1 /* origin: FreeBSD /usr/src/lib/msun/src/e_j1f.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 ponef(float), qonef(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 y1, int sign) 26 { 27 double z,s,c,ss,cc; 28 29 s = sinf(x); 30 if (y1) 31 s = -s; 32 c = cosf(x); 33 cc = s-c; 34 if (ix < 0x7f000000) { 35 ss = -s-c; 36 z = cosf(2*x); 37 if (s*c > 0) 38 cc = z/ss; 39 else 40 ss = z/cc; 41 if (ix < 0x58800000) { 42 if (y1) 43 ss = -ss; 44 cc = ponef(x)*cc-qonef(x)*ss; 45 } 46 } 47 if (sign) 48 cc = -cc; 49 return invsqrtpi*cc/sqrtf(x); 50 } 51 52 /* R0/S0 on [0,2] */ 53 static const float 54 r00 = -6.2500000000e-02, /* 0xbd800000 */ 55 r01 = 1.4070566976e-03, /* 0x3ab86cfd */ 56 r02 = -1.5995563444e-05, /* 0xb7862e36 */ 57 r03 = 4.9672799207e-08, /* 0x335557d2 */ 58 s01 = 1.9153760746e-02, /* 0x3c9ce859 */ 59 s02 = 1.8594678841e-04, /* 0x3942fab6 */ 60 s03 = 1.1771846857e-06, /* 0x359dffc2 */ 61 s04 = 5.0463624390e-09, /* 0x31ad6446 */ 62 s05 = 1.2354227016e-11; /* 0x2d59567e */ 63 64 float j1f(float x) 65 { 66 float z,r,s; 67 uint32_t ix; 68 int sign; 69 70 GET_FLOAT_WORD(ix, x); 71 sign = ix>>31; 72 ix &= 0x7fffffff; 73 if (ix >= 0x7f800000) 74 return 1/(x*x); 75 if (ix >= 0x40000000) /* |x| >= 2 */ 76 return common(ix, fabsf(x), 0, sign); 77 if (ix >= 0x39000000) { /* |x| >= 2**-13 */ 78 z = x*x; 79 r = z*(r00+z*(r01+z*(r02+z*r03))); 80 s = 1+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); 81 z = 0.5f + r/s; 82 } else 83 z = 0.5f; 84 return z*x; 85 } 86 87 static const float U0[5] = { 88 -1.9605709612e-01, /* 0xbe48c331 */ 89 5.0443872809e-02, /* 0x3d4e9e3c */ 90 -1.9125689287e-03, /* 0xbafaaf2a */ 91 2.3525259166e-05, /* 0x37c5581c */ 92 -9.1909917899e-08, /* 0xb3c56003 */ 93 }; 94 static const float V0[5] = { 95 1.9916731864e-02, /* 0x3ca3286a */ 96 2.0255257550e-04, /* 0x3954644b */ 97 1.3560879779e-06, /* 0x35b602d4 */ 98 6.2274145840e-09, /* 0x31d5f8eb */ 99 1.6655924903e-11, /* 0x2d9281cf */ 100 }; 101 102 float y1f(float x) 103 { 104 float z,u,v; 105 uint32_t ix; 106 107 GET_FLOAT_WORD(ix, x); 108 if ((ix & 0x7fffffff) == 0) 109 return -1/0.0f; 110 if (ix>>31) 111 return 0/0.0f; 112 if (ix >= 0x7f800000) 113 return 1/x; 114 if (ix >= 0x40000000) /* |x| >= 2.0 */ 115 return common(ix,x,1,0); 116 if (ix < 0x33000000) /* x < 2**-25 */ 117 return -tpi/x; 118 z = x*x; 119 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); 120 v = 1.0f+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); 121 return x*(u/v) + tpi*(j1f(x)*logf(x)-1.0f/x); 122 } 123 124 /* For x >= 8, the asymptotic expansions of pone is 125 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. 126 * We approximate pone by 127 * pone(x) = 1 + (R/S) 128 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 129 * S = 1 + ps0*s^2 + ... + ps4*s^10 130 * and 131 * | pone(x)-1-R/S | <= 2 ** ( -60.06) 132 */ 133 134 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 135 0.0000000000e+00, /* 0x00000000 */ 136 1.1718750000e-01, /* 0x3df00000 */ 137 1.3239480972e+01, /* 0x4153d4ea */ 138 4.1205184937e+02, /* 0x43ce06a3 */ 139 3.8747453613e+03, /* 0x45722bed */ 140 7.9144794922e+03, /* 0x45f753d6 */ 141 }; 142 static const float ps8[5] = { 143 1.1420736694e+02, /* 0x42e46a2c */ 144 3.6509309082e+03, /* 0x45642ee5 */ 145 3.6956207031e+04, /* 0x47105c35 */ 146 9.7602796875e+04, /* 0x47bea166 */ 147 3.0804271484e+04, /* 0x46f0a88b */ 148 }; 149 150 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 151 1.3199052094e-11, /* 0x2d68333f */ 152 1.1718749255e-01, /* 0x3defffff */ 153 6.8027510643e+00, /* 0x40d9b023 */ 154 1.0830818176e+02, /* 0x42d89dca */ 155 5.1763616943e+02, /* 0x440168b7 */ 156 5.2871520996e+02, /* 0x44042dc6 */ 157 }; 158 static const float ps5[5] = { 159 5.9280597687e+01, /* 0x426d1f55 */ 160 9.9140142822e+02, /* 0x4477d9b1 */ 161 5.3532670898e+03, /* 0x45a74a23 */ 162 7.8446904297e+03, /* 0x45f52586 */ 163 1.5040468750e+03, /* 0x44bc0180 */ 164 }; 165 166 static const float pr3[6] = { 167 3.0250391081e-09, /* 0x314fe10d */ 168 1.1718686670e-01, /* 0x3defffab */ 169 3.9329774380e+00, /* 0x407bb5e7 */ 170 3.5119403839e+01, /* 0x420c7a45 */ 171 9.1055007935e+01, /* 0x42b61c2a */ 172 4.8559066772e+01, /* 0x42423c7c */ 173 }; 174 static const float ps3[5] = { 175 3.4791309357e+01, /* 0x420b2a4d */ 176 3.3676245117e+02, /* 0x43a86198 */ 177 1.0468714600e+03, /* 0x4482dbe3 */ 178 8.9081134033e+02, /* 0x445eb3ed */ 179 1.0378793335e+02, /* 0x42cf936c */ 180 }; 181 182 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 183 1.0771083225e-07, /* 0x33e74ea8 */ 184 1.1717621982e-01, /* 0x3deffa16 */ 185 2.3685150146e+00, /* 0x401795c0 */ 186 1.2242610931e+01, /* 0x4143e1bc */ 187 1.7693971634e+01, /* 0x418d8d41 */ 188 5.0735230446e+00, /* 0x40a25a4d */ 189 }; 190 static const float ps2[5] = { 191 2.1436485291e+01, /* 0x41ab7dec */ 192 1.2529022980e+02, /* 0x42fa9499 */ 193 2.3227647400e+02, /* 0x436846c7 */ 194 1.1767937469e+02, /* 0x42eb5bd7 */ 195 8.3646392822e+00, /* 0x4105d590 */ 196 }; 197 198 static float ponef(float x) 199 { 200 const float *p,*q; 201 float_t z,r,s; 202 uint32_t ix; 203 204 GET_FLOAT_WORD(ix, x); 205 ix &= 0x7fffffff; 206 if (ix >= 0x41000000){p = pr8; q = ps8;} 207 else if (ix >= 0x409173eb){p = pr5; q = ps5;} 208 else if (ix >= 0x4036d917){p = pr3; q = ps3;} 209 else /*ix >= 0x40000000*/ {p = pr2; q = ps2;} 210 z = 1.0f/(x*x); 211 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 212 s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 213 return 1.0f + r/s; 214 } 215 216 /* For x >= 8, the asymptotic expansions of qone is 217 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. 218 * We approximate pone by 219 * qone(x) = s*(0.375 + (R/S)) 220 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 221 * S = 1 + qs1*s^2 + ... + qs6*s^12 222 * and 223 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) 224 */ 225 226 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 227 0.0000000000e+00, /* 0x00000000 */ 228 -1.0253906250e-01, /* 0xbdd20000 */ 229 -1.6271753311e+01, /* 0xc1822c8d */ 230 -7.5960174561e+02, /* 0xc43de683 */ 231 -1.1849806641e+04, /* 0xc639273a */ 232 -4.8438511719e+04, /* 0xc73d3683 */ 233 }; 234 static const float qs8[6] = { 235 1.6139537048e+02, /* 0x43216537 */ 236 7.8253862305e+03, /* 0x45f48b17 */ 237 1.3387534375e+05, /* 0x4802bcd6 */ 238 7.1965775000e+05, /* 0x492fb29c */ 239 6.6660125000e+05, /* 0x4922be94 */ 240 -2.9449025000e+05, /* 0xc88fcb48 */ 241 }; 242 243 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 244 -2.0897993405e-11, /* 0xadb7d219 */ 245 -1.0253904760e-01, /* 0xbdd1fffe */ 246 -8.0564479828e+00, /* 0xc100e736 */ 247 -1.8366960144e+02, /* 0xc337ab6b */ 248 -1.3731937256e+03, /* 0xc4aba633 */ 249 -2.6124443359e+03, /* 0xc523471c */ 250 }; 251 static const float qs5[6] = { 252 8.1276550293e+01, /* 0x42a28d98 */ 253 1.9917987061e+03, /* 0x44f8f98f */ 254 1.7468484375e+04, /* 0x468878f8 */ 255 4.9851425781e+04, /* 0x4742bb6d */ 256 2.7948074219e+04, /* 0x46da5826 */ 257 -4.7191835938e+03, /* 0xc5937978 */ 258 }; 259 260 static const float qr3[6] = { 261 -5.0783124372e-09, /* 0xb1ae7d4f */ 262 -1.0253783315e-01, /* 0xbdd1ff5b */ 263 -4.6101160049e+00, /* 0xc0938612 */ 264 -5.7847221375e+01, /* 0xc267638e */ 265 -2.2824453735e+02, /* 0xc3643e9a */ 266 -2.1921012878e+02, /* 0xc35b35cb */ 267 }; 268 static const float qs3[6] = { 269 4.7665153503e+01, /* 0x423ea91e */ 270 6.7386511230e+02, /* 0x4428775e */ 271 3.3801528320e+03, /* 0x45534272 */ 272 5.5477290039e+03, /* 0x45ad5dd5 */ 273 1.9031191406e+03, /* 0x44ede3d0 */ 274 -1.3520118713e+02, /* 0xc3073381 */ 275 }; 276 277 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 278 -1.7838172539e-07, /* 0xb43f8932 */ 279 -1.0251704603e-01, /* 0xbdd1f475 */ 280 -2.7522056103e+00, /* 0xc0302423 */ 281 -1.9663616180e+01, /* 0xc19d4f16 */ 282 -4.2325313568e+01, /* 0xc2294d1f */ 283 -2.1371921539e+01, /* 0xc1aaf9b2 */ 284 }; 285 static const float qs2[6] = { 286 2.9533363342e+01, /* 0x41ec4454 */ 287 2.5298155212e+02, /* 0x437cfb47 */ 288 7.5750280762e+02, /* 0x443d602e */ 289 7.3939318848e+02, /* 0x4438d92a */ 290 1.5594900513e+02, /* 0x431bf2f2 */ 291 -4.9594988823e+00, /* 0xc09eb437 */ 292 }; 293 294 static float qonef(float x) 295 { 296 const float *p,*q; 297 float_t s,r,z; 298 uint32_t ix; 299 300 GET_FLOAT_WORD(ix, x); 301 ix &= 0x7fffffff; 302 if (ix >= 0x41000000){p = qr8; q = qs8;} 303 else if (ix >= 0x409173eb){p = qr5; q = qs5;} 304 else if (ix >= 0x4036d917){p = qr3; q = qs3;} 305 else /*ix >= 0x40000000*/ {p = qr2; q = qs2;} 306 z = 1.0f/(x*x); 307 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 308 s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 309 return (.375f + r/s)/x; 310 } 311