e_j0f.c revision 97413
1211476Sadrian/* e_j0f.c -- float version of e_j0.c. 2211476Sadrian * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. 3211476Sadrian */ 4211476Sadrian 5211476Sadrian/* 6211476Sadrian * ==================================================== 7211476Sadrian * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 8211476Sadrian * 9211476Sadrian * Developed at SunPro, a Sun Microsystems, Inc. business. 10211476Sadrian * Permission to use, copy, modify, and distribute this 11211476Sadrian * software is freely granted, provided that this notice 12211476Sadrian * is preserved. 13211476Sadrian * ==================================================== 14211476Sadrian */ 15211476Sadrian 16211476Sadrian#ifndef lint 17211476Sadrianstatic char rcsid[] = "$FreeBSD: head/lib/msun/src/e_j0f.c 97413 2002-05-28 18:15:04Z alfred $"; 18211476Sadrian#endif 19211476Sadrian 20211476Sadrian#include "math.h" 21211476Sadrian#include "math_private.h" 22211476Sadrian 23211476Sadrianstatic float pzerof(float), qzerof(float); 24211476Sadrian 25211476Sadrianstatic const float 26211476Sadrianhuge = 1e30, 27211476Sadrianone = 1.0, 28211476Sadrianinvsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 29211476Sadriantpi = 6.3661974669e-01, /* 0x3f22f983 */ 30211476Sadrian /* R0/S0 on [0, 2.00] */ 31211476SadrianR02 = 1.5625000000e-02, /* 0x3c800000 */ 32211476SadrianR03 = -1.8997929874e-04, /* 0xb947352e */ 33211476SadrianR04 = 1.8295404516e-06, /* 0x35f58e88 */ 34211476SadrianR05 = -4.6183270541e-09, /* 0xb19eaf3c */ 35211476SadrianS01 = 1.5619102865e-02, /* 0x3c7fe744 */ 36211476SadrianS02 = 1.1692678527e-04, /* 0x38f53697 */ 37211476SadrianS03 = 5.1354652442e-07, /* 0x3509daa6 */ 38211476SadrianS04 = 1.1661400734e-09; /* 0x30a045e8 */ 39211476Sadrian 40211476Sadrianstatic const float zero = 0.0; 41211476Sadrian 42211476Sadrianfloat 43211476Sadrian__ieee754_j0f(float x) 44211476Sadrian{ 45211476Sadrian float z, s,c,ss,cc,r,u,v; 46211476Sadrian int32_t hx,ix; 47211476Sadrian 48223562Skevlo GET_FLOAT_WORD(hx,x); 49211476Sadrian ix = hx&0x7fffffff; 50211476Sadrian if(ix>=0x7f800000) return one/(x*x); 51211476Sadrian x = fabsf(x); 52211476Sadrian if(ix >= 0x40000000) { /* |x| >= 2.0 */ 53211476Sadrian s = sinf(x); 54211476Sadrian c = cosf(x); 55248844Sadrian ss = s-c; 56253511Sadrian cc = s+c; 57248844Sadrian if(ix<0x7f000000) { /* make sure x+x not overflow */ 58211476Sadrian z = -cosf(x+x); 59211476Sadrian if ((s*c)<zero) cc = z/ss; 60211476Sadrian else ss = z/cc; 61211476Sadrian } 62211476Sadrian /* 63211503Sadrian * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 64211502Sadrian * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 65248844Sadrian */ 66253511Sadrian if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(x); 67211476Sadrian else { 68211476Sadrian u = pzerof(x); v = qzerof(x); 69211476Sadrian z = invsqrtpi*(u*cc-v*ss)/sqrtf(x); 70211476Sadrian } 71211476Sadrian return z; 72211476Sadrian } 73211476Sadrian if(ix<0x39000000) { /* |x| < 2**-13 */ 74211476Sadrian if(huge+x>one) { /* raise inexact if x != 0 */ 75211476Sadrian if(ix<0x32000000) return one; /* |x|<2**-27 */ 76211476Sadrian else return one - (float)0.25*x*x; 77211476Sadrian } 78211476Sadrian } 79211476Sadrian z = x*x; 80211476Sadrian r = z*(R02+z*(R03+z*(R04+z*R05))); 81211476Sadrian s = one+z*(S01+z*(S02+z*(S03+z*S04))); 82211476Sadrian if(ix < 0x3F800000) { /* |x| < 1.00 */ 83211476Sadrian return one + z*((float)-0.25+(r/s)); 84211476Sadrian } else { 85211476Sadrian u = (float)0.5*x; 86211476Sadrian return((one+u)*(one-u)+z*(r/s)); 87211476Sadrian } 88211476Sadrian} 89211476Sadrian 90211476Sadrianstatic const float 91211504Sadrianu00 = -7.3804296553e-02, /* 0xbd9726b5 */ 92211476Sadrianu01 = 1.7666645348e-01, /* 0x3e34e80d */ 93211476Sadrianu02 = -1.3818567619e-02, /* 0xbc626746 */ 94211476Sadrianu03 = 3.4745343146e-04, /* 0x39b62a69 */ 95211476Sadrianu04 = -3.8140706238e-06, /* 0xb67ff53c */ 96211476Sadrianu05 = 1.9559013964e-08, /* 0x32a802ba */ 97211476Sadrianu06 = -3.9820518410e-11, /* 0xae2f21eb */ 98211476Sadrianv01 = 1.2730483897e-02, /* 0x3c509385 */ 99211476Sadrianv02 = 7.6006865129e-05, /* 0x389f65e0 */ 100211476Sadrianv03 = 2.5915085189e-07, /* 0x348b216c */ 101211476Sadrianv04 = 4.4111031494e-10; /* 0x2ff280c2 */ 102211476Sadrian 103211476Sadrianfloat 104211476Sadrian__ieee754_y0f(float x) 105211476Sadrian{ 106211476Sadrian float z, s,c,ss,cc,u,v; 107211476Sadrian int32_t hx,ix; 108211476Sadrian 109211476Sadrian GET_FLOAT_WORD(hx,x); 110211503Sadrian ix = 0x7fffffff&hx; 111211503Sadrian /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */ 112211503Sadrian if(ix>=0x7f800000) return one/(x+x*x); 113211504Sadrian if(ix==0) return -one/zero; 114211503Sadrian if(hx<0) return zero/zero; 115211503Sadrian if(ix >= 0x40000000) { /* |x| >= 2.0 */ 116211503Sadrian /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) 117211503Sadrian * where x0 = x-pi/4 118211503Sadrian * Better formula: 119211503Sadrian * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 120211504Sadrian * = 1/sqrt(2) * (sin(x) + cos(x)) 121211503Sadrian * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 122211503Sadrian * = 1/sqrt(2) * (sin(x) - cos(x)) 123211503Sadrian * To avoid cancellation, use 124211503Sadrian * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 125211503Sadrian * to compute the worse one. 126211503Sadrian */ 127211504Sadrian s = sinf(x); 128211503Sadrian c = cosf(x); 129211503Sadrian ss = s-c; 130211503Sadrian cc = s+c; 131211502Sadrian /* 132211502Sadrian * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 133211502Sadrian * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 134211502Sadrian */ 135211504Sadrian if(ix<0x7f000000) { /* make sure x+x not overflow */ 136211502Sadrian z = -cosf(x+x); 137211502Sadrian if ((s*c)<zero) cc = z/ss; 138211502Sadrian else ss = z/cc; 139211502Sadrian } 140211502Sadrian if(ix>0x80000000) z = (invsqrtpi*ss)/sqrtf(x); 141211502Sadrian else { 142211502Sadrian u = pzerof(x); v = qzerof(x); 143211502Sadrian z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 144211502Sadrian } 145211502Sadrian return z; 146211502Sadrian } 147211502Sadrian if(ix<=0x32000000) { /* x < 2**-27 */ 148248844Sadrian return(u00 + tpi*__ieee754_logf(x)); 149248844Sadrian } 150248844Sadrian z = x*x; 151248844Sadrian u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 152248844Sadrian v = one+z*(v01+z*(v02+z*(v03+z*v04))); 153249118Sadrian return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x))); 154248844Sadrian} 155248844Sadrian 156248844Sadrian/* The asymptotic expansions of pzero is 157248844Sadrian * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 158248844Sadrian * For x >= 2, We approximate pzero by 159249118Sadrian * pzero(x) = 1 + (R/S) 160248844Sadrian * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 161248844Sadrian * S = 1 + pS0*s^2 + ... + pS4*s^10 162211502Sadrian * and 163253511Sadrian * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 164253511Sadrian */ 165253511Sadrianstatic const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 166253511Sadrian 0.0000000000e+00, /* 0x00000000 */ 167253511Sadrian -7.0312500000e-02, /* 0xbd900000 */ 168253511Sadrian -8.0816707611e+00, /* 0xc1014e86 */ 169253511Sadrian -2.5706311035e+02, /* 0xc3808814 */ 170253511Sadrian -2.4852163086e+03, /* 0xc51b5376 */ 171253511Sadrian -5.2530439453e+03, /* 0xc5a4285a */ 172253511Sadrian}; 173253511Sadrianstatic const float pS8[5] = { 174253511Sadrian 1.1653436279e+02, /* 0x42e91198 */ 175253511Sadrian 3.8337448730e+03, /* 0x456f9beb */ 176253511Sadrian 4.0597855469e+04, /* 0x471e95db */ 177253511Sadrian 1.1675296875e+05, /* 0x47e4087c */ 178253511Sadrian 4.7627726562e+04, /* 0x473a0bba */ 179253511Sadrian}; 180253511Sadrianstatic const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 181253511Sadrian -1.1412546255e-11, /* 0xad48c58a */ 182253511Sadrian -7.0312492549e-02, /* 0xbd8fffff */ 183253511Sadrian -4.1596107483e+00, /* 0xc0851b88 */ 184253511Sadrian -6.7674766541e+01, /* 0xc287597b */ 185253511Sadrian -3.3123129272e+02, /* 0xc3a59d9b */ 186253511Sadrian -3.4643338013e+02, /* 0xc3ad3779 */ 187211476Sadrian}; 188211476Sadrianstatic const float pS5[5] = { 189211476Sadrian 6.0753936768e+01, /* 0x42730408 */ 190211476Sadrian 1.0512523193e+03, /* 0x44836813 */ 191211476Sadrian 5.9789707031e+03, /* 0x45bad7c4 */ 192211476Sadrian 9.6254453125e+03, /* 0x461665c8 */ 193211476Sadrian 2.4060581055e+03, /* 0x451660ee */ 194211476Sadrian}; 195211476Sadrian 196211476Sadrianstatic const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 197211476Sadrian -2.5470459075e-09, /* 0xb12f081b */ 198211476Sadrian -7.0311963558e-02, /* 0xbd8fffb8 */ 199211476Sadrian -2.4090321064e+00, /* 0xc01a2d95 */ 200 -2.1965976715e+01, /* 0xc1afba52 */ 201 -5.8079170227e+01, /* 0xc2685112 */ 202 -3.1447946548e+01, /* 0xc1fb9565 */ 203}; 204static const float pS3[5] = { 205 3.5856033325e+01, /* 0x420f6c94 */ 206 3.6151397705e+02, /* 0x43b4c1ca */ 207 1.1936077881e+03, /* 0x44953373 */ 208 1.1279968262e+03, /* 0x448cffe6 */ 209 1.7358093262e+02, /* 0x432d94b8 */ 210}; 211 212static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 213 -8.8753431271e-08, /* 0xb3be98b7 */ 214 -7.0303097367e-02, /* 0xbd8ffb12 */ 215 -1.4507384300e+00, /* 0xbfb9b1cc */ 216 -7.6356959343e+00, /* 0xc0f4579f */ 217 -1.1193166733e+01, /* 0xc1331736 */ 218 -3.2336456776e+00, /* 0xc04ef40d */ 219}; 220static const float pS2[5] = { 221 2.2220300674e+01, /* 0x41b1c32d */ 222 1.3620678711e+02, /* 0x430834f0 */ 223 2.7047027588e+02, /* 0x43873c32 */ 224 1.5387539673e+02, /* 0x4319e01a */ 225 1.4657617569e+01, /* 0x416a859a */ 226}; 227 228 static float pzerof(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>=0x40f71c58){p = pR5; q= pS5;} 237 else if(ix>=0x4036db68){p = pR3; q= pS3;} 238 else if(ix>=0x40000000){p = pR2; q= pS2;} 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 qzero is 247 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 248 * We approximate pzero by 249 * qzero(x) = s*(-1.25 + (R/S)) 250 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 251 * S = 1 + qS0*s^2 + ... + qS5*s^12 252 * and 253 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 254 */ 255static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 256 0.0000000000e+00, /* 0x00000000 */ 257 7.3242187500e-02, /* 0x3d960000 */ 258 1.1768206596e+01, /* 0x413c4a93 */ 259 5.5767340088e+02, /* 0x440b6b19 */ 260 8.8591972656e+03, /* 0x460a6cca */ 261 3.7014625000e+04, /* 0x471096a0 */ 262}; 263static const float qS8[6] = { 264 1.6377603149e+02, /* 0x4323c6aa */ 265 8.0983447266e+03, /* 0x45fd12c2 */ 266 1.4253829688e+05, /* 0x480b3293 */ 267 8.0330925000e+05, /* 0x49441ed4 */ 268 8.4050156250e+05, /* 0x494d3359 */ 269 -3.4389928125e+05, /* 0xc8a7eb69 */ 270}; 271 272static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 273 1.8408595828e-11, /* 0x2da1ec79 */ 274 7.3242180049e-02, /* 0x3d95ffff */ 275 5.8356351852e+00, /* 0x40babd86 */ 276 1.3511157227e+02, /* 0x43071c90 */ 277 1.0272437744e+03, /* 0x448067cd */ 278 1.9899779053e+03, /* 0x44f8bf4b */ 279}; 280static const float qS5[6] = { 281 8.2776611328e+01, /* 0x42a58da0 */ 282 2.0778142090e+03, /* 0x4501dd07 */ 283 1.8847289062e+04, /* 0x46933e94 */ 284 5.6751113281e+04, /* 0x475daf1d */ 285 3.5976753906e+04, /* 0x470c88c1 */ 286 -5.3543427734e+03, /* 0xc5a752be */ 287}; 288 289static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 290 4.3774099900e-09, /* 0x3196681b */ 291 7.3241114616e-02, /* 0x3d95ff70 */ 292 3.3442313671e+00, /* 0x405607e3 */ 293 4.2621845245e+01, /* 0x422a7cc5 */ 294 1.7080809021e+02, /* 0x432acedf */ 295 1.6673394775e+02, /* 0x4326bbe4 */ 296}; 297static const float qS3[6] = { 298 4.8758872986e+01, /* 0x42430916 */ 299 7.0968920898e+02, /* 0x44316c1c */ 300 3.7041481934e+03, /* 0x4567825f */ 301 6.4604252930e+03, /* 0x45c9e367 */ 302 2.5163337402e+03, /* 0x451d4557 */ 303 -1.4924745178e+02, /* 0xc3153f59 */ 304}; 305 306static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 307 1.5044444979e-07, /* 0x342189db */ 308 7.3223426938e-02, /* 0x3d95f62a */ 309 1.9981917143e+00, /* 0x3fffc4bf */ 310 1.4495602608e+01, /* 0x4167edfd */ 311 3.1666231155e+01, /* 0x41fd5471 */ 312 1.6252708435e+01, /* 0x4182058c */ 313}; 314static const float qS2[6] = { 315 3.0365585327e+01, /* 0x41f2ecb8 */ 316 2.6934811401e+02, /* 0x4386ac8f */ 317 8.4478375244e+02, /* 0x44533229 */ 318 8.8293585205e+02, /* 0x445cbbe5 */ 319 2.1266638184e+02, /* 0x4354aa98 */ 320 -5.3109550476e+00, /* 0xc0a9f358 */ 321}; 322 323 static float qzerof(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>=0x40f71c58){p = qR5; q= qS5;} 332 else if(ix>=0x4036db68){p = qR3; q= qS3;} 333 else if(ix>=0x40000000){p = qR2; q= qS2;} 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).125 + r/s)/x; 338} 339