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_math.h> 22 23 #include "math_private.h" 24 25 static float ponef(float), qonef(float); 26 27 static const float 28 huge = 1e30, 29 one = 1.0, 30 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 31 tpi = 6.3661974669e-01, /* 0x3f22f983 */ 32 /* R0/S0 on [0,2] */ 33 r00 = -6.2500000000e-02, /* 0xbd800000 */ 34 r01 = 1.4070566976e-03, /* 0x3ab86cfd */ 35 r02 = -1.5995563444e-05, /* 0xb7862e36 */ 36 r03 = 4.9672799207e-08, /* 0x335557d2 */ 37 s01 = 1.9153760746e-02, /* 0x3c9ce859 */ 38 s02 = 1.8594678841e-04, /* 0x3942fab6 */ 39 s03 = 1.1771846857e-06, /* 0x359dffc2 */ 40 s04 = 5.0463624390e-09, /* 0x31ad6446 */ 41 s05 = 1.2354227016e-11; /* 0x2d59567e */ 42 43 static const float zero = 0.0; 44 45 DLLEXPORT float 46 __ieee754_j1f(float x) 47 { 48 float z, s,c,ss,cc,r,u,v,y; 49 int32_t hx,ix; 50 51 GET_FLOAT_WORD(hx,x); 52 ix = hx&0x7fffffff; 53 if(ix>=0x7f800000) return one/x; 54 y = fabsf(x); 55 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 56 s = sinf(y); 57 c = cosf(y); 58 ss = -s-c; 59 cc = s-c; 60 if(ix<0x7f000000) { /* make sure y+y not overflow */ 61 z = cosf(y+y); 62 if ((s*c)>zero) cc = z/ss; 63 else ss = z/cc; 64 } 65 /* 66 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) 67 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) 68 */ 69 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y); 70 else { 71 u = ponef(y); v = qonef(y); 72 z = invsqrtpi*(u*cc-v*ss)/sqrtf(y); 73 } 74 if(hx<0) return -z; 75 else return z; 76 } 77 if(ix<0x32000000) { /* |x|<2**-27 */ 78 if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */ 79 } 80 z = x*x; 81 r = z*(r00+z*(r01+z*(r02+z*r03))); 82 s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); 83 r *= x; 84 return(x*(float)0.5+r/s); 85 } 86 87 static const float U0[5] = { 88 -1.9605709612e-01, /* 0xbe48c331 */ 89 5.0443872809e-02, /* 0x3d4e9e3c */ 90 -1.9125689287e-03, /* 0xbafaaf2a */ 91 2.3525259166e-05, /* 0x37c5581c */ 92 -9.1909917899e-08, /* 0xb3c56003 */ 93 }; 94 static const float V0[5] = { 95 1.9916731864e-02, /* 0x3ca3286a */ 96 2.0255257550e-04, /* 0x3954644b */ 97 1.3560879779e-06, /* 0x35b602d4 */ 98 6.2274145840e-09, /* 0x31d5f8eb */ 99 1.6655924903e-11, /* 0x2d9281cf */ 100 }; 101 102 DLLEXPORT float 103 __ieee754_y1f(float x) 104 { 105 float z, s,c,ss,cc,u,v; 106 int32_t hx,ix; 107 108 GET_FLOAT_WORD(hx,x); 109 ix = 0x7fffffff&hx; 110 /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ 111 if(ix>=0x7f800000) return one/(x+x*x); 112 if(ix==0) return -one/zero; 113 if(hx<0) return zero/zero; 114 if(ix >= 0x40000000) { /* |x| >= 2.0 */ 115 s = sinf(x); 116 c = cosf(x); 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>0x48000000) z = (invsqrtpi*ss)/sqrtf(x); 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<=0x24800000) { /* x < 2**-54 */ 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*(__ieee754_j1f(x)*__ieee754_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 /* Note: This function is only called for ix>=0x40000000 (see above) */ 226 static float 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 assert(ix>=0x40000000 && ix<=0x48000000); 234 if(ix>=0x41000000) {p = pr8; q= ps8;} 235 else if(ix>=0x40f71c58){p = pr5; q= ps5;} 236 else if(ix>=0x4036db68){p = pr3; q= ps3;} 237 else {p = pr2; q= ps2;} 238 z = one/(x*x); 239 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 240 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 241 return one+ r/s; 242 } 243 244 245 /* For x >= 8, the asymptotic expansions of qone is 246 * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. 247 * We approximate pone by 248 * qone(x) = s*(0.375 + (R/S)) 249 * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 250 * S = 1 + qs1*s^2 + ... + qs6*s^12 251 * and 252 * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) 253 */ 254 255 static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 256 0.0000000000e+00, /* 0x00000000 */ 257 -1.0253906250e-01, /* 0xbdd20000 */ 258 -1.6271753311e+01, /* 0xc1822c8d */ 259 -7.5960174561e+02, /* 0xc43de683 */ 260 -1.1849806641e+04, /* 0xc639273a */ 261 -4.8438511719e+04, /* 0xc73d3683 */ 262 }; 263 static const float qs8[6] = { 264 1.6139537048e+02, /* 0x43216537 */ 265 7.8253862305e+03, /* 0x45f48b17 */ 266 1.3387534375e+05, /* 0x4802bcd6 */ 267 7.1965775000e+05, /* 0x492fb29c */ 268 6.6660125000e+05, /* 0x4922be94 */ 269 -2.9449025000e+05, /* 0xc88fcb48 */ 270 }; 271 272 static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 273 -2.0897993405e-11, /* 0xadb7d219 */ 274 -1.0253904760e-01, /* 0xbdd1fffe */ 275 -8.0564479828e+00, /* 0xc100e736 */ 276 -1.8366960144e+02, /* 0xc337ab6b */ 277 -1.3731937256e+03, /* 0xc4aba633 */ 278 -2.6124443359e+03, /* 0xc523471c */ 279 }; 280 static const float qs5[6] = { 281 8.1276550293e+01, /* 0x42a28d98 */ 282 1.9917987061e+03, /* 0x44f8f98f */ 283 1.7468484375e+04, /* 0x468878f8 */ 284 4.9851425781e+04, /* 0x4742bb6d */ 285 2.7948074219e+04, /* 0x46da5826 */ 286 -4.7191835938e+03, /* 0xc5937978 */ 287 }; 288 289 static const float qr3[6] = { 290 -5.0783124372e-09, /* 0xb1ae7d4f */ 291 -1.0253783315e-01, /* 0xbdd1ff5b */ 292 -4.6101160049e+00, /* 0xc0938612 */ 293 -5.7847221375e+01, /* 0xc267638e */ 294 -2.2824453735e+02, /* 0xc3643e9a */ 295 -2.1921012878e+02, /* 0xc35b35cb */ 296 }; 297 static const float qs3[6] = { 298 4.7665153503e+01, /* 0x423ea91e */ 299 6.7386511230e+02, /* 0x4428775e */ 300 3.3801528320e+03, /* 0x45534272 */ 301 5.5477290039e+03, /* 0x45ad5dd5 */ 302 1.9031191406e+03, /* 0x44ede3d0 */ 303 -1.3520118713e+02, /* 0xc3073381 */ 304 }; 305 306 static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 307 -1.7838172539e-07, /* 0xb43f8932 */ 308 -1.0251704603e-01, /* 0xbdd1f475 */ 309 -2.7522056103e+00, /* 0xc0302423 */ 310 -1.9663616180e+01, /* 0xc19d4f16 */ 311 -4.2325313568e+01, /* 0xc2294d1f */ 312 -2.1371921539e+01, /* 0xc1aaf9b2 */ 313 }; 314 static const float qs2[6] = { 315 2.9533363342e+01, /* 0x41ec4454 */ 316 2.5298155212e+02, /* 0x437cfb47 */ 317 7.5750280762e+02, /* 0x443d602e */ 318 7.3939318848e+02, /* 0x4438d92a */ 319 1.5594900513e+02, /* 0x431bf2f2 */ 320 -4.9594988823e+00, /* 0xc09eb437 */ 321 }; 322 323 /* Note: This function is only called for ix>=0x40000000 (see above) */ 324 static float qonef(float x) 325 { 326 const float *p,*q; 327 float s,r,z; 328 int32_t ix; 329 GET_FLOAT_WORD(ix,x); 330 ix &= 0x7fffffff; 331 assert(ix>=0x40000000 && ix<=0x48000000); 332 if(ix>=0x40200000) {p = qr8; q= qs8;} 333 else if(ix>=0x40f71c58){p = qr5; q= qs5;} 334 else if(ix>=0x4036db68){p = qr3; q= qs3;} 335 else {p = qr2; q= qs2;} 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