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 "math.h" 17 #include "math_private.h" 18 19 static float ponef(float), qonef(float); 20 21 static const float 22 huge = 1e30, 23 one = 1.0, 24 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 25 tpi = 6.3661974669e-01, /* 0x3f22f983 */ 26 /* R0/S0 on [0,2] */ 27 r00 = -6.2500000000e-02, /* 0xbd800000 */ 28 r01 = 1.4070566976e-03, /* 0x3ab86cfd */ 29 r02 = -1.5995563444e-05, /* 0xb7862e36 */ 30 r03 = 4.9672799207e-08, /* 0x335557d2 */ 31 s01 = 1.9153760746e-02, /* 0x3c9ce859 */ 32 s02 = 1.8594678841e-04, /* 0x3942fab6 */ 33 s03 = 1.1771846857e-06, /* 0x359dffc2 */ 34 s04 = 5.0463624390e-09, /* 0x31ad6446 */ 35 s05 = 1.2354227016e-11; /* 0x2d59567e */ 36 37 static const float zero = 0.0; 38 39 float 40 j1f(float x) 41 { 42 float z, s,c,ss,cc,r,u,v,y; 43 int32_t hx,ix; 44 45 GET_FLOAT_WORD(hx,x); 46 ix = hx&0x7fffffff; 47 if(ix>=0x7f800000) return one/x; 48 y = fabsf(x); 49 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 50 s = sinf(y); 51 c = cosf(y); 52 ss = -s-c; 53 cc = s-c; 54 if(ix<0x7f000000) { /* make sure y+y not overflow */ 55 z = cosf(y+y); 56 if ((s*c)>zero) cc = z/ss; 57 else ss = z/cc; 58 } 59 /* 60 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) 61 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) 62 */ 63 if((u_int64_t)ix>0x80000000U) z = (invsqrtpi*cc)/sqrtf(y); 64 else { 65 u = ponef(y); v = qonef(y); 66 z = invsqrtpi*(u*cc-v*ss)/sqrtf(y); 67 } 68 if(hx<0) return -z; 69 else return z; 70 } 71 if(ix<0x32000000) { /* |x|<2**-27 */ 72 if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */ 73 } 74 z = x*x; 75 r = z*(r00+z*(r01+z*(r02+z*r03))); 76 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); 77 r *= x; 78 return(x*(float)0.5+r/s); 79 } 80 81 static const float U0[5] = { 82 -1.9605709612e-01, /* 0xbe48c331 */ 83 5.0443872809e-02, /* 0x3d4e9e3c */ 84 -1.9125689287e-03, /* 0xbafaaf2a */ 85 2.3525259166e-05, /* 0x37c5581c */ 86 -9.1909917899e-08, /* 0xb3c56003 */ 87 }; 88 static const float V0[5] = { 89 1.9916731864e-02, /* 0x3ca3286a */ 90 2.0255257550e-04, /* 0x3954644b */ 91 1.3560879779e-06, /* 0x35b602d4 */ 92 6.2274145840e-09, /* 0x31d5f8eb */ 93 1.6655924903e-11, /* 0x2d9281cf */ 94 }; 95 96 float 97 y1f(float x) 98 { 99 float z, s,c,ss,cc,u,v; 100 int32_t hx,ix; 101 102 GET_FLOAT_WORD(hx,x); 103 ix = 0x7fffffff&hx; 104 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ 105 if(ix>=0x7f800000) return one/(x+x*x); 106 if(ix==0) return -one/zero; 107 if(hx<0) return zero/zero; 108 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 109 s = sinf(x); 110 c = cosf(x); 111 ss = -s-c; 112 cc = s-c; 113 if(ix<0x7f000000) { /* make sure x+x not overflow */ 114 z = cosf(x+x); 115 if ((s*c)>zero) cc = z/ss; 116 else ss = z/cc; 117 } 118 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) 119 * where x0 = x-3pi/4 120 * Better formula: 121 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 122 * = 1/sqrt(2) * (sin(x) - cos(x)) 123 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 124 * = -1/sqrt(2) * (cos(x) + sin(x)) 125 * To avoid cancellation, use 126 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 127 * to compute the worse one. 128 */ 129 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x); 130 else { 131 u = ponef(x); v = qonef(x); 132 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 133 } 134 return z; 135 } 136 if(ix<=0x24800000) { /* x < 2**-54 */ 137 return(-tpi/x); 138 } 139 z = x*x; 140 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); 141 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); 142 return(x*(u/v) + tpi*(j1f(x)*logf(x)-one/x)); 143 } 144 145 /* For x >= 8, the asymptotic expansions of pone is 146 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. 147 * We approximate pone by 148 * pone(x) = 1 + (R/S) 149 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 150 * S = 1 + ps0*s^2 + ... + ps4*s^10 151 * and 152 * | pone(x)-1-R/S | <= 2 ** ( -60.06) 153 */ 154 155 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 156 0.0000000000e+00, /* 0x00000000 */ 157 1.1718750000e-01, /* 0x3df00000 */ 158 1.3239480972e+01, /* 0x4153d4ea */ 159 4.1205184937e+02, /* 0x43ce06a3 */ 160 3.8747453613e+03, /* 0x45722bed */ 161 7.9144794922e+03, /* 0x45f753d6 */ 162 }; 163 static const float ps8[5] = { 164 1.1420736694e+02, /* 0x42e46a2c */ 165 3.6509309082e+03, /* 0x45642ee5 */ 166 3.6956207031e+04, /* 0x47105c35 */ 167 9.7602796875e+04, /* 0x47bea166 */ 168 3.0804271484e+04, /* 0x46f0a88b */ 169 }; 170 171 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 172 1.3199052094e-11, /* 0x2d68333f */ 173 1.1718749255e-01, /* 0x3defffff */ 174 6.8027510643e+00, /* 0x40d9b023 */ 175 1.0830818176e+02, /* 0x42d89dca */ 176 5.1763616943e+02, /* 0x440168b7 */ 177 5.2871520996e+02, /* 0x44042dc6 */ 178 }; 179 static const float ps5[5] = { 180 5.9280597687e+01, /* 0x426d1f55 */ 181 9.9140142822e+02, /* 0x4477d9b1 */ 182 5.3532670898e+03, /* 0x45a74a23 */ 183 7.8446904297e+03, /* 0x45f52586 */ 184 1.5040468750e+03, /* 0x44bc0180 */ 185 }; 186 187 static const float pr3[6] = { 188 3.0250391081e-09, /* 0x314fe10d */ 189 1.1718686670e-01, /* 0x3defffab */ 190 3.9329774380e+00, /* 0x407bb5e7 */ 191 3.5119403839e+01, /* 0x420c7a45 */ 192 9.1055007935e+01, /* 0x42b61c2a */ 193 4.8559066772e+01, /* 0x42423c7c */ 194 }; 195 static const float ps3[5] = { 196 3.4791309357e+01, /* 0x420b2a4d */ 197 3.3676245117e+02, /* 0x43a86198 */ 198 1.0468714600e+03, /* 0x4482dbe3 */ 199 8.9081134033e+02, /* 0x445eb3ed */ 200 1.0378793335e+02, /* 0x42cf936c */ 201 }; 202 203 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 204 1.0771083225e-07, /* 0x33e74ea8 */ 205 1.1717621982e-01, /* 0x3deffa16 */ 206 2.3685150146e+00, /* 0x401795c0 */ 207 1.2242610931e+01, /* 0x4143e1bc */ 208 1.7693971634e+01, /* 0x418d8d41 */ 209 5.0735230446e+00, /* 0x40a25a4d */ 210 }; 211 static const float ps2[5] = { 212 2.1436485291e+01, /* 0x41ab7dec */ 213 1.2529022980e+02, /* 0x42fa9499 */ 214 2.3227647400e+02, /* 0x436846c7 */ 215 1.1767937469e+02, /* 0x42eb5bd7 */ 216 8.3646392822e+00, /* 0x4105d590 */ 217 }; 218 219 static float 220 ponef(float x) 221 { 222 const float *p,*q; 223 float z,r,s; 224 int32_t ix; 225 GET_FLOAT_WORD(ix,x); 226 ix &= 0x7fffffff; 227 if(ix>=0x41000000) {p = pr8; q= ps8;} 228 else if(ix>=0x40f71c58){p = pr5; q= ps5;} 229 else if(ix>=0x4036db68){p = pr3; q= ps3;} 230 else /*if(ix>=0x40000000)*/ {p = pr2; q= ps2;} 231 z = one/(x*x); 232 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 233 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 234 return one+ r/s; 235 } 236 237 238 /* For x >= 8, the asymptotic expansions of qone is 239 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. 240 * We approximate pone by 241 * qone(x) = s*(0.375 + (R/S)) 242 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 243 * S = 1 + qs1*s^2 + ... + qs6*s^12 244 * and 245 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) 246 */ 247 248 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 249 0.0000000000e+00, /* 0x00000000 */ 250 -1.0253906250e-01, /* 0xbdd20000 */ 251 -1.6271753311e+01, /* 0xc1822c8d */ 252 -7.5960174561e+02, /* 0xc43de683 */ 253 -1.1849806641e+04, /* 0xc639273a */ 254 -4.8438511719e+04, /* 0xc73d3683 */ 255 }; 256 static const float qs8[6] = { 257 1.6139537048e+02, /* 0x43216537 */ 258 7.8253862305e+03, /* 0x45f48b17 */ 259 1.3387534375e+05, /* 0x4802bcd6 */ 260 7.1965775000e+05, /* 0x492fb29c */ 261 6.6660125000e+05, /* 0x4922be94 */ 262 -2.9449025000e+05, /* 0xc88fcb48 */ 263 }; 264 265 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 266 -2.0897993405e-11, /* 0xadb7d219 */ 267 -1.0253904760e-01, /* 0xbdd1fffe */ 268 -8.0564479828e+00, /* 0xc100e736 */ 269 -1.8366960144e+02, /* 0xc337ab6b */ 270 -1.3731937256e+03, /* 0xc4aba633 */ 271 -2.6124443359e+03, /* 0xc523471c */ 272 }; 273 static const float qs5[6] = { 274 8.1276550293e+01, /* 0x42a28d98 */ 275 1.9917987061e+03, /* 0x44f8f98f */ 276 1.7468484375e+04, /* 0x468878f8 */ 277 4.9851425781e+04, /* 0x4742bb6d */ 278 2.7948074219e+04, /* 0x46da5826 */ 279 -4.7191835938e+03, /* 0xc5937978 */ 280 }; 281 282 static const float qr3[6] = { 283 -5.0783124372e-09, /* 0xb1ae7d4f */ 284 -1.0253783315e-01, /* 0xbdd1ff5b */ 285 -4.6101160049e+00, /* 0xc0938612 */ 286 -5.7847221375e+01, /* 0xc267638e */ 287 -2.2824453735e+02, /* 0xc3643e9a */ 288 -2.1921012878e+02, /* 0xc35b35cb */ 289 }; 290 static const float qs3[6] = { 291 4.7665153503e+01, /* 0x423ea91e */ 292 6.7386511230e+02, /* 0x4428775e */ 293 3.3801528320e+03, /* 0x45534272 */ 294 5.5477290039e+03, /* 0x45ad5dd5 */ 295 1.9031191406e+03, /* 0x44ede3d0 */ 296 -1.3520118713e+02, /* 0xc3073381 */ 297 }; 298 299 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 300 -1.7838172539e-07, /* 0xb43f8932 */ 301 -1.0251704603e-01, /* 0xbdd1f475 */ 302 -2.7522056103e+00, /* 0xc0302423 */ 303 -1.9663616180e+01, /* 0xc19d4f16 */ 304 -4.2325313568e+01, /* 0xc2294d1f */ 305 -2.1371921539e+01, /* 0xc1aaf9b2 */ 306 }; 307 static const float qs2[6] = { 308 2.9533363342e+01, /* 0x41ec4454 */ 309 2.5298155212e+02, /* 0x437cfb47 */ 310 7.5750280762e+02, /* 0x443d602e */ 311 7.3939318848e+02, /* 0x4438d92a */ 312 1.5594900513e+02, /* 0x431bf2f2 */ 313 -4.9594988823e+00, /* 0xc09eb437 */ 314 }; 315 316 static float 317 qonef(float x) 318 { 319 const float *p,*q; 320 float s,r,z; 321 int32_t ix; 322 GET_FLOAT_WORD(ix,x); 323 ix &= 0x7fffffff; 324 if(ix>=0x40200000) {p = qr8; q= qs8;} 325 else if(ix>=0x40f71c58){p = qr5; q= qs5;} 326 else if(ix>=0x4036db68){p = qr3; q= qs3;} 327 else /*if(ix>=0x40000000)*/ {p = qr2; q= qs2;} 328 z = one/(x*x); 329 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 330 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 331 return ((float).375 + r/s)/x; 332 } 333