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+/* origin: FreeBSD /usr/src/lib/msun/src/e_jn.c */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunSoft, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+/*
+ * jn(n, x), yn(n, x)
+ * floating point Bessel's function of the 1st and 2nd kind
+ * of order n
+ *
+ * Special cases:
+ * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
+ * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
+ * Note 2. About jn(n,x), yn(n,x)
+ * For n=0, j0(x) is called,
+ * for n=1, j1(x) is called,
+ * for n<x, forward recursion us used starting
+ * from values of j0(x) and j1(x).
+ * for n>x, a continued fraction approximation to
+ * j(n,x)/j(n-1,x) is evaluated and then backward
+ * recursion is used starting from a supposed value
+ * for j(n,x). The resulting value of j(0,x) is
+ * compared with the actual value to correct the
+ * supposed value of j(n,x).
+ *
+ * yn(n,x) is similar in all respects, except
+ * that forward recursion is used for all
+ * values of n>1.
+ *
+ */
+
+#include "libm.h"
+
+static const double
+invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
+two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
+one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
+
+static const double zero = 0.00000000000000000000e+00;
+
+double jn(int n, double x)
+{
+ int32_t i,hx,ix,lx,sgn;
+ double a, b, temp, di;
+ double z, w;
+
+ /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
+ * Thus, J(-n,x) = J(n,-x)
+ */
+ EXTRACT_WORDS(hx, lx, x);
+ ix = 0x7fffffff & hx;
+ /* if J(n,NaN) is NaN */
+ if ((ix|((uint32_t)(lx|-lx))>>31) > 0x7ff00000)
+ return x+x;
+ if (n < 0) {
+ n = -n;
+ x = -x;
+ hx ^= 0x80000000;
+ }
+ if (n == 0) return j0(x);
+ if (n == 1) return j1(x);
+
+ sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
+ x = fabs(x);
+ if ((ix|lx) == 0 || ix >= 0x7ff00000) /* if x is 0 or inf */
+ b = zero;
+ else if ((double)n <= x) {
+ /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
+ if (ix >= 0x52D00000) { /* x > 2**302 */
+ /* (x >> n**2)
+ * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ * Let s=sin(x), c=cos(x),
+ * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
+ *
+ * n sin(xn)*sqt2 cos(xn)*sqt2
+ * ----------------------------------
+ * 0 s-c c+s
+ * 1 -s-c -c+s
+ * 2 -s+c -c-s
+ * 3 s+c c-s
+ */
+ switch(n&3) {
+ case 0: temp = cos(x)+sin(x); break;
+ case 1: temp = -cos(x)+sin(x); break;
+ case 2: temp = -cos(x)-sin(x); break;
+ case 3: temp = cos(x)-sin(x); break;
+ }
+ b = invsqrtpi*temp/sqrt(x);
+ } else {
+ a = j0(x);
+ b = j1(x);
+ for (i=1; i<n; i++){
+ temp = b;
+ b = b*((double)(i+i)/x) - a; /* avoid underflow */
+ a = temp;
+ }
+ }
+ } else {
+ if (ix < 0x3e100000) { /* x < 2**-29 */
+ /* x is tiny, return the first Taylor expansion of J(n,x)
+ * J(n,x) = 1/n!*(x/2)^n - ...
+ */
+ if (n > 33) /* underflow */
+ b = zero;
+ else {
+ temp = x*0.5;
+ b = temp;
+ for (a=one,i=2; i<=n; i++) {
+ a *= (double)i; /* a = n! */
+ b *= temp; /* b = (x/2)^n */
+ }
+ b = b/a;
+ }
+ } else {
+ /* use backward recurrence */
+ /* x x^2 x^2
+ * J(n,x)/J(n-1,x) = ---- ------ ------ .....
+ * 2n - 2(n+1) - 2(n+2)
+ *
+ * 1 1 1
+ * (for large x) = ---- ------ ------ .....
+ * 2n 2(n+1) 2(n+2)
+ * -- - ------ - ------ -
+ * x x x
+ *
+ * Let w = 2n/x and h=2/x, then the above quotient
+ * is equal to the continued fraction:
+ * 1
+ * = -----------------------
+ * 1
+ * w - -----------------
+ * 1
+ * w+h - ---------
+ * w+2h - ...
+ *
+ * To determine how many terms needed, let
+ * Q(0) = w, Q(1) = w(w+h) - 1,
+ * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
+ * When Q(k) > 1e4 good for single
+ * When Q(k) > 1e9 good for double
+ * When Q(k) > 1e17 good for quadruple
+ */
+ /* determine k */
+ double t,v;
+ double q0,q1,h,tmp;
+ int32_t k,m;
+
+ w = (n+n)/(double)x; h = 2.0/(double)x;
+ q0 = w;
+ z = w+h;
+ q1 = w*z - 1.0;
+ k = 1;
+ while (q1 < 1.0e9) {
+ k += 1;
+ z += h;
+ tmp = z*q1 - q0;
+ q0 = q1;
+ q1 = tmp;
+ }
+ m = n+n;
+ for (t=zero, i = 2*(n+k); i>=m; i -= 2)
+ t = one/(i/x-t);
+ a = t;
+ b = one;
+ /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
+ * Hence, if n*(log(2n/x)) > ...
+ * single 8.8722839355e+01
+ * double 7.09782712893383973096e+02
+ * long double 1.1356523406294143949491931077970765006170e+04
+ * then recurrent value may overflow and the result is
+ * likely underflow to zero
+ */
+ tmp = n;
+ v = two/x;
+ tmp = tmp*log(fabs(v*tmp));
+ if (tmp < 7.09782712893383973096e+02) {
+ for (i=n-1,di=(double)(i+i); i>0; i--) {
+ temp = b;
+ b *= di;
+ b = b/x - a;
+ a = temp;
+ di -= two;
+ }
+ } else {
+ for (i=n-1,di=(double)(i+i); i>0; i--) {
+ temp = b;
+ b *= di;
+ b = b/x - a;
+ a = temp;
+ di -= two;
+ /* scale b to avoid spurious overflow */
+ if (b > 1e100) {
+ a /= b;
+ t /= b;
+ b = one;
+ }
+ }
+ }
+ z = j0(x);
+ w = j1(x);
+ if (fabs(z) >= fabs(w))
+ b = t*z/b;
+ else
+ b = t*w/a;
+ }
+ }
+ if (sgn==1) return -b;
+ return b;
+}
+
+
+
+double yn(int n, double x)
+{
+ int32_t i,hx,ix,lx;
+ int32_t sign;
+ double a, b, temp;
+
+ EXTRACT_WORDS(hx, lx, x);
+ ix = 0x7fffffff & hx;
+ /* if Y(n,NaN) is NaN */
+ if ((ix|((uint32_t)(lx|-lx))>>31) > 0x7ff00000)
+ return x+x;
+ if ((ix|lx) == 0)
+ return -one/zero;
+ if (hx < 0)
+ return zero/zero;
+ sign = 1;
+ if (n < 0) {
+ n = -n;
+ sign = 1 - ((n&1)<<1);
+ }
+ if (n == 0)
+ return y0(x);
+ if (n == 1)
+ return sign*y1(x);
+ if (ix == 0x7ff00000)
+ return zero;
+ if (ix >= 0x52D00000) { /* x > 2**302 */
+ /* (x >> n**2)
+ * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ * Let s=sin(x), c=cos(x),
+ * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
+ *
+ * n sin(xn)*sqt2 cos(xn)*sqt2
+ * ----------------------------------
+ * 0 s-c c+s
+ * 1 -s-c -c+s
+ * 2 -s+c -c-s
+ * 3 s+c c-s
+ */
+ switch(n&3) {
+ case 0: temp = sin(x)-cos(x); break;
+ case 1: temp = -sin(x)-cos(x); break;
+ case 2: temp = -sin(x)+cos(x); break;
+ case 3: temp = sin(x)+cos(x); break;
+ }
+ b = invsqrtpi*temp/sqrt(x);
+ } else {
+ uint32_t high;
+ a = y0(x);
+ b = y1(x);
+ /* quit if b is -inf */
+ GET_HIGH_WORD(high, b);
+ for (i=1; i<n && high!=0xfff00000; i++){
+ temp = b;
+ b = ((double)(i+i)/x)*b - a;
+ GET_HIGH_WORD(high, b);
+ a = temp;
+ }
+ }
+ if (sign > 0) return b;
+ return -b;
+}