1141296Sdas/* @(#)e_jn.c 1.4 95/01/18 */ 22116Sjkh/* 32116Sjkh * ==================================================== 42116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 52116Sjkh * 6141296Sdas * Developed at SunSoft, a Sun Microsystems, Inc. business. 72116Sjkh * Permission to use, copy, modify, and distribute this 8336196Smarkj * software is freely granted, provided that this notice 92116Sjkh * is preserved. 102116Sjkh * ==================================================== 112116Sjkh */ 122116Sjkh 13176451Sdas#include <sys/cdefs.h> 14176451Sdas__FBSDID("$FreeBSD: stable/11/lib/msun/src/e_jn.c 347068 2019-05-03 22:56:50Z peterj $"); 152116Sjkh 162116Sjkh/* 172116Sjkh * __ieee754_jn(n, x), __ieee754_yn(n, x) 182116Sjkh * floating point Bessel's function of the 1st and 2nd kind 192116Sjkh * of order n 20336196Smarkj * 212116Sjkh * Special cases: 222116Sjkh * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 232116Sjkh * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 242116Sjkh * Note 2. About jn(n,x), yn(n,x) 252116Sjkh * For n=0, j0(x) is called, 262116Sjkh * for n=1, j1(x) is called, 272116Sjkh * for n<x, forward recursion us used starting 282116Sjkh * from values of j0(x) and j1(x). 292116Sjkh * for n>x, a continued fraction approximation to 302116Sjkh * j(n,x)/j(n-1,x) is evaluated and then backward 312116Sjkh * recursion is used starting from a supposed value 322116Sjkh * for j(n,x). The resulting value of j(0,x) is 332116Sjkh * compared with the actual value to correct the 342116Sjkh * supposed value of j(n,x). 352116Sjkh * 362116Sjkh * yn(n,x) is similar in all respects, except 372116Sjkh * that forward recursion is used for all 382116Sjkh * values of n>1. 392116Sjkh */ 402116Sjkh 412116Sjkh#include "math.h" 422116Sjkh#include "math_private.h" 432116Sjkh 44279856Skarglstatic const volatile double vone = 1, vzero = 0; 45279856Skargl 462116Sjkhstatic const double 472116Sjkhinvsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 482116Sjkhtwo = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 492116Sjkhone = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ 502116Sjkh 512116Sjkhstatic const double zero = 0.00000000000000000000e+00; 522116Sjkh 5397413Salfreddouble 5497413Salfred__ieee754_jn(int n, double x) 552116Sjkh{ 562116Sjkh int32_t i,hx,ix,lx, sgn; 57347068Speterj double a, b, c, s, temp, di; 582116Sjkh double z, w; 592116Sjkh 602116Sjkh /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 612116Sjkh * Thus, J(-n,x) = J(n,-x) 622116Sjkh */ 632116Sjkh EXTRACT_WORDS(hx,lx,x); 642116Sjkh ix = 0x7fffffff&hx; 652116Sjkh /* if J(n,NaN) is NaN */ 662116Sjkh if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 67336196Smarkj if(n<0){ 682116Sjkh n = -n; 692116Sjkh x = -x; 702116Sjkh hx ^= 0x80000000; 712116Sjkh } 722116Sjkh if(n==0) return(__ieee754_j0(x)); 732116Sjkh if(n==1) return(__ieee754_j1(x)); 742116Sjkh sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 752116Sjkh x = fabs(x); 762116Sjkh if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 772116Sjkh b = zero; 78336196Smarkj else if((double)n<=x) { 792116Sjkh /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 802116Sjkh if(ix>=0x52D00000) { /* x > 2**302 */ 81336196Smarkj /* (x >> n**2) 822116Sjkh * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 832116Sjkh * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 84336196Smarkj * Let s=sin(x), c=cos(x), 85336196Smarkj * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then 862116Sjkh * 872116Sjkh * n sin(xn)*sqt2 cos(xn)*sqt2 882116Sjkh * ---------------------------------- 892116Sjkh * 0 s-c c+s 902116Sjkh * 1 -s-c -c+s 912116Sjkh * 2 -s+c -c-s 922116Sjkh * 3 s+c c-s 932116Sjkh */ 94347068Speterj sincos(x, &s, &c); 952116Sjkh switch(n&3) { 96347068Speterj case 0: temp = c+s; break; 97347068Speterj case 1: temp = -c+s; break; 98347068Speterj case 2: temp = -c-s; break; 99347068Speterj case 3: temp = c-s; break; 1002116Sjkh } 1012116Sjkh b = invsqrtpi*temp/sqrt(x); 102336196Smarkj } else { 1032116Sjkh a = __ieee754_j0(x); 1042116Sjkh b = __ieee754_j1(x); 1052116Sjkh for(i=1;i<n;i++){ 1062116Sjkh temp = b; 1072116Sjkh b = b*((double)(i+i)/x) - a; /* avoid underflow */ 1082116Sjkh a = temp; 1092116Sjkh } 1102116Sjkh } 1112116Sjkh } else { 1122116Sjkh if(ix<0x3e100000) { /* x < 2**-29 */ 113336196Smarkj /* x is tiny, return the first Taylor expansion of J(n,x) 1142116Sjkh * J(n,x) = 1/n!*(x/2)^n - ... 1152116Sjkh */ 1162116Sjkh if(n>33) /* underflow */ 1172116Sjkh b = zero; 1182116Sjkh else { 1192116Sjkh temp = x*0.5; b = temp; 1202116Sjkh for (a=one,i=2;i<=n;i++) { 1212116Sjkh a *= (double)i; /* a = n! */ 1222116Sjkh b *= temp; /* b = (x/2)^n */ 1232116Sjkh } 1242116Sjkh b = b/a; 1252116Sjkh } 1262116Sjkh } else { 1272116Sjkh /* use backward recurrence */ 128336196Smarkj /* x x^2 x^2 1292116Sjkh * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 1302116Sjkh * 2n - 2(n+1) - 2(n+2) 1312116Sjkh * 132336196Smarkj * 1 1 1 1332116Sjkh * (for large x) = ---- ------ ------ ..... 1342116Sjkh * 2n 2(n+1) 2(n+2) 135336196Smarkj * -- - ------ - ------ - 1362116Sjkh * x x x 1372116Sjkh * 1382116Sjkh * Let w = 2n/x and h=2/x, then the above quotient 1392116Sjkh * is equal to the continued fraction: 1402116Sjkh * 1 1412116Sjkh * = ----------------------- 1422116Sjkh * 1 1432116Sjkh * w - ----------------- 1442116Sjkh * 1 1452116Sjkh * w+h - --------- 1462116Sjkh * w+2h - ... 1472116Sjkh * 1482116Sjkh * To determine how many terms needed, let 1492116Sjkh * Q(0) = w, Q(1) = w(w+h) - 1, 1502116Sjkh * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 151336196Smarkj * When Q(k) > 1e4 good for single 152336196Smarkj * When Q(k) > 1e9 good for double 153336196Smarkj * When Q(k) > 1e17 good for quadruple 1542116Sjkh */ 1552116Sjkh /* determine k */ 1562116Sjkh double t,v; 1572116Sjkh double q0,q1,h,tmp; int32_t k,m; 1582116Sjkh w = (n+n)/(double)x; h = 2.0/(double)x; 1592116Sjkh q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 1602116Sjkh while(q1<1.0e9) { 1612116Sjkh k += 1; z += h; 1622116Sjkh tmp = z*q1 - q0; 1632116Sjkh q0 = q1; 1642116Sjkh q1 = tmp; 1652116Sjkh } 1662116Sjkh m = n+n; 1672116Sjkh for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 1682116Sjkh a = t; 1692116Sjkh b = one; 1702116Sjkh /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 1712116Sjkh * Hence, if n*(log(2n/x)) > ... 1722116Sjkh * single 8.8722839355e+01 1732116Sjkh * double 7.09782712893383973096e+02 1742116Sjkh * long double 1.1356523406294143949491931077970765006170e+04 1758870Srgrimes * then recurrent value may overflow and the result is 1762116Sjkh * likely underflow to zero 1772116Sjkh */ 1782116Sjkh tmp = n; 1792116Sjkh v = two/x; 1802116Sjkh tmp = tmp*__ieee754_log(fabs(v*tmp)); 1812116Sjkh if(tmp<7.09782712893383973096e+02) { 1822116Sjkh for(i=n-1,di=(double)(i+i);i>0;i--){ 1832116Sjkh temp = b; 1842116Sjkh b *= di; 1852116Sjkh b = b/x - a; 1862116Sjkh a = temp; 1872116Sjkh di -= two; 1882116Sjkh } 1892116Sjkh } else { 1902116Sjkh for(i=n-1,di=(double)(i+i);i>0;i--){ 1912116Sjkh temp = b; 1922116Sjkh b *= di; 1932116Sjkh b = b/x - a; 1942116Sjkh a = temp; 1952116Sjkh di -= two; 1962116Sjkh /* scale b to avoid spurious overflow */ 1972116Sjkh if(b>1e100) { 1982116Sjkh a /= b; 1992116Sjkh t /= b; 2002116Sjkh b = one; 2012116Sjkh } 2022116Sjkh } 2032116Sjkh } 204215237Suqs z = __ieee754_j0(x); 205215237Suqs w = __ieee754_j1(x); 206215237Suqs if (fabs(z) >= fabs(w)) 207215237Suqs b = (t*z/b); 208215237Suqs else 209215237Suqs b = (t*w/a); 2102116Sjkh } 2112116Sjkh } 2122116Sjkh if(sgn==1) return -b; else return b; 2132116Sjkh} 2142116Sjkh 21597413Salfreddouble 21697413Salfred__ieee754_yn(int n, double x) 2172116Sjkh{ 2182116Sjkh int32_t i,hx,ix,lx; 2192116Sjkh int32_t sign; 220347068Speterj double a, b, c, s, temp; 2212116Sjkh 2222116Sjkh EXTRACT_WORDS(hx,lx,x); 2232116Sjkh ix = 0x7fffffff&hx; 224279856Skargl /* yn(n,NaN) = NaN */ 2252116Sjkh if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 226279856Skargl /* yn(n,+-0) = -inf and raise divide-by-zero exception. */ 227279856Skargl if((ix|lx)==0) return -one/vzero; 228279856Skargl /* yn(n,x<0) = NaN and raise invalid exception. */ 229279856Skargl if(hx<0) return vzero/vzero; 2302116Sjkh sign = 1; 2312116Sjkh if(n<0){ 2322116Sjkh n = -n; 2337658Sbde sign = 1 - ((n&1)<<1); 2342116Sjkh } 2352116Sjkh if(n==0) return(__ieee754_y0(x)); 2362116Sjkh if(n==1) return(sign*__ieee754_y1(x)); 2372116Sjkh if(ix==0x7ff00000) return zero; 2382116Sjkh if(ix>=0x52D00000) { /* x > 2**302 */ 239336196Smarkj /* (x >> n**2) 2402116Sjkh * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 2412116Sjkh * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 242336196Smarkj * Let s=sin(x), c=cos(x), 243336196Smarkj * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then 2442116Sjkh * 2452116Sjkh * n sin(xn)*sqt2 cos(xn)*sqt2 2462116Sjkh * ---------------------------------- 2472116Sjkh * 0 s-c c+s 2482116Sjkh * 1 -s-c -c+s 2492116Sjkh * 2 -s+c -c-s 2502116Sjkh * 3 s+c c-s 2512116Sjkh */ 252347068Speterj sincos(x, &s, &c); 2532116Sjkh switch(n&3) { 254347068Speterj case 0: temp = s-c; break; 255347068Speterj case 1: temp = -s-c; break; 256347068Speterj case 2: temp = -s+c; break; 257347068Speterj case 3: temp = s+c; break; 2582116Sjkh } 2592116Sjkh b = invsqrtpi*temp/sqrt(x); 2602116Sjkh } else { 2612116Sjkh u_int32_t high; 2622116Sjkh a = __ieee754_y0(x); 2632116Sjkh b = __ieee754_y1(x); 2642116Sjkh /* quit if b is -inf */ 2652116Sjkh GET_HIGH_WORD(high,b); 2668870Srgrimes for(i=1;i<n&&high!=0xfff00000;i++){ 2672116Sjkh temp = b; 2682116Sjkh b = ((double)(i+i)/x)*b - a; 2692116Sjkh GET_HIGH_WORD(high,b); 2702116Sjkh a = temp; 2712116Sjkh } 2722116Sjkh } 2732116Sjkh if(sign>0) return b; else return -b; 2742116Sjkh} 275