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 /* 20 * See e_j1.c for complete comments. 21 */ 22 23 #include "math.h" 24 #include "math_private.h" 25 26 static float ponef(float), qonef(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] */ 36 r00 = -6.2500000000e-02, /* 0xbd800000 */ 37 r01 = 1.4070566976e-03, /* 0x3ab86cfd */ 38 r02 = -1.5995563444e-05, /* 0xb7862e36 */ 39 r03 = 4.9672799207e-08, /* 0x335557d2 */ 40 s01 = 1.9153760746e-02, /* 0x3c9ce859 */ 41 s02 = 1.8594678841e-04, /* 0x3942fab6 */ 42 s03 = 1.1771846857e-06, /* 0x359dffc2 */ 43 s04 = 5.0463624390e-09, /* 0x31ad6446 */ 44 s05 = 1.2354227016e-11; /* 0x2d59567e */ 45 46 static const float zero = 0.0; 47 48 float 49 __ieee754_j1f(float x) 50 { 51 float z, s,c,ss,cc,r,u,v,y; 52 int32_t hx,ix; 53 54 GET_FLOAT_WORD(hx,x); 55 ix = hx&0x7fffffff; 56 if(ix>=0x7f800000) return one/x; 57 y = fabsf(x); 58 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 59 s = sinf(y); 60 c = cosf(y); 61 ss = -s-c; 62 cc = s-c; 63 if(ix<0x7f000000) { /* make sure y+y not overflow */ 64 z = cosf(y+y); 65 if ((s*c)>zero) cc = z/ss; 66 else ss = z/cc; 67 } 68 /* 69 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) 70 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) 71 */ 72 if(ix>0x58000000) z = (invsqrtpi*cc)/sqrtf(y); /* |x|>2**49 */ 73 else { 74 u = ponef(y); v = qonef(y); 75 z = invsqrtpi*(u*cc-v*ss)/sqrtf(y); 76 } 77 if(hx<0) return -z; 78 else return z; 79 } 80 if(ix<0x39000000) { /* |x|<2**-13 */ 81 if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */ 82 } 83 z = x*x; 84 r = z*(r00+z*(r01+z*(r02+z*r03))); 85 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); 86 r *= x; 87 return(x*(float)0.5+r/s); 88 } 89 90 static const float U0[5] = { 91 -1.9605709612e-01, /* 0xbe48c331 */ 92 5.0443872809e-02, /* 0x3d4e9e3c */ 93 -1.9125689287e-03, /* 0xbafaaf2a */ 94 2.3525259166e-05, /* 0x37c5581c */ 95 -9.1909917899e-08, /* 0xb3c56003 */ 96 }; 97 static const float V0[5] = { 98 1.9916731864e-02, /* 0x3ca3286a */ 99 2.0255257550e-04, /* 0x3954644b */ 100 1.3560879779e-06, /* 0x35b602d4 */ 101 6.2274145840e-09, /* 0x31d5f8eb */ 102 1.6655924903e-11, /* 0x2d9281cf */ 103 }; 104 105 float 106 __ieee754_y1f(float x) 107 { 108 float z, s,c,ss,cc,u,v; 109 int32_t hx,ix; 110 111 GET_FLOAT_WORD(hx,x); 112 ix = 0x7fffffff&hx; 113 if(ix>=0x7f800000) return vone/(x+x*x); 114 if(ix==0) return -one/vzero; 115 if(hx<0) return vzero/vzero; 116 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 117 s = sinf(x); 118 c = cosf(x); 119 ss = -s-c; 120 cc = s-c; 121 if(ix<0x7f000000) { /* make sure x+x not overflow */ 122 z = cosf(x+x); 123 if ((s*c)>zero) cc = z/ss; 124 else ss = z/cc; 125 } 126 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) 127 * where x0 = x-3pi/4 128 * Better formula: 129 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 130 * = 1/sqrt(2) * (sin(x) - cos(x)) 131 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 132 * = -1/sqrt(2) * (cos(x) + sin(x)) 133 * To avoid cancellation, use 134 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 135 * to compute the worse one. 136 */ 137 if(ix>0x58000000) z = (invsqrtpi*ss)/sqrtf(x); /* |x|>2**49 */ 138 else { 139 u = ponef(x); v = qonef(x); 140 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 141 } 142 return z; 143 } 144 if(ix<=0x33000000) { /* x < 2**-25 */ 145 return(-tpi/x); 146 } 147 z = x*x; 148 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); 149 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); 150 return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x)); 151 } 152 153 /* For x >= 8, the asymptotic expansions of pone is 154 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. 155 * We approximate pone by 156 * pone(x) = 1 + (R/S) 157 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 158 * S = 1 + ps0*s^2 + ... + ps4*s^10 159 * and 160 * | pone(x)-1-R/S | <= 2 ** ( -60.06) 161 */ 162 163 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 164 0.0000000000e+00, /* 0x00000000 */ 165 1.1718750000e-01, /* 0x3df00000 */ 166 1.3239480972e+01, /* 0x4153d4ea */ 167 4.1205184937e+02, /* 0x43ce06a3 */ 168 3.8747453613e+03, /* 0x45722bed */ 169 7.9144794922e+03, /* 0x45f753d6 */ 170 }; 171 static const float ps8[5] = { 172 1.1420736694e+02, /* 0x42e46a2c */ 173 3.6509309082e+03, /* 0x45642ee5 */ 174 3.6956207031e+04, /* 0x47105c35 */ 175 9.7602796875e+04, /* 0x47bea166 */ 176 3.0804271484e+04, /* 0x46f0a88b */ 177 }; 178 179 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 180 1.3199052094e-11, /* 0x2d68333f */ 181 1.1718749255e-01, /* 0x3defffff */ 182 6.8027510643e+00, /* 0x40d9b023 */ 183 1.0830818176e+02, /* 0x42d89dca */ 184 5.1763616943e+02, /* 0x440168b7 */ 185 5.2871520996e+02, /* 0x44042dc6 */ 186 }; 187 static const float ps5[5] = { 188 5.9280597687e+01, /* 0x426d1f55 */ 189 9.9140142822e+02, /* 0x4477d9b1 */ 190 5.3532670898e+03, /* 0x45a74a23 */ 191 7.8446904297e+03, /* 0x45f52586 */ 192 1.5040468750e+03, /* 0x44bc0180 */ 193 }; 194 195 static const float pr3[6] = { 196 3.0250391081e-09, /* 0x314fe10d */ 197 1.1718686670e-01, /* 0x3defffab */ 198 3.9329774380e+00, /* 0x407bb5e7 */ 199 3.5119403839e+01, /* 0x420c7a45 */ 200 9.1055007935e+01, /* 0x42b61c2a */ 201 4.8559066772e+01, /* 0x42423c7c */ 202 }; 203 static const float ps3[5] = { 204 3.4791309357e+01, /* 0x420b2a4d */ 205 3.3676245117e+02, /* 0x43a86198 */ 206 1.0468714600e+03, /* 0x4482dbe3 */ 207 8.9081134033e+02, /* 0x445eb3ed */ 208 1.0378793335e+02, /* 0x42cf936c */ 209 }; 210 211 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 212 1.0771083225e-07, /* 0x33e74ea8 */ 213 1.1717621982e-01, /* 0x3deffa16 */ 214 2.3685150146e+00, /* 0x401795c0 */ 215 1.2242610931e+01, /* 0x4143e1bc */ 216 1.7693971634e+01, /* 0x418d8d41 */ 217 5.0735230446e+00, /* 0x40a25a4d */ 218 }; 219 static const float ps2[5] = { 220 2.1436485291e+01, /* 0x41ab7dec */ 221 1.2529022980e+02, /* 0x42fa9499 */ 222 2.3227647400e+02, /* 0x436846c7 */ 223 1.1767937469e+02, /* 0x42eb5bd7 */ 224 8.3646392822e+00, /* 0x4105d590 */ 225 }; 226 227 static __inline float 228 ponef(float x) 229 { 230 const float *p,*q; 231 float z,r,s; 232 int32_t ix; 233 GET_FLOAT_WORD(ix,x); 234 ix &= 0x7fffffff; 235 if(ix>=0x41000000) {p = pr8; q= ps8;} 236 else if(ix>=0x409173eb){p = pr5; q= ps5;} 237 else if(ix>=0x4036d917){p = pr3; q= ps3;} 238 else {p = pr2; q= ps2;} /* ix>=0x40000000 */ 239 z = one/(x*x); 240 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 241 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 242 return one+ r/s; 243 } 244 245 246 /* For x >= 8, the asymptotic expansions of qone is 247 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. 248 * We approximate pone by 249 * qone(x) = s*(0.375 + (R/S)) 250 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 251 * S = 1 + qs1*s^2 + ... + qs6*s^12 252 * and 253 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) 254 */ 255 256 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 257 0.0000000000e+00, /* 0x00000000 */ 258 -1.0253906250e-01, /* 0xbdd20000 */ 259 -1.6271753311e+01, /* 0xc1822c8d */ 260 -7.5960174561e+02, /* 0xc43de683 */ 261 -1.1849806641e+04, /* 0xc639273a */ 262 -4.8438511719e+04, /* 0xc73d3683 */ 263 }; 264 static const float qs8[6] = { 265 1.6139537048e+02, /* 0x43216537 */ 266 7.8253862305e+03, /* 0x45f48b17 */ 267 1.3387534375e+05, /* 0x4802bcd6 */ 268 7.1965775000e+05, /* 0x492fb29c */ 269 6.6660125000e+05, /* 0x4922be94 */ 270 -2.9449025000e+05, /* 0xc88fcb48 */ 271 }; 272 273 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 274 -2.0897993405e-11, /* 0xadb7d219 */ 275 -1.0253904760e-01, /* 0xbdd1fffe */ 276 -8.0564479828e+00, /* 0xc100e736 */ 277 -1.8366960144e+02, /* 0xc337ab6b */ 278 -1.3731937256e+03, /* 0xc4aba633 */ 279 -2.6124443359e+03, /* 0xc523471c */ 280 }; 281 static const float qs5[6] = { 282 8.1276550293e+01, /* 0x42a28d98 */ 283 1.9917987061e+03, /* 0x44f8f98f */ 284 1.7468484375e+04, /* 0x468878f8 */ 285 4.9851425781e+04, /* 0x4742bb6d */ 286 2.7948074219e+04, /* 0x46da5826 */ 287 -4.7191835938e+03, /* 0xc5937978 */ 288 }; 289 290 static const float qr3[6] = { 291 -5.0783124372e-09, /* 0xb1ae7d4f */ 292 -1.0253783315e-01, /* 0xbdd1ff5b */ 293 -4.6101160049e+00, /* 0xc0938612 */ 294 -5.7847221375e+01, /* 0xc267638e */ 295 -2.2824453735e+02, /* 0xc3643e9a */ 296 -2.1921012878e+02, /* 0xc35b35cb */ 297 }; 298 static const float qs3[6] = { 299 4.7665153503e+01, /* 0x423ea91e */ 300 6.7386511230e+02, /* 0x4428775e */ 301 3.3801528320e+03, /* 0x45534272 */ 302 5.5477290039e+03, /* 0x45ad5dd5 */ 303 1.9031191406e+03, /* 0x44ede3d0 */ 304 -1.3520118713e+02, /* 0xc3073381 */ 305 }; 306 307 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 308 -1.7838172539e-07, /* 0xb43f8932 */ 309 -1.0251704603e-01, /* 0xbdd1f475 */ 310 -2.7522056103e+00, /* 0xc0302423 */ 311 -1.9663616180e+01, /* 0xc19d4f16 */ 312 -4.2325313568e+01, /* 0xc2294d1f */ 313 -2.1371921539e+01, /* 0xc1aaf9b2 */ 314 }; 315 static const float qs2[6] = { 316 2.9533363342e+01, /* 0x41ec4454 */ 317 2.5298155212e+02, /* 0x437cfb47 */ 318 7.5750280762e+02, /* 0x443d602e */ 319 7.3939318848e+02, /* 0x4438d92a */ 320 1.5594900513e+02, /* 0x431bf2f2 */ 321 -4.9594988823e+00, /* 0xc09eb437 */ 322 }; 323 324 static __inline float 325 qonef(float x) 326 { 327 const float *p,*q; 328 float s,r,z; 329 int32_t ix; 330 GET_FLOAT_WORD(ix,x); 331 ix &= 0x7fffffff; 332 if(ix>=0x41000000) {p = qr8; q= qs8;} 333 else if(ix>=0x409173eb){p = qr5; q= qs5;} 334 else if(ix>=0x4036d917){p = qr3; q= qs3;} 335 else {p = qr2; q= qs2;} /* ix>=0x40000000 */ 336 z = one/(x*x); 337 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 338 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 339 return ((float).375 + r/s)/x; 340 } 341