1 /* e_j1f.c -- float version of e_j1.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 #include "math.h" 20 #include "math_private.h" 21 22 static float ponef(float), qonef(float); 23 24 static const float 25 huge = 1e30, 26 one = 1.0, 27 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 28 tpi = 6.3661974669e-01, /* 0x3f22f983 */ 29 /* R0/S0 on [0,2] */ 30 r00 = -6.2500000000e-02, /* 0xbd800000 */ 31 r01 = 1.4070566976e-03, /* 0x3ab86cfd */ 32 r02 = -1.5995563444e-05, /* 0xb7862e36 */ 33 r03 = 4.9672799207e-08, /* 0x335557d2 */ 34 s01 = 1.9153760746e-02, /* 0x3c9ce859 */ 35 s02 = 1.8594678841e-04, /* 0x3942fab6 */ 36 s03 = 1.1771846857e-06, /* 0x359dffc2 */ 37 s04 = 5.0463624390e-09, /* 0x31ad6446 */ 38 s05 = 1.2354227016e-11; /* 0x2d59567e */ 39 40 static const float zero = 0.0; 41 42 float 43 __ieee754_j1f(float x) 44 { 45 float z, s,c,ss,cc,r,u,v,y; 46 int32_t hx,ix; 47 48 GET_FLOAT_WORD(hx,x); 49 ix = hx&0x7fffffff; 50 if(ix>=0x7f800000) return one/x; 51 y = fabsf(x); 52 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 53 s = sinf(y); 54 c = cosf(y); 55 ss = -s-c; 56 cc = s-c; 57 if(ix<0x7f000000) { /* make sure y+y not overflow */ 58 z = cosf(y+y); 59 if ((s*c)>zero) cc = z/ss; 60 else ss = z/cc; 61 } 62 /* 63 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) 64 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) 65 */ 66 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y); 67 else { 68 u = ponef(y); v = qonef(y); 69 z = invsqrtpi*(u*cc-v*ss)/sqrtf(y); 70 } 71 if(hx<0) return -z; 72 else return z; 73 } 74 if(ix<0x32000000) { /* |x|<2**-27 */ 75 if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */ 76 } 77 z = x*x; 78 r = z*(r00+z*(r01+z*(r02+z*r03))); 79 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); 80 r *= x; 81 return(x*(float)0.5+r/s); 82 } 83 84 static const float U0[5] = { 85 -1.9605709612e-01, /* 0xbe48c331 */ 86 5.0443872809e-02, /* 0x3d4e9e3c */ 87 -1.9125689287e-03, /* 0xbafaaf2a */ 88 2.3525259166e-05, /* 0x37c5581c */ 89 -9.1909917899e-08, /* 0xb3c56003 */ 90 }; 91 static const float V0[5] = { 92 1.9916731864e-02, /* 0x3ca3286a */ 93 2.0255257550e-04, /* 0x3954644b */ 94 1.3560879779e-06, /* 0x35b602d4 */ 95 6.2274145840e-09, /* 0x31d5f8eb */ 96 1.6655924903e-11, /* 0x2d9281cf */ 97 }; 98 99 float 100 __ieee754_y1f(float x) 101 { 102 float z, s,c,ss,cc,u,v; 103 int32_t hx,ix; 104 105 GET_FLOAT_WORD(hx,x); 106 ix = 0x7fffffff&hx; 107 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ 108 if(ix>=0x7f800000) return one/(x+x*x); 109 if(ix==0) return -one/zero; 110 if(hx<0) return zero/zero; 111 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 112 s = sinf(x); 113 c = cosf(x); 114 ss = -s-c; 115 cc = s-c; 116 if(ix<0x7f000000) { /* make sure x+x not overflow */ 117 z = cosf(x+x); 118 if ((s*c)>zero) cc = z/ss; 119 else ss = z/cc; 120 } 121 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) 122 * where x0 = x-3pi/4 123 * Better formula: 124 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 125 * = 1/sqrt(2) * (sin(x) - cos(x)) 126 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 127 * = -1/sqrt(2) * (cos(x) + sin(x)) 128 * To avoid cancellation, use 129 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 130 * to compute the worse one. 131 */ 132 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x); 133 else { 134 u = ponef(x); v = qonef(x); 135 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 136 } 137 return z; 138 } 139 if(ix<=0x24800000) { /* x < 2**-54 */ 140 return(-tpi/x); 141 } 142 z = x*x; 143 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); 144 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); 145 return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x)); 146 } 147 148 /* For x >= 8, the asymptotic expansions of pone is 149 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. 150 * We approximate pone by 151 * pone(x) = 1 + (R/S) 152 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 153 * S = 1 + ps0*s^2 + ... + ps4*s^10 154 * and 155 * | pone(x)-1-R/S | <= 2 ** ( -60.06) 156 */ 157 158 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 159 0.0000000000e+00, /* 0x00000000 */ 160 1.1718750000e-01, /* 0x3df00000 */ 161 1.3239480972e+01, /* 0x4153d4ea */ 162 4.1205184937e+02, /* 0x43ce06a3 */ 163 3.8747453613e+03, /* 0x45722bed */ 164 7.9144794922e+03, /* 0x45f753d6 */ 165 }; 166 static const float ps8[5] = { 167 1.1420736694e+02, /* 0x42e46a2c */ 168 3.6509309082e+03, /* 0x45642ee5 */ 169 3.6956207031e+04, /* 0x47105c35 */ 170 9.7602796875e+04, /* 0x47bea166 */ 171 3.0804271484e+04, /* 0x46f0a88b */ 172 }; 173 174 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 175 1.3199052094e-11, /* 0x2d68333f */ 176 1.1718749255e-01, /* 0x3defffff */ 177 6.8027510643e+00, /* 0x40d9b023 */ 178 1.0830818176e+02, /* 0x42d89dca */ 179 5.1763616943e+02, /* 0x440168b7 */ 180 5.2871520996e+02, /* 0x44042dc6 */ 181 }; 182 static const float ps5[5] = { 183 5.9280597687e+01, /* 0x426d1f55 */ 184 9.9140142822e+02, /* 0x4477d9b1 */ 185 5.3532670898e+03, /* 0x45a74a23 */ 186 7.8446904297e+03, /* 0x45f52586 */ 187 1.5040468750e+03, /* 0x44bc0180 */ 188 }; 189 190 static const float pr3[6] = { 191 3.0250391081e-09, /* 0x314fe10d */ 192 1.1718686670e-01, /* 0x3defffab */ 193 3.9329774380e+00, /* 0x407bb5e7 */ 194 3.5119403839e+01, /* 0x420c7a45 */ 195 9.1055007935e+01, /* 0x42b61c2a */ 196 4.8559066772e+01, /* 0x42423c7c */ 197 }; 198 static const float ps3[5] = { 199 3.4791309357e+01, /* 0x420b2a4d */ 200 3.3676245117e+02, /* 0x43a86198 */ 201 1.0468714600e+03, /* 0x4482dbe3 */ 202 8.9081134033e+02, /* 0x445eb3ed */ 203 1.0378793335e+02, /* 0x42cf936c */ 204 }; 205 206 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 207 1.0771083225e-07, /* 0x33e74ea8 */ 208 1.1717621982e-01, /* 0x3deffa16 */ 209 2.3685150146e+00, /* 0x401795c0 */ 210 1.2242610931e+01, /* 0x4143e1bc */ 211 1.7693971634e+01, /* 0x418d8d41 */ 212 5.0735230446e+00, /* 0x40a25a4d */ 213 }; 214 static const float ps2[5] = { 215 2.1436485291e+01, /* 0x41ab7dec */ 216 1.2529022980e+02, /* 0x42fa9499 */ 217 2.3227647400e+02, /* 0x436846c7 */ 218 1.1767937469e+02, /* 0x42eb5bd7 */ 219 8.3646392822e+00, /* 0x4105d590 */ 220 }; 221 222 static float ponef(float x) 223 { 224 const float *p,*q; 225 float z,r,s; 226 int32_t ix; 227 GET_FLOAT_WORD(ix,x); 228 ix &= 0x7fffffff; 229 if(ix>=0x41000000) {p = pr8; q= ps8;} 230 else if(ix>=0x40f71c58){p = pr5; q= ps5;} 231 else if(ix>=0x4036db68){p = pr3; q= ps3;} 232 else if(ix>=0x40000000){p = pr2; q= ps2;} 233 z = one/(x*x); 234 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 235 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 236 return one+ r/s; 237 } 238 239 240 /* For x >= 8, the asymptotic expansions of qone is 241 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. 242 * We approximate pone by 243 * qone(x) = s*(0.375 + (R/S)) 244 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 245 * S = 1 + qs1*s^2 + ... + qs6*s^12 246 * and 247 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) 248 */ 249 250 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 251 0.0000000000e+00, /* 0x00000000 */ 252 -1.0253906250e-01, /* 0xbdd20000 */ 253 -1.6271753311e+01, /* 0xc1822c8d */ 254 -7.5960174561e+02, /* 0xc43de683 */ 255 -1.1849806641e+04, /* 0xc639273a */ 256 -4.8438511719e+04, /* 0xc73d3683 */ 257 }; 258 static const float qs8[6] = { 259 1.6139537048e+02, /* 0x43216537 */ 260 7.8253862305e+03, /* 0x45f48b17 */ 261 1.3387534375e+05, /* 0x4802bcd6 */ 262 7.1965775000e+05, /* 0x492fb29c */ 263 6.6660125000e+05, /* 0x4922be94 */ 264 -2.9449025000e+05, /* 0xc88fcb48 */ 265 }; 266 267 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 268 -2.0897993405e-11, /* 0xadb7d219 */ 269 -1.0253904760e-01, /* 0xbdd1fffe */ 270 -8.0564479828e+00, /* 0xc100e736 */ 271 -1.8366960144e+02, /* 0xc337ab6b */ 272 -1.3731937256e+03, /* 0xc4aba633 */ 273 -2.6124443359e+03, /* 0xc523471c */ 274 }; 275 static const float qs5[6] = { 276 8.1276550293e+01, /* 0x42a28d98 */ 277 1.9917987061e+03, /* 0x44f8f98f */ 278 1.7468484375e+04, /* 0x468878f8 */ 279 4.9851425781e+04, /* 0x4742bb6d */ 280 2.7948074219e+04, /* 0x46da5826 */ 281 -4.7191835938e+03, /* 0xc5937978 */ 282 }; 283 284 static const float qr3[6] = { 285 -5.0783124372e-09, /* 0xb1ae7d4f */ 286 -1.0253783315e-01, /* 0xbdd1ff5b */ 287 -4.6101160049e+00, /* 0xc0938612 */ 288 -5.7847221375e+01, /* 0xc267638e */ 289 -2.2824453735e+02, /* 0xc3643e9a */ 290 -2.1921012878e+02, /* 0xc35b35cb */ 291 }; 292 static const float qs3[6] = { 293 4.7665153503e+01, /* 0x423ea91e */ 294 6.7386511230e+02, /* 0x4428775e */ 295 3.3801528320e+03, /* 0x45534272 */ 296 5.5477290039e+03, /* 0x45ad5dd5 */ 297 1.9031191406e+03, /* 0x44ede3d0 */ 298 -1.3520118713e+02, /* 0xc3073381 */ 299 }; 300 301 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 302 -1.7838172539e-07, /* 0xb43f8932 */ 303 -1.0251704603e-01, /* 0xbdd1f475 */ 304 -2.7522056103e+00, /* 0xc0302423 */ 305 -1.9663616180e+01, /* 0xc19d4f16 */ 306 -4.2325313568e+01, /* 0xc2294d1f */ 307 -2.1371921539e+01, /* 0xc1aaf9b2 */ 308 }; 309 static const float qs2[6] = { 310 2.9533363342e+01, /* 0x41ec4454 */ 311 2.5298155212e+02, /* 0x437cfb47 */ 312 7.5750280762e+02, /* 0x443d602e */ 313 7.3939318848e+02, /* 0x4438d92a */ 314 1.5594900513e+02, /* 0x431bf2f2 */ 315 -4.9594988823e+00, /* 0xc09eb437 */ 316 }; 317 318 static float qonef(float x) 319 { 320 const float *p,*q; 321 float s,r,z; 322 int32_t ix; 323 GET_FLOAT_WORD(ix,x); 324 ix &= 0x7fffffff; 325 if(ix>=0x40200000) {p = qr8; q= qs8;} 326 else if(ix>=0x40f71c58){p = qr5; q= qs5;} 327 else if(ix>=0x4036db68){p = qr3; q= qs3;} 328 else if(ix>=0x40000000){p = qr2; q= qs2;} 329 z = one/(x*x); 330 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 331 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 332 return ((float).375 + r/s)/x; 333 } 334