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 __inline 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 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 sincosf(y, &s, &c); 60 ss = -s-c; 61 cc = s-c; 62 if(ix<0x7f000000) { /* make sure y+y not overflow */ 63 z = cosf(y+y); 64 if ((s*c)>zero) cc = z/ss; 65 else ss = z/cc; 66 } 67 /* 68 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) 69 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) 70 */ 71 if(ix>0x58000000) z = (invsqrtpi*cc)/sqrtf(y); /* |x|>2**49 */ 72 else { 73 u = ponef(y); v = qonef(y); 74 z = invsqrtpi*(u*cc-v*ss)/sqrtf(y); 75 } 76 if(hx<0) return -z; 77 else return z; 78 } 79 if(ix<0x39000000) { /* |x|<2**-13 */ 80 if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */ 81 } 82 z = x*x; 83 r = z*(r00+z*(r01+z*(r02+z*r03))); 84 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); 85 r *= x; 86 return(x*(float)0.5+r/s); 87 } 88 89 static const float U0[5] = { 90 -1.9605709612e-01, /* 0xbe48c331 */ 91 5.0443872809e-02, /* 0x3d4e9e3c */ 92 -1.9125689287e-03, /* 0xbafaaf2a */ 93 2.3525259166e-05, /* 0x37c5581c */ 94 -9.1909917899e-08, /* 0xb3c56003 */ 95 }; 96 static const float V0[5] = { 97 1.9916731864e-02, /* 0x3ca3286a */ 98 2.0255257550e-04, /* 0x3954644b */ 99 1.3560879779e-06, /* 0x35b602d4 */ 100 6.2274145840e-09, /* 0x31d5f8eb */ 101 1.6655924903e-11, /* 0x2d9281cf */ 102 }; 103 104 float 105 y1f(float x) 106 { 107 float z, s,c,ss,cc,u,v; 108 int32_t hx,ix; 109 110 GET_FLOAT_WORD(hx,x); 111 ix = 0x7fffffff&hx; 112 if(ix>=0x7f800000) return vone/(x+x*x); 113 if(ix==0) return -one/vzero; 114 if(hx<0) return vzero/vzero; 115 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 116 sincosf(x, &s, &c); 117 ss = -s-c; 118 cc = s-c; 119 if(ix<0x7f000000) { /* make sure x+x not overflow */ 120 z = cosf(x+x); 121 if ((s*c)>zero) cc = z/ss; 122 else ss = z/cc; 123 } 124 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) 125 * where x0 = x-3pi/4 126 * Better formula: 127 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 128 * = 1/sqrt(2) * (sin(x) - cos(x)) 129 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 130 * = -1/sqrt(2) * (cos(x) + sin(x)) 131 * To avoid cancellation, use 132 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 133 * to compute the worse one. 134 */ 135 if(ix>0x58000000) z = (invsqrtpi*ss)/sqrtf(x); /* |x|>2**49 */ 136 else { 137 u = ponef(x); v = qonef(x); 138 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 139 } 140 return z; 141 } 142 if(ix<=0x33000000) { /* x < 2**-25 */ 143 return(-tpi/x); 144 } 145 z = x*x; 146 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); 147 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); 148 return(x*(u/v) + tpi*(j1f(x)*logf(x)-one/x)); 149 } 150 151 /* For x >= 8, the asymptotic expansions of pone is 152 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. 153 * We approximate pone by 154 * pone(x) = 1 + (R/S) 155 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 156 * S = 1 + ps0*s^2 + ... + ps4*s^10 157 * and 158 * | pone(x)-1-R/S | <= 2 ** ( -60.06) 159 */ 160 161 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 162 0.0000000000e+00, /* 0x00000000 */ 163 1.1718750000e-01, /* 0x3df00000 */ 164 1.3239480972e+01, /* 0x4153d4ea */ 165 4.1205184937e+02, /* 0x43ce06a3 */ 166 3.8747453613e+03, /* 0x45722bed */ 167 7.9144794922e+03, /* 0x45f753d6 */ 168 }; 169 static const float ps8[5] = { 170 1.1420736694e+02, /* 0x42e46a2c */ 171 3.6509309082e+03, /* 0x45642ee5 */ 172 3.6956207031e+04, /* 0x47105c35 */ 173 9.7602796875e+04, /* 0x47bea166 */ 174 3.0804271484e+04, /* 0x46f0a88b */ 175 }; 176 177 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 178 1.3199052094e-11, /* 0x2d68333f */ 179 1.1718749255e-01, /* 0x3defffff */ 180 6.8027510643e+00, /* 0x40d9b023 */ 181 1.0830818176e+02, /* 0x42d89dca */ 182 5.1763616943e+02, /* 0x440168b7 */ 183 5.2871520996e+02, /* 0x44042dc6 */ 184 }; 185 static const float ps5[5] = { 186 5.9280597687e+01, /* 0x426d1f55 */ 187 9.9140142822e+02, /* 0x4477d9b1 */ 188 5.3532670898e+03, /* 0x45a74a23 */ 189 7.8446904297e+03, /* 0x45f52586 */ 190 1.5040468750e+03, /* 0x44bc0180 */ 191 }; 192 193 static const float pr3[6] = { 194 3.0250391081e-09, /* 0x314fe10d */ 195 1.1718686670e-01, /* 0x3defffab */ 196 3.9329774380e+00, /* 0x407bb5e7 */ 197 3.5119403839e+01, /* 0x420c7a45 */ 198 9.1055007935e+01, /* 0x42b61c2a */ 199 4.8559066772e+01, /* 0x42423c7c */ 200 }; 201 static const float ps3[5] = { 202 3.4791309357e+01, /* 0x420b2a4d */ 203 3.3676245117e+02, /* 0x43a86198 */ 204 1.0468714600e+03, /* 0x4482dbe3 */ 205 8.9081134033e+02, /* 0x445eb3ed */ 206 1.0378793335e+02, /* 0x42cf936c */ 207 }; 208 209 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 210 1.0771083225e-07, /* 0x33e74ea8 */ 211 1.1717621982e-01, /* 0x3deffa16 */ 212 2.3685150146e+00, /* 0x401795c0 */ 213 1.2242610931e+01, /* 0x4143e1bc */ 214 1.7693971634e+01, /* 0x418d8d41 */ 215 5.0735230446e+00, /* 0x40a25a4d */ 216 }; 217 static const float ps2[5] = { 218 2.1436485291e+01, /* 0x41ab7dec */ 219 1.2529022980e+02, /* 0x42fa9499 */ 220 2.3227647400e+02, /* 0x436846c7 */ 221 1.1767937469e+02, /* 0x42eb5bd7 */ 222 8.3646392822e+00, /* 0x4105d590 */ 223 }; 224 225 static __inline float 226 ponef(float x) 227 { 228 const float *p,*q; 229 float z,r,s; 230 int32_t ix; 231 GET_FLOAT_WORD(ix,x); 232 ix &= 0x7fffffff; 233 if(ix>=0x41000000) {p = pr8; q= ps8;} 234 else if(ix>=0x409173eb){p = pr5; q= ps5;} 235 else if(ix>=0x4036d917){p = pr3; q= ps3;} 236 else {p = pr2; q= ps2;} /* ix>=0x40000000 */ 237 z = one/(x*x); 238 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 239 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 240 return one+ r/s; 241 } 242 243 244 /* For x >= 8, the asymptotic expansions of qone is 245 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. 246 * We approximate pone by 247 * qone(x) = s*(0.375 + (R/S)) 248 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 249 * S = 1 + qs1*s^2 + ... + qs6*s^12 250 * and 251 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) 252 */ 253 254 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 255 0.0000000000e+00, /* 0x00000000 */ 256 -1.0253906250e-01, /* 0xbdd20000 */ 257 -1.6271753311e+01, /* 0xc1822c8d */ 258 -7.5960174561e+02, /* 0xc43de683 */ 259 -1.1849806641e+04, /* 0xc639273a */ 260 -4.8438511719e+04, /* 0xc73d3683 */ 261 }; 262 static const float qs8[6] = { 263 1.6139537048e+02, /* 0x43216537 */ 264 7.8253862305e+03, /* 0x45f48b17 */ 265 1.3387534375e+05, /* 0x4802bcd6 */ 266 7.1965775000e+05, /* 0x492fb29c */ 267 6.6660125000e+05, /* 0x4922be94 */ 268 -2.9449025000e+05, /* 0xc88fcb48 */ 269 }; 270 271 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 272 -2.0897993405e-11, /* 0xadb7d219 */ 273 -1.0253904760e-01, /* 0xbdd1fffe */ 274 -8.0564479828e+00, /* 0xc100e736 */ 275 -1.8366960144e+02, /* 0xc337ab6b */ 276 -1.3731937256e+03, /* 0xc4aba633 */ 277 -2.6124443359e+03, /* 0xc523471c */ 278 }; 279 static const float qs5[6] = { 280 8.1276550293e+01, /* 0x42a28d98 */ 281 1.9917987061e+03, /* 0x44f8f98f */ 282 1.7468484375e+04, /* 0x468878f8 */ 283 4.9851425781e+04, /* 0x4742bb6d */ 284 2.7948074219e+04, /* 0x46da5826 */ 285 -4.7191835938e+03, /* 0xc5937978 */ 286 }; 287 288 static const float qr3[6] = { 289 -5.0783124372e-09, /* 0xb1ae7d4f */ 290 -1.0253783315e-01, /* 0xbdd1ff5b */ 291 -4.6101160049e+00, /* 0xc0938612 */ 292 -5.7847221375e+01, /* 0xc267638e */ 293 -2.2824453735e+02, /* 0xc3643e9a */ 294 -2.1921012878e+02, /* 0xc35b35cb */ 295 }; 296 static const float qs3[6] = { 297 4.7665153503e+01, /* 0x423ea91e */ 298 6.7386511230e+02, /* 0x4428775e */ 299 3.3801528320e+03, /* 0x45534272 */ 300 5.5477290039e+03, /* 0x45ad5dd5 */ 301 1.9031191406e+03, /* 0x44ede3d0 */ 302 -1.3520118713e+02, /* 0xc3073381 */ 303 }; 304 305 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 306 -1.7838172539e-07, /* 0xb43f8932 */ 307 -1.0251704603e-01, /* 0xbdd1f475 */ 308 -2.7522056103e+00, /* 0xc0302423 */ 309 -1.9663616180e+01, /* 0xc19d4f16 */ 310 -4.2325313568e+01, /* 0xc2294d1f */ 311 -2.1371921539e+01, /* 0xc1aaf9b2 */ 312 }; 313 static const float qs2[6] = { 314 2.9533363342e+01, /* 0x41ec4454 */ 315 2.5298155212e+02, /* 0x437cfb47 */ 316 7.5750280762e+02, /* 0x443d602e */ 317 7.3939318848e+02, /* 0x4438d92a */ 318 1.5594900513e+02, /* 0x431bf2f2 */ 319 -4.9594988823e+00, /* 0xc09eb437 */ 320 }; 321 322 static __inline float 323 qonef(float x) 324 { 325 const float *p,*q; 326 float s,r,z; 327 int32_t ix; 328 GET_FLOAT_WORD(ix,x); 329 ix &= 0x7fffffff; 330 if(ix>=0x41000000) {p = qr8; q= qs8;} 331 else if(ix>=0x409173eb){p = qr5; q= qs5;} 332 else if(ix>=0x4036d917){p = qr3; q= qs3;} 333 else {p = qr2; q= qs2;} /* ix>=0x40000000 */ 334 z = one/(x*x); 335 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 336 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 337 return ((float).375 + r/s)/x; 338 } 339