12116Sjkh/* e_j1f.c -- float version of e_j1.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("$FreeBSD$"); 182116Sjkh 192116Sjkh#include "math.h" 202116Sjkh#include "math_private.h" 212116Sjkh 222116Sjkhstatic float ponef(float), qonef(float); 232116Sjkh 248870Srgrimesstatic const float 252116Sjkhhuge = 1e30, 262116Sjkhone = 1.0, 272116Sjkhinvsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ 282116Sjkhtpi = 6.3661974669e-01, /* 0x3f22f983 */ 292116Sjkh /* R0/S0 on [0,2] */ 302116Sjkhr00 = -6.2500000000e-02, /* 0xbd800000 */ 312116Sjkhr01 = 1.4070566976e-03, /* 0x3ab86cfd */ 322116Sjkhr02 = -1.5995563444e-05, /* 0xb7862e36 */ 332116Sjkhr03 = 4.9672799207e-08, /* 0x335557d2 */ 342116Sjkhs01 = 1.9153760746e-02, /* 0x3c9ce859 */ 352116Sjkhs02 = 1.8594678841e-04, /* 0x3942fab6 */ 362116Sjkhs03 = 1.1771846857e-06, /* 0x359dffc2 */ 372116Sjkhs04 = 5.0463624390e-09, /* 0x31ad6446 */ 382116Sjkhs05 = 1.2354227016e-11; /* 0x2d59567e */ 392116Sjkh 402116Sjkhstatic const float zero = 0.0; 412116Sjkh 4297413Salfredfloat 4397413Salfred__ieee754_j1f(float x) 442116Sjkh{ 452116Sjkh float z, s,c,ss,cc,r,u,v,y; 462116Sjkh int32_t hx,ix; 472116Sjkh 482116Sjkh GET_FLOAT_WORD(hx,x); 492116Sjkh ix = hx&0x7fffffff; 502116Sjkh if(ix>=0x7f800000) return one/x; 512116Sjkh y = fabsf(x); 522116Sjkh if(ix >= 0x40000000) { /* |x| >= 2.0 */ 532116Sjkh s = sinf(y); 542116Sjkh c = cosf(y); 552116Sjkh ss = -s-c; 562116Sjkh cc = s-c; 572116Sjkh if(ix<0x7f000000) { /* make sure y+y not overflow */ 582116Sjkh z = cosf(y+y); 592116Sjkh if ((s*c)>zero) cc = z/ss; 602116Sjkh else ss = z/cc; 612116Sjkh } 622116Sjkh /* 632116Sjkh * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) 642116Sjkh * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) 652116Sjkh */ 662116Sjkh if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y); 672116Sjkh else { 682116Sjkh u = ponef(y); v = qonef(y); 692116Sjkh z = invsqrtpi*(u*cc-v*ss)/sqrtf(y); 702116Sjkh } 712116Sjkh if(hx<0) return -z; 722116Sjkh else return z; 732116Sjkh } 742116Sjkh if(ix<0x32000000) { /* |x|<2**-27 */ 752116Sjkh if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */ 762116Sjkh } 772116Sjkh z = x*x; 782116Sjkh r = z*(r00+z*(r01+z*(r02+z*r03))); 792116Sjkh s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); 802116Sjkh r *= x; 812116Sjkh return(x*(float)0.5+r/s); 822116Sjkh} 832116Sjkh 842116Sjkhstatic const float U0[5] = { 852116Sjkh -1.9605709612e-01, /* 0xbe48c331 */ 862116Sjkh 5.0443872809e-02, /* 0x3d4e9e3c */ 872116Sjkh -1.9125689287e-03, /* 0xbafaaf2a */ 882116Sjkh 2.3525259166e-05, /* 0x37c5581c */ 892116Sjkh -9.1909917899e-08, /* 0xb3c56003 */ 902116Sjkh}; 912116Sjkhstatic const float V0[5] = { 922116Sjkh 1.9916731864e-02, /* 0x3ca3286a */ 932116Sjkh 2.0255257550e-04, /* 0x3954644b */ 942116Sjkh 1.3560879779e-06, /* 0x35b602d4 */ 952116Sjkh 6.2274145840e-09, /* 0x31d5f8eb */ 962116Sjkh 1.6655924903e-11, /* 0x2d9281cf */ 972116Sjkh}; 982116Sjkh 9997413Salfredfloat 10097413Salfred__ieee754_y1f(float x) 1012116Sjkh{ 1022116Sjkh float z, s,c,ss,cc,u,v; 1032116Sjkh int32_t hx,ix; 1042116Sjkh 1052116Sjkh GET_FLOAT_WORD(hx,x); 1062116Sjkh ix = 0x7fffffff&hx; 1072116Sjkh /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ 1088870Srgrimes if(ix>=0x7f800000) return one/(x+x*x); 1092116Sjkh if(ix==0) return -one/zero; 1102116Sjkh if(hx<0) return zero/zero; 1112116Sjkh if(ix >= 0x40000000) { /* |x| >= 2.0 */ 1122116Sjkh s = sinf(x); 1132116Sjkh c = cosf(x); 1142116Sjkh ss = -s-c; 1152116Sjkh cc = s-c; 1162116Sjkh if(ix<0x7f000000) { /* make sure x+x not overflow */ 1172116Sjkh z = cosf(x+x); 1182116Sjkh if ((s*c)>zero) cc = z/ss; 1192116Sjkh else ss = z/cc; 1202116Sjkh } 1212116Sjkh /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) 1222116Sjkh * where x0 = x-3pi/4 1232116Sjkh * Better formula: 1242116Sjkh * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 1252116Sjkh * = 1/sqrt(2) * (sin(x) - cos(x)) 1262116Sjkh * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 1272116Sjkh * = -1/sqrt(2) * (cos(x) + sin(x)) 1282116Sjkh * To avoid cancellation, use 1292116Sjkh * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 1302116Sjkh * to compute the worse one. 1312116Sjkh */ 1322116Sjkh if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x); 1332116Sjkh else { 1342116Sjkh u = ponef(x); v = qonef(x); 1352116Sjkh z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); 1362116Sjkh } 1372116Sjkh return z; 1388870Srgrimes } 1392116Sjkh if(ix<=0x24800000) { /* x < 2**-54 */ 1402116Sjkh return(-tpi/x); 1418870Srgrimes } 1422116Sjkh z = x*x; 1432116Sjkh u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); 1442116Sjkh v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); 1452116Sjkh return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x)); 1462116Sjkh} 1472116Sjkh 1482116Sjkh/* For x >= 8, the asymptotic expansions of pone is 1492116Sjkh * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. 1502116Sjkh * We approximate pone by 1512116Sjkh * pone(x) = 1 + (R/S) 1522116Sjkh * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 1532116Sjkh * S = 1 + ps0*s^2 + ... + ps4*s^10 1542116Sjkh * and 1552116Sjkh * | pone(x)-1-R/S | <= 2 ** ( -60.06) 1562116Sjkh */ 1572116Sjkh 1582116Sjkhstatic const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 1592116Sjkh 0.0000000000e+00, /* 0x00000000 */ 1602116Sjkh 1.1718750000e-01, /* 0x3df00000 */ 1612116Sjkh 1.3239480972e+01, /* 0x4153d4ea */ 1622116Sjkh 4.1205184937e+02, /* 0x43ce06a3 */ 1632116Sjkh 3.8747453613e+03, /* 0x45722bed */ 1642116Sjkh 7.9144794922e+03, /* 0x45f753d6 */ 1652116Sjkh}; 1662116Sjkhstatic const float ps8[5] = { 1672116Sjkh 1.1420736694e+02, /* 0x42e46a2c */ 1682116Sjkh 3.6509309082e+03, /* 0x45642ee5 */ 1692116Sjkh 3.6956207031e+04, /* 0x47105c35 */ 1702116Sjkh 9.7602796875e+04, /* 0x47bea166 */ 1712116Sjkh 3.0804271484e+04, /* 0x46f0a88b */ 1722116Sjkh}; 1732116Sjkh 1742116Sjkhstatic const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 1752116Sjkh 1.3199052094e-11, /* 0x2d68333f */ 1762116Sjkh 1.1718749255e-01, /* 0x3defffff */ 1772116Sjkh 6.8027510643e+00, /* 0x40d9b023 */ 1782116Sjkh 1.0830818176e+02, /* 0x42d89dca */ 1792116Sjkh 5.1763616943e+02, /* 0x440168b7 */ 1802116Sjkh 5.2871520996e+02, /* 0x44042dc6 */ 1812116Sjkh}; 1822116Sjkhstatic const float ps5[5] = { 1832116Sjkh 5.9280597687e+01, /* 0x426d1f55 */ 1842116Sjkh 9.9140142822e+02, /* 0x4477d9b1 */ 1852116Sjkh 5.3532670898e+03, /* 0x45a74a23 */ 1862116Sjkh 7.8446904297e+03, /* 0x45f52586 */ 1872116Sjkh 1.5040468750e+03, /* 0x44bc0180 */ 1882116Sjkh}; 1892116Sjkh 1902116Sjkhstatic const float pr3[6] = { 1912116Sjkh 3.0250391081e-09, /* 0x314fe10d */ 1922116Sjkh 1.1718686670e-01, /* 0x3defffab */ 1932116Sjkh 3.9329774380e+00, /* 0x407bb5e7 */ 1942116Sjkh 3.5119403839e+01, /* 0x420c7a45 */ 1952116Sjkh 9.1055007935e+01, /* 0x42b61c2a */ 1962116Sjkh 4.8559066772e+01, /* 0x42423c7c */ 1972116Sjkh}; 1982116Sjkhstatic const float ps3[5] = { 1992116Sjkh 3.4791309357e+01, /* 0x420b2a4d */ 2002116Sjkh 3.3676245117e+02, /* 0x43a86198 */ 2012116Sjkh 1.0468714600e+03, /* 0x4482dbe3 */ 2022116Sjkh 8.9081134033e+02, /* 0x445eb3ed */ 2032116Sjkh 1.0378793335e+02, /* 0x42cf936c */ 2042116Sjkh}; 2052116Sjkh 2062116Sjkhstatic const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 2072116Sjkh 1.0771083225e-07, /* 0x33e74ea8 */ 2082116Sjkh 1.1717621982e-01, /* 0x3deffa16 */ 2092116Sjkh 2.3685150146e+00, /* 0x401795c0 */ 2102116Sjkh 1.2242610931e+01, /* 0x4143e1bc */ 2112116Sjkh 1.7693971634e+01, /* 0x418d8d41 */ 2122116Sjkh 5.0735230446e+00, /* 0x40a25a4d */ 2132116Sjkh}; 2142116Sjkhstatic const float ps2[5] = { 2152116Sjkh 2.1436485291e+01, /* 0x41ab7dec */ 2162116Sjkh 1.2529022980e+02, /* 0x42fa9499 */ 2172116Sjkh 2.3227647400e+02, /* 0x436846c7 */ 2182116Sjkh 1.1767937469e+02, /* 0x42eb5bd7 */ 2192116Sjkh 8.3646392822e+00, /* 0x4105d590 */ 2202116Sjkh}; 2212116Sjkh 2222116Sjkh static float ponef(float x) 2232116Sjkh{ 2242116Sjkh const float *p,*q; 2252116Sjkh float z,r,s; 2262116Sjkh int32_t ix; 2272116Sjkh GET_FLOAT_WORD(ix,x); 2282116Sjkh ix &= 0x7fffffff; 2292116Sjkh if(ix>=0x41000000) {p = pr8; q= ps8;} 2302116Sjkh else if(ix>=0x40f71c58){p = pr5; q= ps5;} 2312116Sjkh else if(ix>=0x4036db68){p = pr3; q= ps3;} 2322116Sjkh else if(ix>=0x40000000){p = pr2; q= ps2;} 2332116Sjkh z = one/(x*x); 2342116Sjkh r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 2352116Sjkh s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); 2362116Sjkh return one+ r/s; 2372116Sjkh} 2382116Sjkh 2398870Srgrimes 2402116Sjkh/* For x >= 8, the asymptotic expansions of qone is 2412116Sjkh * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. 2422116Sjkh * We approximate pone by 2432116Sjkh * qone(x) = s*(0.375 + (R/S)) 2442116Sjkh * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 2452116Sjkh * S = 1 + qs1*s^2 + ... + qs6*s^12 2462116Sjkh * and 2472116Sjkh * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) 2482116Sjkh */ 2492116Sjkh 2502116Sjkhstatic const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ 2512116Sjkh 0.0000000000e+00, /* 0x00000000 */ 2522116Sjkh -1.0253906250e-01, /* 0xbdd20000 */ 2532116Sjkh -1.6271753311e+01, /* 0xc1822c8d */ 2542116Sjkh -7.5960174561e+02, /* 0xc43de683 */ 2552116Sjkh -1.1849806641e+04, /* 0xc639273a */ 2562116Sjkh -4.8438511719e+04, /* 0xc73d3683 */ 2572116Sjkh}; 2582116Sjkhstatic const float qs8[6] = { 2592116Sjkh 1.6139537048e+02, /* 0x43216537 */ 2602116Sjkh 7.8253862305e+03, /* 0x45f48b17 */ 2612116Sjkh 1.3387534375e+05, /* 0x4802bcd6 */ 2622116Sjkh 7.1965775000e+05, /* 0x492fb29c */ 2632116Sjkh 6.6660125000e+05, /* 0x4922be94 */ 2642116Sjkh -2.9449025000e+05, /* 0xc88fcb48 */ 2652116Sjkh}; 2662116Sjkh 2672116Sjkhstatic const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ 2682116Sjkh -2.0897993405e-11, /* 0xadb7d219 */ 2692116Sjkh -1.0253904760e-01, /* 0xbdd1fffe */ 2702116Sjkh -8.0564479828e+00, /* 0xc100e736 */ 2712116Sjkh -1.8366960144e+02, /* 0xc337ab6b */ 2722116Sjkh -1.3731937256e+03, /* 0xc4aba633 */ 2732116Sjkh -2.6124443359e+03, /* 0xc523471c */ 2742116Sjkh}; 2752116Sjkhstatic const float qs5[6] = { 2762116Sjkh 8.1276550293e+01, /* 0x42a28d98 */ 2772116Sjkh 1.9917987061e+03, /* 0x44f8f98f */ 2782116Sjkh 1.7468484375e+04, /* 0x468878f8 */ 2792116Sjkh 4.9851425781e+04, /* 0x4742bb6d */ 2802116Sjkh 2.7948074219e+04, /* 0x46da5826 */ 2812116Sjkh -4.7191835938e+03, /* 0xc5937978 */ 2822116Sjkh}; 2832116Sjkh 2842116Sjkhstatic const float qr3[6] = { 2852116Sjkh -5.0783124372e-09, /* 0xb1ae7d4f */ 2862116Sjkh -1.0253783315e-01, /* 0xbdd1ff5b */ 2872116Sjkh -4.6101160049e+00, /* 0xc0938612 */ 2882116Sjkh -5.7847221375e+01, /* 0xc267638e */ 2892116Sjkh -2.2824453735e+02, /* 0xc3643e9a */ 2902116Sjkh -2.1921012878e+02, /* 0xc35b35cb */ 2912116Sjkh}; 2922116Sjkhstatic const float qs3[6] = { 2932116Sjkh 4.7665153503e+01, /* 0x423ea91e */ 2942116Sjkh 6.7386511230e+02, /* 0x4428775e */ 2952116Sjkh 3.3801528320e+03, /* 0x45534272 */ 2962116Sjkh 5.5477290039e+03, /* 0x45ad5dd5 */ 2972116Sjkh 1.9031191406e+03, /* 0x44ede3d0 */ 2982116Sjkh -1.3520118713e+02, /* 0xc3073381 */ 2992116Sjkh}; 3002116Sjkh 3012116Sjkhstatic const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ 3022116Sjkh -1.7838172539e-07, /* 0xb43f8932 */ 3032116Sjkh -1.0251704603e-01, /* 0xbdd1f475 */ 3042116Sjkh -2.7522056103e+00, /* 0xc0302423 */ 3052116Sjkh -1.9663616180e+01, /* 0xc19d4f16 */ 3062116Sjkh -4.2325313568e+01, /* 0xc2294d1f */ 3072116Sjkh -2.1371921539e+01, /* 0xc1aaf9b2 */ 3082116Sjkh}; 3092116Sjkhstatic const float qs2[6] = { 3102116Sjkh 2.9533363342e+01, /* 0x41ec4454 */ 3112116Sjkh 2.5298155212e+02, /* 0x437cfb47 */ 3122116Sjkh 7.5750280762e+02, /* 0x443d602e */ 3132116Sjkh 7.3939318848e+02, /* 0x4438d92a */ 3142116Sjkh 1.5594900513e+02, /* 0x431bf2f2 */ 3152116Sjkh -4.9594988823e+00, /* 0xc09eb437 */ 3162116Sjkh}; 3172116Sjkh 3182116Sjkh static float qonef(float x) 3192116Sjkh{ 3202116Sjkh const float *p,*q; 3212116Sjkh float s,r,z; 3222116Sjkh int32_t ix; 3232116Sjkh GET_FLOAT_WORD(ix,x); 3242116Sjkh ix &= 0x7fffffff; 3252116Sjkh if(ix>=0x40200000) {p = qr8; q= qs8;} 3262116Sjkh else if(ix>=0x40f71c58){p = qr5; q= qs5;} 3272116Sjkh else if(ix>=0x4036db68){p = qr3; q= qs3;} 3282116Sjkh else if(ix>=0x40000000){p = qr2; q= qs2;} 3292116Sjkh z = one/(x*x); 3302116Sjkh r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); 3312116Sjkh s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); 3322116Sjkh return ((float).375 + r/s)/x; 3332116Sjkh} 334