1 /* origin: FreeBSD /usr/src/lib/msun/src/e_j0f.c */ 2 /* 3 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. 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 #define _GNU_SOURCE 17 #include "libm.h" 18 19 static float pzerof(float), qzerof(float); 20 21 static const float 22 invsqrtpi = 5.6418961287e-01, /* 0x3f106ebb */ 23 tpi = 6.3661974669e-01; /* 0x3f22f983 */ 24 25 static float common(uint32_t ix, float x, int y0) 26 { 27 float z,s,c,ss,cc; 28 /* 29 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 30 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 31 */ 32 s = sinf(x); 33 c = cosf(x); 34 if (y0) 35 c = -c; 36 cc = s+c; 37 if (ix < 0x7f000000) { 38 ss = s-c; 39 z = -cosf(2*x); 40 if (s*c < 0) 41 cc = z/ss; 42 else 43 ss = z/cc; 44 if (ix < 0x58800000) { 45 if (y0) 46 ss = -ss; 47 cc = pzerof(x)*cc-qzerof(x)*ss; 48 } 49 } 50 return invsqrtpi*cc/sqrtf(x); 51 } 52 53 /* R0/S0 on [0, 2.00] */ 54 static const float 55 R02 = 1.5625000000e-02, /* 0x3c800000 */ 56 R03 = -1.8997929874e-04, /* 0xb947352e */ 57 R04 = 1.8295404516e-06, /* 0x35f58e88 */ 58 R05 = -4.6183270541e-09, /* 0xb19eaf3c */ 59 S01 = 1.5619102865e-02, /* 0x3c7fe744 */ 60 S02 = 1.1692678527e-04, /* 0x38f53697 */ 61 S03 = 5.1354652442e-07, /* 0x3509daa6 */ 62 S04 = 1.1661400734e-09; /* 0x30a045e8 */ 63 64 float j0f(float x) 65 { 66 float z,r,s; 67 uint32_t ix; 68 69 GET_FLOAT_WORD(ix, x); 70 ix &= 0x7fffffff; 71 if (ix >= 0x7f800000) 72 return 1/(x*x); 73 x = fabsf(x); 74 75 if (ix >= 0x40000000) { /* |x| >= 2 */ 76 /* large ulp error near zeros */ 77 return common(ix, x, 0); 78 } 79 if (ix >= 0x3a000000) { /* |x| >= 2**-11 */ 80 /* up to 4ulp error near 2 */ 81 z = x*x; 82 r = z*(R02+z*(R03+z*(R04+z*R05))); 83 s = 1+z*(S01+z*(S02+z*(S03+z*S04))); 84 return (1+x/2)*(1-x/2) + z*(r/s); 85 } 86 if (ix >= 0x21800000) /* |x| >= 2**-60 */ 87 x = 0.25f*x*x; 88 return 1 - x; 89 } 90 91 static const float 92 u00 = -7.3804296553e-02, /* 0xbd9726b5 */ 93 u01 = 1.7666645348e-01, /* 0x3e34e80d */ 94 u02 = -1.3818567619e-02, /* 0xbc626746 */ 95 u03 = 3.4745343146e-04, /* 0x39b62a69 */ 96 u04 = -3.8140706238e-06, /* 0xb67ff53c */ 97 u05 = 1.9559013964e-08, /* 0x32a802ba */ 98 u06 = -3.9820518410e-11, /* 0xae2f21eb */ 99 v01 = 1.2730483897e-02, /* 0x3c509385 */ 100 v02 = 7.6006865129e-05, /* 0x389f65e0 */ 101 v03 = 2.5915085189e-07, /* 0x348b216c */ 102 v04 = 4.4111031494e-10; /* 0x2ff280c2 */ 103 104 float y0f(float x) 105 { 106 float z,u,v; 107 uint32_t ix; 108 109 GET_FLOAT_WORD(ix, x); 110 if ((ix & 0x7fffffff) == 0) 111 return -1/0.0f; 112 if (ix>>31) 113 return 0/0.0f; 114 if (ix >= 0x7f800000) 115 return 1/x; 116 if (ix >= 0x40000000) { /* |x| >= 2.0 */ 117 /* large ulp error near zeros */ 118 return common(ix,x,1); 119 } 120 if (ix >= 0x39000000) { /* x >= 2**-13 */ 121 /* large ulp error at x ~= 0.89 */ 122 z = x*x; 123 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 124 v = 1+z*(v01+z*(v02+z*(v03+z*v04))); 125 return u/v + tpi*(j0f(x)*logf(x)); 126 } 127 return u00 + tpi*logf(x); 128 } 129 130 /* The asymptotic expansions of pzero is 131 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 132 * For x >= 2, We approximate pzero by 133 * pzero(x) = 1 + (R/S) 134 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 135 * S = 1 + pS0*s^2 + ... + pS4*s^10 136 * and 137 * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 138 */ 139 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 140 0.0000000000e+00, /* 0x00000000 */ 141 -7.0312500000e-02, /* 0xbd900000 */ 142 -8.0816707611e+00, /* 0xc1014e86 */ 143 -2.5706311035e+02, /* 0xc3808814 */ 144 -2.4852163086e+03, /* 0xc51b5376 */ 145 -5.2530439453e+03, /* 0xc5a4285a */ 146 }; 147 static const float pS8[5] = { 148 1.1653436279e+02, /* 0x42e91198 */ 149 3.8337448730e+03, /* 0x456f9beb */ 150 4.0597855469e+04, /* 0x471e95db */ 151 1.1675296875e+05, /* 0x47e4087c */ 152 4.7627726562e+04, /* 0x473a0bba */ 153 }; 154 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 155 -1.1412546255e-11, /* 0xad48c58a */ 156 -7.0312492549e-02, /* 0xbd8fffff */ 157 -4.1596107483e+00, /* 0xc0851b88 */ 158 -6.7674766541e+01, /* 0xc287597b */ 159 -3.3123129272e+02, /* 0xc3a59d9b */ 160 -3.4643338013e+02, /* 0xc3ad3779 */ 161 }; 162 static const float pS5[5] = { 163 6.0753936768e+01, /* 0x42730408 */ 164 1.0512523193e+03, /* 0x44836813 */ 165 5.9789707031e+03, /* 0x45bad7c4 */ 166 9.6254453125e+03, /* 0x461665c8 */ 167 2.4060581055e+03, /* 0x451660ee */ 168 }; 169 170 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 171 -2.5470459075e-09, /* 0xb12f081b */ 172 -7.0311963558e-02, /* 0xbd8fffb8 */ 173 -2.4090321064e+00, /* 0xc01a2d95 */ 174 -2.1965976715e+01, /* 0xc1afba52 */ 175 -5.8079170227e+01, /* 0xc2685112 */ 176 -3.1447946548e+01, /* 0xc1fb9565 */ 177 }; 178 static const float pS3[5] = { 179 3.5856033325e+01, /* 0x420f6c94 */ 180 3.6151397705e+02, /* 0x43b4c1ca */ 181 1.1936077881e+03, /* 0x44953373 */ 182 1.1279968262e+03, /* 0x448cffe6 */ 183 1.7358093262e+02, /* 0x432d94b8 */ 184 }; 185 186 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 187 -8.8753431271e-08, /* 0xb3be98b7 */ 188 -7.0303097367e-02, /* 0xbd8ffb12 */ 189 -1.4507384300e+00, /* 0xbfb9b1cc */ 190 -7.6356959343e+00, /* 0xc0f4579f */ 191 -1.1193166733e+01, /* 0xc1331736 */ 192 -3.2336456776e+00, /* 0xc04ef40d */ 193 }; 194 static const float pS2[5] = { 195 2.2220300674e+01, /* 0x41b1c32d */ 196 1.3620678711e+02, /* 0x430834f0 */ 197 2.7047027588e+02, /* 0x43873c32 */ 198 1.5387539673e+02, /* 0x4319e01a */ 199 1.4657617569e+01, /* 0x416a859a */ 200 }; 201 202 static float pzerof(float x) 203 { 204 const float *p,*q; 205 float_t z,r,s; 206 uint32_t ix; 207 208 GET_FLOAT_WORD(ix, x); 209 ix &= 0x7fffffff; 210 if (ix >= 0x41000000){p = pR8; q = pS8;} 211 else if (ix >= 0x409173eb){p = pR5; q = pS5;} 212 else if (ix >= 0x4036d917){p = pR3; q = pS3;} 213 else /*ix >= 0x40000000*/ {p = pR2; q = pS2;} 214 z = 1.0f/(x*x); 215 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 216 s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 217 return 1.0f + r/s; 218 } 219 220 221 /* For x >= 8, the asymptotic expansions of qzero is 222 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 223 * We approximate pzero by 224 * qzero(x) = s*(-1.25 + (R/S)) 225 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 226 * S = 1 + qS0*s^2 + ... + qS5*s^12 227 * and 228 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 229 */ 230 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 231 0.0000000000e+00, /* 0x00000000 */ 232 7.3242187500e-02, /* 0x3d960000 */ 233 1.1768206596e+01, /* 0x413c4a93 */ 234 5.5767340088e+02, /* 0x440b6b19 */ 235 8.8591972656e+03, /* 0x460a6cca */ 236 3.7014625000e+04, /* 0x471096a0 */ 237 }; 238 static const float qS8[6] = { 239 1.6377603149e+02, /* 0x4323c6aa */ 240 8.0983447266e+03, /* 0x45fd12c2 */ 241 1.4253829688e+05, /* 0x480b3293 */ 242 8.0330925000e+05, /* 0x49441ed4 */ 243 8.4050156250e+05, /* 0x494d3359 */ 244 -3.4389928125e+05, /* 0xc8a7eb69 */ 245 }; 246 247 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 248 1.8408595828e-11, /* 0x2da1ec79 */ 249 7.3242180049e-02, /* 0x3d95ffff */ 250 5.8356351852e+00, /* 0x40babd86 */ 251 1.3511157227e+02, /* 0x43071c90 */ 252 1.0272437744e+03, /* 0x448067cd */ 253 1.9899779053e+03, /* 0x44f8bf4b */ 254 }; 255 static const float qS5[6] = { 256 8.2776611328e+01, /* 0x42a58da0 */ 257 2.0778142090e+03, /* 0x4501dd07 */ 258 1.8847289062e+04, /* 0x46933e94 */ 259 5.6751113281e+04, /* 0x475daf1d */ 260 3.5976753906e+04, /* 0x470c88c1 */ 261 -5.3543427734e+03, /* 0xc5a752be */ 262 }; 263 264 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 265 4.3774099900e-09, /* 0x3196681b */ 266 7.3241114616e-02, /* 0x3d95ff70 */ 267 3.3442313671e+00, /* 0x405607e3 */ 268 4.2621845245e+01, /* 0x422a7cc5 */ 269 1.7080809021e+02, /* 0x432acedf */ 270 1.6673394775e+02, /* 0x4326bbe4 */ 271 }; 272 static const float qS3[6] = { 273 4.8758872986e+01, /* 0x42430916 */ 274 7.0968920898e+02, /* 0x44316c1c */ 275 3.7041481934e+03, /* 0x4567825f */ 276 6.4604252930e+03, /* 0x45c9e367 */ 277 2.5163337402e+03, /* 0x451d4557 */ 278 -1.4924745178e+02, /* 0xc3153f59 */ 279 }; 280 281 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 282 1.5044444979e-07, /* 0x342189db */ 283 7.3223426938e-02, /* 0x3d95f62a */ 284 1.9981917143e+00, /* 0x3fffc4bf */ 285 1.4495602608e+01, /* 0x4167edfd */ 286 3.1666231155e+01, /* 0x41fd5471 */ 287 1.6252708435e+01, /* 0x4182058c */ 288 }; 289 static const float qS2[6] = { 290 3.0365585327e+01, /* 0x41f2ecb8 */ 291 2.6934811401e+02, /* 0x4386ac8f */ 292 8.4478375244e+02, /* 0x44533229 */ 293 8.8293585205e+02, /* 0x445cbbe5 */ 294 2.1266638184e+02, /* 0x4354aa98 */ 295 -5.3109550476e+00, /* 0xc0a9f358 */ 296 }; 297 298 static float qzerof(float x) 299 { 300 const float *p,*q; 301 float_t s,r,z; 302 uint32_t ix; 303 304 GET_FLOAT_WORD(ix, x); 305 ix &= 0x7fffffff; 306 if (ix >= 0x41000000){p = qR8; q = qS8;} 307 else if (ix >= 0x409173eb){p = qR5; q = qS5;} 308 else if (ix >= 0x4036d917){p = qR3; q = qS3;} 309 else /*ix >= 0x40000000*/ {p = qR2; q = qS2;} 310 z = 1.0f/(x*x); 311 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 312 s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 313 return (-.125f + r/s)/x; 314 } 315