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 <assert.h> 17 18 #include "cdefs-compat.h" 19 //__FBSDID("$FreeBSD: src/lib/msun/src/e_j1f.c,v 1.8 2008/02/22 02:30:35 das Exp $"); 20 21 #include "openlibm.h" 22 #include "math_private.h" 23 24 static float ponef(float), qonef(float); 25 26 static const float 27 huge = 1e30, 28 one = 1.0, 29 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 30 tpi = 6.3661974669e-01, /* 0x3f22f983 */ 31 /* R0/S0 on [0,2] */ 32 r00 = -6.2500000000e-02, /* 0xbd800000 */ 33 r01 = 1.4070566976e-03, /* 0x3ab86cfd */ 34 r02 = -1.5995563444e-05, /* 0xb7862e36 */ 35 r03 = 4.9672799207e-08, /* 0x335557d2 */ 36 s01 = 1.9153760746e-02, /* 0x3c9ce859 */ 37 s02 = 1.8594678841e-04, /* 0x3942fab6 */ 38 s03 = 1.1771846857e-06, /* 0x359dffc2 */ 39 s04 = 5.0463624390e-09, /* 0x31ad6446 */ 40 s05 = 1.2354227016e-11; /* 0x2d59567e */ 41 42 static const float zero = 0.0; 43 44 DLLEXPORT float 45 __ieee754_j1f(float x) 46 { 47 float z, s,c,ss,cc,r,u,v,y; 48 int32_t hx,ix; 49 50 GET_FLOAT_WORD(hx,x); 51 ix = hx&0x7fffffff; 52 if(ix>=0x7f800000) return one/x; 53 y = fabsf(x); 54 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 55 s = sinf(y); 56 c = cosf(y); 57 ss = -s-c; 58 cc = s-c; 59 if(ix<0x7f000000) { /* make sure y+y not overflow */ 60 z = cosf(y+y); 61 if ((s*c)>zero) cc = z/ss; 62 else ss = z/cc; 63 } 64 /* 65 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) 66 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) 67 */ 68 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y); 69 else { 70 u = ponef(y); v = qonef(y); 71 z = invsqrtpi*(u*cc-v*ss)/sqrtf(y); 72 } 73 if(hx<0) return -z; 74 else return z; 75 } 76 if(ix<0x32000000) { /* |x|<2**-27 */ 77 if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */ 78 } 79 z = x*x; 80 r = z*(r00+z*(r01+z*(r02+z*r03))); 81 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); 82 r *= x; 83 return(x*(float)0.5+r/s); 84 } 85 86 static const float U0[5] = { 87 -1.9605709612e-01, /* 0xbe48c331 */ 88 5.0443872809e-02, /* 0x3d4e9e3c */ 89 -1.9125689287e-03, /* 0xbafaaf2a */ 90 2.3525259166e-05, /* 0x37c5581c */ 91 -9.1909917899e-08, /* 0xb3c56003 */ 92 }; 93 static const float V0[5] = { 94 1.9916731864e-02, /* 0x3ca3286a */ 95 2.0255257550e-04, /* 0x3954644b */ 96 1.3560879779e-06, /* 0x35b602d4 */ 97 6.2274145840e-09, /* 0x31d5f8eb */ 98 1.6655924903e-11, /* 0x2d9281cf */ 99 }; 100 101 DLLEXPORT float 102 __ieee754_y1f(float x) 103 { 104 float z, s,c,ss,cc,u,v; 105 int32_t hx,ix; 106 107 GET_FLOAT_WORD(hx,x); 108 ix = 0x7fffffff&hx; 109 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ 110 if(ix>=0x7f800000) return one/(x+x*x); 111 if(ix==0) return -one/zero; 112 if(hx<0) return zero/zero; 113 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 114 s = sinf(x); 115 c = cosf(x); 116 ss = -s-c; 117 cc = s-c; 118 if(ix<0x7f000000) { /* make sure x+x not overflow */ 119 z = cosf(x+x); 120 if ((s*c)>zero) cc = z/ss; 121 else ss = z/cc; 122 } 123 /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) 124 * where x0 = x-3pi/4 125 * Better formula: 126 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 127 * = 1/sqrt(2) * (sin(x) - cos(x)) 128 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 129 * = -1/sqrt(2) * (cos(x) + sin(x)) 130 * To avoid cancellation, use 131 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 132 * to compute the worse one. 133 */ 134 if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x); 135 else { 136 u = ponef(x); v = qonef(x); 137 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 138 } 139 return z; 140 } 141 if(ix<=0x24800000) { /* x < 2**-54 */ 142 return(-tpi/x); 143 } 144 z = x*x; 145 u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); 146 v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); 147 return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x)); 148 } 149 150 /* For x >= 8, the asymptotic expansions of pone is 151 * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. 152 * We approximate pone by 153 * pone(x) = 1 + (R/S) 154 * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 155 * S = 1 + ps0*s^2 + ... + ps4*s^10 156 * and 157 * | pone(x)-1-R/S | <= 2 ** ( -60.06) 158 */ 159 160 static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 161 0.0000000000e+00, /* 0x00000000 */ 162 1.1718750000e-01, /* 0x3df00000 */ 163 1.3239480972e+01, /* 0x4153d4ea */ 164 4.1205184937e+02, /* 0x43ce06a3 */ 165 3.8747453613e+03, /* 0x45722bed */ 166 7.9144794922e+03, /* 0x45f753d6 */ 167 }; 168 static const float ps8[5] = { 169 1.1420736694e+02, /* 0x42e46a2c */ 170 3.6509309082e+03, /* 0x45642ee5 */ 171 3.6956207031e+04, /* 0x47105c35 */ 172 9.7602796875e+04, /* 0x47bea166 */ 173 3.0804271484e+04, /* 0x46f0a88b */ 174 }; 175 176 static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 177 1.3199052094e-11, /* 0x2d68333f */ 178 1.1718749255e-01, /* 0x3defffff */ 179 6.8027510643e+00, /* 0x40d9b023 */ 180 1.0830818176e+02, /* 0x42d89dca */ 181 5.1763616943e+02, /* 0x440168b7 */ 182 5.2871520996e+02, /* 0x44042dc6 */ 183 }; 184 static const float ps5[5] = { 185 5.9280597687e+01, /* 0x426d1f55 */ 186 9.9140142822e+02, /* 0x4477d9b1 */ 187 5.3532670898e+03, /* 0x45a74a23 */ 188 7.8446904297e+03, /* 0x45f52586 */ 189 1.5040468750e+03, /* 0x44bc0180 */ 190 }; 191 192 static const float pr3[6] = { 193 3.0250391081e-09, /* 0x314fe10d */ 194 1.1718686670e-01, /* 0x3defffab */ 195 3.9329774380e+00, /* 0x407bb5e7 */ 196 3.5119403839e+01, /* 0x420c7a45 */ 197 9.1055007935e+01, /* 0x42b61c2a */ 198 4.8559066772e+01, /* 0x42423c7c */ 199 }; 200 static const float ps3[5] = { 201 3.4791309357e+01, /* 0x420b2a4d */ 202 3.3676245117e+02, /* 0x43a86198 */ 203 1.0468714600e+03, /* 0x4482dbe3 */ 204 8.9081134033e+02, /* 0x445eb3ed */ 205 1.0378793335e+02, /* 0x42cf936c */ 206 }; 207 208 static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 209 1.0771083225e-07, /* 0x33e74ea8 */ 210 1.1717621982e-01, /* 0x3deffa16 */ 211 2.3685150146e+00, /* 0x401795c0 */ 212 1.2242610931e+01, /* 0x4143e1bc */ 213 1.7693971634e+01, /* 0x418d8d41 */ 214 5.0735230446e+00, /* 0x40a25a4d */ 215 }; 216 static const float ps2[5] = { 217 2.1436485291e+01, /* 0x41ab7dec */ 218 1.2529022980e+02, /* 0x42fa9499 */ 219 2.3227647400e+02, /* 0x436846c7 */ 220 1.1767937469e+02, /* 0x42eb5bd7 */ 221 8.3646392822e+00, /* 0x4105d590 */ 222 }; 223 224 /* Note: This function is only called for ix>=0x40000000 (see above) */ 225 static float ponef(float x) 226 { 227 const float *p,*q; 228 float z,r,s; 229 int32_t ix; 230 GET_FLOAT_WORD(ix,x); 231 ix &= 0x7fffffff; 232 assert(ix>=0x40000000 && ix<=0x48000000); 233 if(ix>=0x41000000) {p = pr8; q= ps8;} 234 else if(ix>=0x40f71c58){p = pr5; q= ps5;} 235 else if(ix>=0x4036db68){p = pr3; q= ps3;} 236 else {p = pr2; q= ps2;} 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 /* Note: This function is only called for ix>=0x40000000 (see above) */ 323 static float 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 assert(ix>=0x40000000 && ix<=0x48000000); 331 if(ix>=0x40200000) {p = qr8; q= qs8;} 332 else if(ix>=0x40f71c58){p = qr5; q= qs5;} 333 else if(ix>=0x4036db68){p = qr3; q= qs3;} 334 else {p = qr2; q= qs2;} 335 z = one/(x*x); 336 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 337 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 338 return ((float).375 + r/s)/x; 339 } 340