3293 */ 272 8.0330925000e+05, /* 0x49441ed4 */ 273 8.4050156250e+05, /* 0x494d3359 */ 274 -3.4389928125e+05, /* 0xc8a7eb69 */ 275 }; 276 277 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 278 1.8408595828e-11, /* 0x2da1ec79 */ 279 7.3242180049e-02, /* 0x3d95ffff */ 280 5.8356351852e+00, /* 0x40babd86 */ 281 1.3511157227e+02, /* 0x43071c90 */ 282 1.0272437744e+03, /* 0x448067cd */ 283 1.9899779053e+03, /* 0x44f8bf4b */ 284 }; 285 static const float qS5[6] = { 286 8.2776611328e+01, /* 0x42a58da0 */ 287 2.0778142090e+03, /* 0x4501dd07 */ 288 1.8847289062e+04, /* 0x46933e94 */ 289 5.6751113281e+04, /* 0x475daf1d */ 290 3.5976753906e+04, /* 0x470c88c1 */ 291 -5.3543427734e+03, /* 0xc5a752be */ 292 }; 293 294 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 295 4.3774099900e-09, /* 0x3196681b */ 296 7.3241114616e-02, /* 0x3d95ff70 */ 297 3.3442313671e+00, /* 0x405607e3 */ 298 4.2621845245e+01, /* 0x422a7cc5 */ 299 1.7080809021e+02, /* 0x432acedf */ 300 1.6673394775e+02, /* 0x4326bbe4 */ 301 }; 302 static const float qS3[6] = { 303 4.8758872986e+01, /* 0x42430916 */ 304 7.0968920898e+02, /* 0x44316c1c */ 305 3.7041481934e+03, /* 0x4567825f */ 306 6.4604252930e+03, /* 0x45c9e367 */ 307 2.5163337402e+03, /* 0x451d4557 */ 308 -1.4924745178e+02, /* 0xc3153f59 */ 309 }; 310 311 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 312 1.5044444979e-07, /* 0x342189db */ 313 7.3223426938e-02, /* 0x3d95f62a */ 314 1.9981917143e+00, /* 0x3fffc4bf */ 315 1.4495602608e+01, /* 0x4167edfd */ 316 3.1666231155e+01, /* 0x41fd5471 */ 317 1.6252708435e+01, /* 0x4182058c */ 318 }; 319 static const float qS2[6] = { 320 3.0365585327e+01, /* 0x41f2ecb8 */ 321 2.6934811401e+02, /* 0x4386ac8f */ 322 8.4478375244e+02, /* 0x44533229 */ 323 8.8293585205e+02, /* 0x445cbbe5 */ 324 2.1266638184e+02, /* 0x4354aa98 */ 325 -5.3109550476e+00, /* 0xc0a9f358 */ 326 }; 327 328 static __inline float 329 qzerof(float x) 330 { 331 const float *p,*q; 332 float s,r,z; 333 int32_t ix; 334 GET_FLOAT_WORD(ix,x); 335 ix &= 0x7fffffff; 336 if(ix>=0x41000000) {p = qR8; q= qS8;} 337 else if(ix>=0x409173eb){p = qR5; q= qS5;} 338 else if(ix>=0x4036d917){p = qR3; q= qS3;} 339 else {p = qR2; q= qS2;} /* ix>=0x40000000 */ 340 z = one/(x*x); 341 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 342 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 343 return (-(float).125 + r/s)/x; 344 } 345