e_j0f.c revision 284810
1169695Skan/* e_j0f.c -- float version of e_j0.c. 2169695Skan * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. 3169695Skan */ 4169695Skan 5169695Skan/* 6169695Skan * ==================================================== 7169695Skan * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 8169695Skan * 9169695Skan * Developed at SunPro, a Sun Microsystems, Inc. business. 10169695Skan * Permission to use, copy, modify, and distribute this 11169695Skan * software is freely granted, provided that this notice 12169695Skan * is preserved. 13169695Skan * ==================================================== 14169695Skan */ 15169695Skan 16169695Skan#include <sys/cdefs.h> 17169695Skan__FBSDID("$FreeBSD: stable/10/lib/msun/src/e_j0f.c 284810 2015-06-25 13:01:10Z tijl $"); 18169695Skan 19169695Skan/* 20169695Skan * See e_j0.c for complete comments. 21169695Skan */ 22169695Skan 23169695Skan#include "math.h" 24169695Skan#include "math_private.h" 25169695Skan 26169695Skanstatic __inline float pzerof(float), qzerof(float); 27169695Skan 28169695Skanstatic const volatile float vone = 1, vzero = 0; 29169695Skan 30169695Skanstatic const float 31169695Skanhuge = 1e30, 32169695Skanone = 1.0, 33169695Skaninvsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 34169695Skantpi = 6.3661974669e-01, /* 0x3f22f983 */ 35169695Skan /* R0/S0 on [0, 2.00] */ 36169695SkanR02 = 1.5625000000e-02, /* 0x3c800000 */ 37169695SkanR03 = -1.8997929874e-04, /* 0xb947352e */ 38169695SkanR04 = 1.8295404516e-06, /* 0x35f58e88 */ 39169695SkanR05 = -4.6183270541e-09, /* 0xb19eaf3c */ 40169695SkanS01 = 1.5619102865e-02, /* 0x3c7fe744 */ 41169695SkanS02 = 1.1692678527e-04, /* 0x38f53697 */ 42169695SkanS03 = 5.1354652442e-07, /* 0x3509daa6 */ 43169695SkanS04 = 1.1661400734e-09; /* 0x30a045e8 */ 44169695Skan 45169695Skanstatic const float zero = 0.0; 46169695Skan 47169695Skanfloat 48169695Skan__ieee754_j0f(float x) 49169695Skan{ 50169695Skan float z, s,c,ss,cc,r,u,v; 51169695Skan int32_t hx,ix; 52169695Skan 53169695Skan GET_FLOAT_WORD(hx,x); 54169695Skan ix = hx&0x7fffffff; 55169695Skan if(ix>=0x7f800000) return one/(x*x); 56169695Skan x = fabsf(x); 57169695Skan if(ix >= 0x40000000) { /* |x| >= 2.0 */ 58169695Skan s = sinf(x); 59169695Skan c = cosf(x); 60169695Skan ss = s-c; 61169695Skan cc = s+c; 62169695Skan if(ix<0x7f000000) { /* make sure x+x not overflow */ 63169695Skan z = -cosf(x+x); 64169695Skan if ((s*c)<zero) cc = z/ss; 65169695Skan else ss = z/cc; 66169695Skan } 67169695Skan /* 68169695Skan * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 69169695Skan * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 70169695Skan */ 71169695Skan if(ix>0x58000000) z = (invsqrtpi*cc)/sqrtf(x); /* |x|>2**49 */ 72169695Skan else { 73169695Skan u = pzerof(x); v = qzerof(x); 74169695Skan z = invsqrtpi*(u*cc-v*ss)/sqrtf(x); 75169695Skan } 76169695Skan return z; 77169695Skan } 78169695Skan if(ix<0x3b000000) { /* |x| < 2**-9 */ 79169695Skan if(huge+x>one) { /* raise inexact if x != 0 */ 80169695Skan if(ix<0x39800000) return one; /* |x|<2**-12 */ 81169695Skan else return one - x*x/4; 82169695Skan } 83169695Skan } 84169695Skan z = x*x; 85169695Skan r = z*(R02+z*(R03+z*(R04+z*R05))); 86169695Skan s = one+z*(S01+z*(S02+z*(S03+z*S04))); 87169695Skan if(ix < 0x3F800000) { /* |x| < 1.00 */ 88169695Skan return one + z*((float)-0.25+(r/s)); 89169695Skan } else { 90169695Skan u = (float)0.5*x; 91169695Skan return((one+u)*(one-u)+z*(r/s)); 92169695Skan } 93169695Skan} 94169695Skan 95169695Skanstatic const float 96169695Skanu00 = -7.3804296553e-02, /* 0xbd9726b5 */ 97169695Skanu01 = 1.7666645348e-01, /* 0x3e34e80d */ 98169695Skanu02 = -1.3818567619e-02, /* 0xbc626746 */ 99169695Skanu03 = 3.4745343146e-04, /* 0x39b62a69 */ 100169695Skanu04 = -3.8140706238e-06, /* 0xb67ff53c */ 101169695Skanu05 = 1.9559013964e-08, /* 0x32a802ba */ 102169695Skanu06 = -3.9820518410e-11, /* 0xae2f21eb */ 103169695Skanv01 = 1.2730483897e-02, /* 0x3c509385 */ 104169695Skanv02 = 7.6006865129e-05, /* 0x389f65e0 */ 105169695Skanv03 = 2.5915085189e-07, /* 0x348b216c */ 106169695Skanv04 = 4.4111031494e-10; /* 0x2ff280c2 */ 107169695Skan 108169695Skanfloat 109169695Skan__ieee754_y0f(float x) 110169695Skan{ 111169695Skan float z, s,c,ss,cc,u,v; 112169695Skan int32_t hx,ix; 113169695Skan 114169695Skan GET_FLOAT_WORD(hx,x); 115169695Skan ix = 0x7fffffff&hx; 116169695Skan if(ix>=0x7f800000) return vone/(x+x*x); 117169695Skan if(ix==0) return -one/vzero; 118169695Skan if(hx<0) return vzero/vzero; 119169695Skan if(ix >= 0x40000000) { /* |x| >= 2.0 */ 120169695Skan /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) 121169695Skan * where x0 = x-pi/4 122169695Skan * Better formula: 123169695Skan * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 124169695Skan * = 1/sqrt(2) * (sin(x) + cos(x)) 125169695Skan * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 126169695Skan * = 1/sqrt(2) * (sin(x) - cos(x)) 127169695Skan * To avoid cancellation, use 128169695Skan * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 129169695Skan * to compute the worse one. 130169695Skan */ 131169695Skan s = sinf(x); 132169695Skan c = cosf(x); 133169695Skan ss = s-c; 134169695Skan cc = s+c; 135169695Skan /* 136169695Skan * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 137169695Skan * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 138169695Skan */ 139169695Skan if(ix<0x7f000000) { /* make sure x+x not overflow */ 140169695Skan z = -cosf(x+x); 141169695Skan if ((s*c)<zero) cc = z/ss; 142169695Skan else ss = z/cc; 143169695Skan } 144169695Skan if(ix>0x58000000) z = (invsqrtpi*ss)/sqrtf(x); /* |x|>2**49 */ 145169695Skan else { 146169695Skan u = pzerof(x); v = qzerof(x); 147169695Skan z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 148169695Skan } 149169695Skan return z; 150169695Skan } 151169695Skan if(ix<=0x39000000) { /* x < 2**-13 */ 152169695Skan return(u00 + tpi*__ieee754_logf(x)); 153169695Skan } 154169695Skan z = x*x; 155169695Skan u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 156169695Skan v = one+z*(v01+z*(v02+z*(v03+z*v04))); 157169695Skan return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x))); 158169695Skan} 159169695Skan 160169695Skan/* The asymptotic expansions of pzero is 161169695Skan * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 162169695Skan * For x >= 2, We approximate pzero by 163169695Skan * pzero(x) = 1 + (R/S) 164169695Skan * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 165169695Skan * S = 1 + pS0*s^2 + ... + pS4*s^10 166169695Skan * and 167169695Skan * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 168169695Skan */ 169169695Skanstatic const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 170169695Skan 0.0000000000e+00, /* 0x00000000 */ 171169695Skan -7.0312500000e-02, /* 0xbd900000 */ 172169695Skan -8.0816707611e+00, /* 0xc1014e86 */ 173169695Skan -2.5706311035e+02, /* 0xc3808814 */ 174169695Skan -2.4852163086e+03, /* 0xc51b5376 */ 175169695Skan -5.2530439453e+03, /* 0xc5a4285a */ 176169695Skan}; 177169695Skanstatic const float pS8[5] = { 178169695Skan 1.1653436279e+02, /* 0x42e91198 */ 179169695Skan 3.8337448730e+03, /* 0x456f9beb */ 180169695Skan 4.0597855469e+04, /* 0x471e95db */ 181169695Skan 1.1675296875e+05, /* 0x47e4087c */ 182169695Skan 4.7627726562e+04, /* 0x473a0bba */ 183169695Skan}; 184169695Skanstatic const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 185169695Skan -1.1412546255e-11, /* 0xad48c58a */ 186169695Skan -7.0312492549e-02, /* 0xbd8fffff */ 187169695Skan -4.1596107483e+00, /* 0xc0851b88 */ 188169695Skan -6.7674766541e+01, /* 0xc287597b */ 189169695Skan -3.3123129272e+02, /* 0xc3a59d9b */ 190169695Skan -3.4643338013e+02, /* 0xc3ad3779 */ 191169695Skan}; 192169695Skanstatic const float pS5[5] = { 193169695Skan 6.0753936768e+01, /* 0x42730408 */ 194169695Skan 1.0512523193e+03, /* 0x44836813 */ 195169695Skan 5.9789707031e+03, /* 0x45bad7c4 */ 196169695Skan 9.6254453125e+03, /* 0x461665c8 */ 197169695Skan 2.4060581055e+03, /* 0x451660ee */ 198169695Skan}; 199169695Skan 200169695Skanstatic const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 201169695Skan -2.5470459075e-09, /* 0xb12f081b */ 202169695Skan -7.0311963558e-02, /* 0xbd8fffb8 */ 203169695Skan -2.4090321064e+00, /* 0xc01a2d95 */ 204169695Skan -2.1965976715e+01, /* 0xc1afba52 */ 205169695Skan -5.8079170227e+01, /* 0xc2685112 */ 206169695Skan -3.1447946548e+01, /* 0xc1fb9565 */ 207169695Skan}; 208169695Skanstatic const float pS3[5] = { 209169695Skan 3.5856033325e+01, /* 0x420f6c94 */ 210169695Skan 3.6151397705e+02, /* 0x43b4c1ca */ 211169695Skan 1.1936077881e+03, /* 0x44953373 */ 212169695Skan 1.1279968262e+03, /* 0x448cffe6 */ 213169695Skan 1.7358093262e+02, /* 0x432d94b8 */ 214169695Skan}; 215169695Skan 216169695Skanstatic const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 217169695Skan -8.8753431271e-08, /* 0xb3be98b7 */ 218169695Skan -7.0303097367e-02, /* 0xbd8ffb12 */ 219169695Skan -1.4507384300e+00, /* 0xbfb9b1cc */ 220169695Skan -7.6356959343e+00, /* 0xc0f4579f */ 221169695Skan -1.1193166733e+01, /* 0xc1331736 */ 222169695Skan -3.2336456776e+00, /* 0xc04ef40d */ 223169695Skan}; 224169695Skanstatic const float pS2[5] = { 225169695Skan 2.2220300674e+01, /* 0x41b1c32d */ 226169695Skan 1.3620678711e+02, /* 0x430834f0 */ 227169695Skan 2.7047027588e+02, /* 0x43873c32 */ 228169695Skan 1.5387539673e+02, /* 0x4319e01a */ 229169695Skan 1.4657617569e+01, /* 0x416a859a */ 230169695Skan}; 231169695Skan 232169695Skanstatic __inline float 233169695Skanpzerof(float x) 234169695Skan{ 235169695Skan const float *p,*q; 236169695Skan float z,r,s; 237169695Skan int32_t ix; 238169695Skan GET_FLOAT_WORD(ix,x); 239169695Skan ix &= 0x7fffffff; 240169695Skan if(ix>=0x41000000) {p = pR8; q= pS8;} 241169695Skan else if(ix>=0x409173eb){p = pR5; q= pS5;} 242169695Skan else if(ix>=0x4036d917){p = pR3; q= pS3;} 243169695Skan else {p = pR2; q= pS2;} /* ix>=0x40000000 */ 244169695Skan z = one/(x*x); 245169695Skan r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 246169695Skan s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 247169695Skan return one+ r/s; 248169695Skan} 249169695Skan 250169695Skan 251169695Skan/* For x >= 8, the asymptotic expansions of qzero is 252169695Skan * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 253169695Skan * We approximate pzero by 254169695Skan * qzero(x) = s*(-1.25 + (R/S)) 255169695Skan * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 256169695Skan * S = 1 + qS0*s^2 + ... + qS5*s^12 257169695Skan * and 258169695Skan * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 259169695Skan */ 260169695Skanstatic const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 261169695Skan 0.0000000000e+00, /* 0x00000000 */ 262169695Skan 7.3242187500e-02, /* 0x3d960000 */ 263169695Skan 1.1768206596e+01, /* 0x413c4a93 */ 264169695Skan 5.5767340088e+02, /* 0x440b6b19 */ 265169695Skan 8.8591972656e+03, /* 0x460a6cca */ 266169695Skan 3.7014625000e+04, /* 0x471096a0 */ 267169695Skan}; 268169695Skanstatic const float qS8[6] = { 269169695Skan 1.6377603149e+02, /* 0x4323c6aa */ 270169695Skan 8.0983447266e+03, /* 0x45fd12c2 */ 271169695Skan 1.4253829688e+05, /* 0x480b3293 */ 272169695Skan 8.0330925000e+05, /* 0x49441ed4 */ 273169695Skan 8.4050156250e+05, /* 0x494d3359 */ 274169695Skan -3.4389928125e+05, /* 0xc8a7eb69 */ 275169695Skan}; 276169695Skan 277169695Skanstatic const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 278169695Skan 1.8408595828e-11, /* 0x2da1ec79 */ 279169695Skan 7.3242180049e-02, /* 0x3d95ffff */ 280169695Skan 5.8356351852e+00, /* 0x40babd86 */ 281169695Skan 1.3511157227e+02, /* 0x43071c90 */ 282169695Skan 1.0272437744e+03, /* 0x448067cd */ 283169695Skan 1.9899779053e+03, /* 0x44f8bf4b */ 284169695Skan}; 285169695Skanstatic const float qS5[6] = { 286169695Skan 8.2776611328e+01, /* 0x42a58da0 */ 287169695Skan 2.0778142090e+03, /* 0x4501dd07 */ 288169695Skan 1.8847289062e+04, /* 0x46933e94 */ 289169695Skan 5.6751113281e+04, /* 0x475daf1d */ 290169695Skan 3.5976753906e+04, /* 0x470c88c1 */ 291169695Skan -5.3543427734e+03, /* 0xc5a752be */ 292169695Skan}; 293169695Skan 294169695Skanstatic const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 295169695Skan 4.3774099900e-09, /* 0x3196681b */ 296169695Skan 7.3241114616e-02, /* 0x3d95ff70 */ 297169695Skan 3.3442313671e+00, /* 0x405607e3 */ 298169695Skan 4.2621845245e+01, /* 0x422a7cc5 */ 299169695Skan 1.7080809021e+02, /* 0x432acedf */ 300169695Skan 1.6673394775e+02, /* 0x4326bbe4 */ 301169695Skan}; 302169695Skanstatic const float qS3[6] = { 303169695Skan 4.8758872986e+01, /* 0x42430916 */ 304169695Skan 7.0968920898e+02, /* 0x44316c1c */ 305169695Skan 3.7041481934e+03, /* 0x4567825f */ 306169695Skan 6.4604252930e+03, /* 0x45c9e367 */ 307169695Skan 2.5163337402e+03, /* 0x451d4557 */ 308169695Skan -1.4924745178e+02, /* 0xc3153f59 */ 309169695Skan}; 310169695Skan 311169695Skanstatic const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 312169695Skan 1.5044444979e-07, /* 0x342189db */ 313169695Skan 7.3223426938e-02, /* 0x3d95f62a */ 314169695Skan 1.9981917143e+00, /* 0x3fffc4bf */ 315169695Skan 1.4495602608e+01, /* 0x4167edfd */ 316169695Skan 3.1666231155e+01, /* 0x41fd5471 */ 317169695Skan 1.6252708435e+01, /* 0x4182058c */ 318169695Skan}; 319169695Skanstatic const float qS2[6] = { 320169695Skan 3.0365585327e+01, /* 0x41f2ecb8 */ 321169695Skan 2.6934811401e+02, /* 0x4386ac8f */ 322169695Skan 8.4478375244e+02, /* 0x44533229 */ 323169695Skan 8.8293585205e+02, /* 0x445cbbe5 */ 324169695Skan 2.1266638184e+02, /* 0x4354aa98 */ 325169695Skan -5.3109550476e+00, /* 0xc0a9f358 */ 326169695Skan}; 327169695Skan 328169695Skanstatic __inline float 329169695Skanqzerof(float x) 330169695Skan{ 331169695Skan const float *p,*q; 332169695Skan float s,r,z; 333169695Skan int32_t ix; 334169695Skan GET_FLOAT_WORD(ix,x); 335169695Skan ix &= 0x7fffffff; 336169695Skan if(ix>=0x41000000) {p = qR8; q= qS8;} 337169695Skan else if(ix>=0x409173eb){p = qR5; q= qS5;} 338169695Skan else if(ix>=0x4036d917){p = qR3; q= qS3;} 339169695Skan else {p = qR2; q= qS2;} /* ix>=0x40000000 */ 340169695Skan z = one/(x*x); 341169695Skan r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 342169695Skan s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 343169695Skan return (-(float).125 + r/s)/x; 344169695Skan} 345169695Skan