12116Sjkh/* e_j0f.c -- float version of e_j0.c. 22116Sjkh * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. 32116Sjkh */ 42116Sjkh 52116Sjkh/* 62116Sjkh * ==================================================== 72116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 82116Sjkh * 92116Sjkh * Developed at SunPro, a Sun Microsystems, Inc. business. 102116Sjkh * Permission to use, copy, modify, and distribute this 118870Srgrimes * software is freely granted, provided that this notice 122116Sjkh * is preserved. 132116Sjkh * ==================================================== 142116Sjkh */ 152116Sjkh 16176451Sdas#include <sys/cdefs.h> 17176451Sdas__FBSDID("$FreeBSD: releng/11.0/lib/msun/src/e_j0f.c 283032 2015-05-17 16:27:06Z kargl $"); 182116Sjkh 19279856Skargl/* 20279856Skargl * See e_j0.c for complete comments. 21279856Skargl */ 22279856Skargl 232116Sjkh#include "math.h" 242116Sjkh#include "math_private.h" 252116Sjkh 26283032Skarglstatic __inline float pzerof(float), qzerof(float); 272116Sjkh 28279856Skarglstatic const volatile float vone = 1, vzero = 0; 29279856Skargl 308870Srgrimesstatic const float 312116Sjkhhuge = 1e30, 322116Sjkhone = 1.0, 332116Sjkhinvsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 342116Sjkhtpi = 6.3661974669e-01, /* 0x3f22f983 */ 352116Sjkh /* R0/S0 on [0, 2.00] */ 362116SjkhR02 = 1.5625000000e-02, /* 0x3c800000 */ 372116SjkhR03 = -1.8997929874e-04, /* 0xb947352e */ 382116SjkhR04 = 1.8295404516e-06, /* 0x35f58e88 */ 392116SjkhR05 = -4.6183270541e-09, /* 0xb19eaf3c */ 402116SjkhS01 = 1.5619102865e-02, /* 0x3c7fe744 */ 412116SjkhS02 = 1.1692678527e-04, /* 0x38f53697 */ 422116SjkhS03 = 5.1354652442e-07, /* 0x3509daa6 */ 432116SjkhS04 = 1.1661400734e-09; /* 0x30a045e8 */ 442116Sjkh 452116Sjkhstatic const float zero = 0.0; 462116Sjkh 4797413Salfredfloat 4897413Salfred__ieee754_j0f(float x) 492116Sjkh{ 502116Sjkh float z, s,c,ss,cc,r,u,v; 512116Sjkh int32_t hx,ix; 522116Sjkh 532116Sjkh GET_FLOAT_WORD(hx,x); 542116Sjkh ix = hx&0x7fffffff; 552116Sjkh if(ix>=0x7f800000) return one/(x*x); 562116Sjkh x = fabsf(x); 572116Sjkh if(ix >= 0x40000000) { /* |x| >= 2.0 */ 582116Sjkh s = sinf(x); 592116Sjkh c = cosf(x); 602116Sjkh ss = s-c; 612116Sjkh cc = s+c; 622116Sjkh if(ix<0x7f000000) { /* make sure x+x not overflow */ 632116Sjkh z = -cosf(x+x); 642116Sjkh if ((s*c)<zero) cc = z/ss; 652116Sjkh else ss = z/cc; 662116Sjkh } 672116Sjkh /* 682116Sjkh * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 692116Sjkh * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 702116Sjkh */ 71279491Skargl if(ix>0x58000000) z = (invsqrtpi*cc)/sqrtf(x); /* |x|>2**49 */ 722116Sjkh else { 732116Sjkh u = pzerof(x); v = qzerof(x); 742116Sjkh z = invsqrtpi*(u*cc-v*ss)/sqrtf(x); 752116Sjkh } 762116Sjkh return z; 772116Sjkh } 78275614Skargl if(ix<0x3b000000) { /* |x| < 2**-9 */ 792116Sjkh if(huge+x>one) { /* raise inexact if x != 0 */ 80275518Skargl if(ix<0x39800000) return one; /* |x|<2**-12 */ 81275518Skargl else return one - x*x/4; 822116Sjkh } 832116Sjkh } 842116Sjkh z = x*x; 852116Sjkh r = z*(R02+z*(R03+z*(R04+z*R05))); 862116Sjkh s = one+z*(S01+z*(S02+z*(S03+z*S04))); 872116Sjkh if(ix < 0x3F800000) { /* |x| < 1.00 */ 882116Sjkh return one + z*((float)-0.25+(r/s)); 892116Sjkh } else { 902116Sjkh u = (float)0.5*x; 912116Sjkh return((one+u)*(one-u)+z*(r/s)); 922116Sjkh } 932116Sjkh} 942116Sjkh 952116Sjkhstatic const float 962116Sjkhu00 = -7.3804296553e-02, /* 0xbd9726b5 */ 972116Sjkhu01 = 1.7666645348e-01, /* 0x3e34e80d */ 982116Sjkhu02 = -1.3818567619e-02, /* 0xbc626746 */ 992116Sjkhu03 = 3.4745343146e-04, /* 0x39b62a69 */ 1002116Sjkhu04 = -3.8140706238e-06, /* 0xb67ff53c */ 1012116Sjkhu05 = 1.9559013964e-08, /* 0x32a802ba */ 1022116Sjkhu06 = -3.9820518410e-11, /* 0xae2f21eb */ 1032116Sjkhv01 = 1.2730483897e-02, /* 0x3c509385 */ 1042116Sjkhv02 = 7.6006865129e-05, /* 0x389f65e0 */ 1052116Sjkhv03 = 2.5915085189e-07, /* 0x348b216c */ 1062116Sjkhv04 = 4.4111031494e-10; /* 0x2ff280c2 */ 1072116Sjkh 10897413Salfredfloat 10997413Salfred__ieee754_y0f(float x) 1102116Sjkh{ 1112116Sjkh float z, s,c,ss,cc,u,v; 1122116Sjkh int32_t hx,ix; 1132116Sjkh 1142116Sjkh GET_FLOAT_WORD(hx,x); 1152116Sjkh ix = 0x7fffffff&hx; 116279856Skargl if(ix>=0x7f800000) return vone/(x+x*x); 117279856Skargl if(ix==0) return -one/vzero; 118279856Skargl if(hx<0) return vzero/vzero; 1192116Sjkh if(ix >= 0x40000000) { /* |x| >= 2.0 */ 1202116Sjkh /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) 1212116Sjkh * where x0 = x-pi/4 1222116Sjkh * Better formula: 1232116Sjkh * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) 1242116Sjkh * = 1/sqrt(2) * (sin(x) + cos(x)) 1252116Sjkh * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 1262116Sjkh * = 1/sqrt(2) * (sin(x) - cos(x)) 1272116Sjkh * To avoid cancellation, use 1282116Sjkh * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 1292116Sjkh * to compute the worse one. 1302116Sjkh */ 1312116Sjkh s = sinf(x); 1322116Sjkh c = cosf(x); 1332116Sjkh ss = s-c; 1342116Sjkh cc = s+c; 1352116Sjkh /* 1362116Sjkh * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 1372116Sjkh * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 1382116Sjkh */ 1392116Sjkh if(ix<0x7f000000) { /* make sure x+x not overflow */ 1402116Sjkh z = -cosf(x+x); 1412116Sjkh if ((s*c)<zero) cc = z/ss; 1422116Sjkh else ss = z/cc; 1432116Sjkh } 144279491Skargl if(ix>0x58000000) z = (invsqrtpi*ss)/sqrtf(x); /* |x|>2**49 */ 1452116Sjkh else { 1462116Sjkh u = pzerof(x); v = qzerof(x); 1472116Sjkh z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 1482116Sjkh } 1492116Sjkh return z; 1502116Sjkh } 151279491Skargl if(ix<=0x39000000) { /* x < 2**-13 */ 1522116Sjkh return(u00 + tpi*__ieee754_logf(x)); 1532116Sjkh } 1542116Sjkh z = x*x; 1552116Sjkh u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); 1562116Sjkh v = one+z*(v01+z*(v02+z*(v03+z*v04))); 1572116Sjkh return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x))); 1582116Sjkh} 1592116Sjkh 1602116Sjkh/* The asymptotic expansions of pzero is 1612116Sjkh * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. 1622116Sjkh * For x >= 2, We approximate pzero by 1632116Sjkh * pzero(x) = 1 + (R/S) 1642116Sjkh * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 1652116Sjkh * S = 1 + pS0*s^2 + ... + pS4*s^10 1662116Sjkh * and 1672116Sjkh * | pzero(x)-1-R/S | <= 2 ** ( -60.26) 1682116Sjkh */ 1692116Sjkhstatic const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 1702116Sjkh 0.0000000000e+00, /* 0x00000000 */ 1712116Sjkh -7.0312500000e-02, /* 0xbd900000 */ 1722116Sjkh -8.0816707611e+00, /* 0xc1014e86 */ 1732116Sjkh -2.5706311035e+02, /* 0xc3808814 */ 1742116Sjkh -2.4852163086e+03, /* 0xc51b5376 */ 1752116Sjkh -5.2530439453e+03, /* 0xc5a4285a */ 1762116Sjkh}; 1772116Sjkhstatic const float pS8[5] = { 1782116Sjkh 1.1653436279e+02, /* 0x42e91198 */ 1792116Sjkh 3.8337448730e+03, /* 0x456f9beb */ 1802116Sjkh 4.0597855469e+04, /* 0x471e95db */ 1812116Sjkh 1.1675296875e+05, /* 0x47e4087c */ 1822116Sjkh 4.7627726562e+04, /* 0x473a0bba */ 1832116Sjkh}; 1842116Sjkhstatic const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 1852116Sjkh -1.1412546255e-11, /* 0xad48c58a */ 1862116Sjkh -7.0312492549e-02, /* 0xbd8fffff */ 1872116Sjkh -4.1596107483e+00, /* 0xc0851b88 */ 1882116Sjkh -6.7674766541e+01, /* 0xc287597b */ 1892116Sjkh -3.3123129272e+02, /* 0xc3a59d9b */ 1902116Sjkh -3.4643338013e+02, /* 0xc3ad3779 */ 1912116Sjkh}; 1922116Sjkhstatic const float pS5[5] = { 1932116Sjkh 6.0753936768e+01, /* 0x42730408 */ 1942116Sjkh 1.0512523193e+03, /* 0x44836813 */ 1952116Sjkh 5.9789707031e+03, /* 0x45bad7c4 */ 1962116Sjkh 9.6254453125e+03, /* 0x461665c8 */ 1972116Sjkh 2.4060581055e+03, /* 0x451660ee */ 1982116Sjkh}; 1992116Sjkh 2002116Sjkhstatic const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 2012116Sjkh -2.5470459075e-09, /* 0xb12f081b */ 2022116Sjkh -7.0311963558e-02, /* 0xbd8fffb8 */ 2032116Sjkh -2.4090321064e+00, /* 0xc01a2d95 */ 2042116Sjkh -2.1965976715e+01, /* 0xc1afba52 */ 2052116Sjkh -5.8079170227e+01, /* 0xc2685112 */ 2062116Sjkh -3.1447946548e+01, /* 0xc1fb9565 */ 2072116Sjkh}; 2082116Sjkhstatic const float pS3[5] = { 2092116Sjkh 3.5856033325e+01, /* 0x420f6c94 */ 2102116Sjkh 3.6151397705e+02, /* 0x43b4c1ca */ 2112116Sjkh 1.1936077881e+03, /* 0x44953373 */ 2122116Sjkh 1.1279968262e+03, /* 0x448cffe6 */ 2132116Sjkh 1.7358093262e+02, /* 0x432d94b8 */ 2142116Sjkh}; 2152116Sjkh 2162116Sjkhstatic const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 2172116Sjkh -8.8753431271e-08, /* 0xb3be98b7 */ 2182116Sjkh -7.0303097367e-02, /* 0xbd8ffb12 */ 2192116Sjkh -1.4507384300e+00, /* 0xbfb9b1cc */ 2202116Sjkh -7.6356959343e+00, /* 0xc0f4579f */ 2212116Sjkh -1.1193166733e+01, /* 0xc1331736 */ 2222116Sjkh -3.2336456776e+00, /* 0xc04ef40d */ 2232116Sjkh}; 2242116Sjkhstatic const float pS2[5] = { 2252116Sjkh 2.2220300674e+01, /* 0x41b1c32d */ 2262116Sjkh 1.3620678711e+02, /* 0x430834f0 */ 2272116Sjkh 2.7047027588e+02, /* 0x43873c32 */ 2282116Sjkh 1.5387539673e+02, /* 0x4319e01a */ 2292116Sjkh 1.4657617569e+01, /* 0x416a859a */ 2302116Sjkh}; 2312116Sjkh 232279493Skarglstatic __inline float 233279493Skarglpzerof(float x) 2342116Sjkh{ 2352116Sjkh const float *p,*q; 2362116Sjkh float z,r,s; 2372116Sjkh int32_t ix; 2382116Sjkh GET_FLOAT_WORD(ix,x); 2392116Sjkh ix &= 0x7fffffff; 2402116Sjkh if(ix>=0x41000000) {p = pR8; q= pS8;} 241279491Skargl else if(ix>=0x409173eb){p = pR5; q= pS5;} 242279491Skargl else if(ix>=0x4036d917){p = pR3; q= pS3;} 243279127Spfg else {p = pR2; q= pS2;} /* ix>=0x40000000 */ 2442116Sjkh z = one/(x*x); 2452116Sjkh r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 2462116Sjkh s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 2472116Sjkh return one+ r/s; 2482116Sjkh} 2492116Sjkh 2508870Srgrimes 2512116Sjkh/* For x >= 8, the asymptotic expansions of qzero is 2522116Sjkh * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. 2532116Sjkh * We approximate pzero by 2542116Sjkh * qzero(x) = s*(-1.25 + (R/S)) 2552116Sjkh * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 2562116Sjkh * S = 1 + qS0*s^2 + ... + qS5*s^12 2572116Sjkh * and 2582116Sjkh * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) 2592116Sjkh */ 2602116Sjkhstatic const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 2612116Sjkh 0.0000000000e+00, /* 0x00000000 */ 2622116Sjkh 7.3242187500e-02, /* 0x3d960000 */ 2632116Sjkh 1.1768206596e+01, /* 0x413c4a93 */ 2642116Sjkh 5.5767340088e+02, /* 0x440b6b19 */ 2652116Sjkh 8.8591972656e+03, /* 0x460a6cca */ 2662116Sjkh 3.7014625000e+04, /* 0x471096a0 */ 2672116Sjkh}; 2682116Sjkhstatic const float qS8[6] = { 2692116Sjkh 1.6377603149e+02, /* 0x4323c6aa */ 2702116Sjkh 8.0983447266e+03, /* 0x45fd12c2 */ 2712116Sjkh 1.4253829688e+05, /* 0x480b3293 */ 2722116Sjkh 8.0330925000e+05, /* 0x49441ed4 */ 2732116Sjkh 8.4050156250e+05, /* 0x494d3359 */ 2742116Sjkh -3.4389928125e+05, /* 0xc8a7eb69 */ 2752116Sjkh}; 2762116Sjkh 2772116Sjkhstatic const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 2782116Sjkh 1.8408595828e-11, /* 0x2da1ec79 */ 2792116Sjkh 7.3242180049e-02, /* 0x3d95ffff */ 2802116Sjkh 5.8356351852e+00, /* 0x40babd86 */ 2812116Sjkh 1.3511157227e+02, /* 0x43071c90 */ 2822116Sjkh 1.0272437744e+03, /* 0x448067cd */ 2832116Sjkh 1.9899779053e+03, /* 0x44f8bf4b */ 2842116Sjkh}; 2852116Sjkhstatic const float qS5[6] = { 2862116Sjkh 8.2776611328e+01, /* 0x42a58da0 */ 2872116Sjkh 2.0778142090e+03, /* 0x4501dd07 */ 2882116Sjkh 1.8847289062e+04, /* 0x46933e94 */ 2892116Sjkh 5.6751113281e+04, /* 0x475daf1d */ 2902116Sjkh 3.5976753906e+04, /* 0x470c88c1 */ 2912116Sjkh -5.3543427734e+03, /* 0xc5a752be */ 2922116Sjkh}; 2932116Sjkh 2942116Sjkhstatic const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ 2952116Sjkh 4.3774099900e-09, /* 0x3196681b */ 2962116Sjkh 7.3241114616e-02, /* 0x3d95ff70 */ 2972116Sjkh 3.3442313671e+00, /* 0x405607e3 */ 2982116Sjkh 4.2621845245e+01, /* 0x422a7cc5 */ 2992116Sjkh 1.7080809021e+02, /* 0x432acedf */ 3002116Sjkh 1.6673394775e+02, /* 0x4326bbe4 */ 3012116Sjkh}; 3022116Sjkhstatic const float qS3[6] = { 3032116Sjkh 4.8758872986e+01, /* 0x42430916 */ 3042116Sjkh 7.0968920898e+02, /* 0x44316c1c */ 3052116Sjkh 3.7041481934e+03, /* 0x4567825f */ 3062116Sjkh 6.4604252930e+03, /* 0x45c9e367 */ 3072116Sjkh 2.5163337402e+03, /* 0x451d4557 */ 3082116Sjkh -1.4924745178e+02, /* 0xc3153f59 */ 3092116Sjkh}; 3102116Sjkh 3112116Sjkhstatic const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 3122116Sjkh 1.5044444979e-07, /* 0x342189db */ 3132116Sjkh 7.3223426938e-02, /* 0x3d95f62a */ 3142116Sjkh 1.9981917143e+00, /* 0x3fffc4bf */ 3152116Sjkh 1.4495602608e+01, /* 0x4167edfd */ 3162116Sjkh 3.1666231155e+01, /* 0x41fd5471 */ 3172116Sjkh 1.6252708435e+01, /* 0x4182058c */ 3182116Sjkh}; 3192116Sjkhstatic const float qS2[6] = { 3202116Sjkh 3.0365585327e+01, /* 0x41f2ecb8 */ 3212116Sjkh 2.6934811401e+02, /* 0x4386ac8f */ 3222116Sjkh 8.4478375244e+02, /* 0x44533229 */ 3232116Sjkh 8.8293585205e+02, /* 0x445cbbe5 */ 3242116Sjkh 2.1266638184e+02, /* 0x4354aa98 */ 3252116Sjkh -5.3109550476e+00, /* 0xc0a9f358 */ 3262116Sjkh}; 3272116Sjkh 328279493Skarglstatic __inline float 329279493Skarglqzerof(float x) 3302116Sjkh{ 3312116Sjkh const float *p,*q; 3322116Sjkh float s,r,z; 3332116Sjkh int32_t ix; 3342116Sjkh GET_FLOAT_WORD(ix,x); 3352116Sjkh ix &= 0x7fffffff; 3362116Sjkh if(ix>=0x41000000) {p = qR8; q= qS8;} 337279491Skargl else if(ix>=0x409173eb){p = qR5; q= qS5;} 338279491Skargl else if(ix>=0x4036d917){p = qR3; q= qS3;} 339279240Spfg else {p = qR2; q= qS2;} /* ix>=0x40000000 */ 3402116Sjkh z = one/(x*x); 3412116Sjkh r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 3422116Sjkh s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 3432116Sjkh return (-(float).125 + r/s)/x; 3442116Sjkh} 345