Add a "fork" of musl as //fusl.

fusl/src/math/jn.c

Index: fusl/src/math/jn.c
diff --git a/fusl/src/math/jn.c b/fusl/src/math/jn.c
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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 is 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 */
+
+double jn(int n, double x)
+{
+	uint32_t ix, lx;
+	int nm1, i, sign;
+	double a, b, temp;
+
+	EXTRACT_WORDS(ix, lx, x);
+	sign = ix>>31;
+	ix &= 0x7fffffff;
+
+	if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */
+		return x;
+
+	/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
+	 * Thus, J(-n,x) = J(n,-x)
+	 */
+	/* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */
+	if (n == 0)
+		return j0(x);
+	if (n < 0) {
+		nm1 = -(n+1);
+		x = -x;
+		sign ^= 1;
+	} else
+		nm1 = n-1;
+	if (nm1 == 0)
+		return j1(x);
+
+	sign &= n;  /* even n: 0, odd n: signbit(x) */
+	x = fabs(x);
+	if ((ix|lx) == 0 || ix == 0x7ff00000)  /* if x is 0 or inf */
+		b = 0.0;
+	else if (nm1 < 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(nm1&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;
+			default:
+			case 3: temp =  cos(x)+sin(x); break;
+			}
+			b = invsqrtpi*temp/sqrt(x);
+		} else {
+			a = j0(x);
+			b = j1(x);
+			for (i=0; i<nm1; ) {
+				i++;
+				temp = b;
+				b = b*(2.0*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 (nm1 > 32)  /* underflow */
+				b = 0.0;
+			else {
+				temp = x*0.5;
+				b = temp;
+				a = 1.0;
+				for (i=2; i<=nm1+1; 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,q0,q1,w,h,z,tmp,nf;
+			int k;
+
+			nf = nm1 + 1.0;
+			w = 2*nf/x;
+			h = 2/x;
+			z = w+h;
+			q0 = w;
+			q1 = w*z - 1.0;
+			k = 1;
+			while (q1 < 1.0e9) {
+				k += 1;
+				z += h;
+				tmp = z*q1 - q0;
+				q0 = q1;
+				q1 = tmp;
+			}
+			for (t=0.0, i=k; i>=0; i--)
+				t = 1/(2*(i+nf)/x - t);
+			a = t;
+			b = 1.0;
+			/*  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 = nf*log(fabs(w));
+			if (tmp < 7.09782712893383973096e+02) {
+				for (i=nm1; i>0; i--) {
+					temp = b;
+					b = b*(2.0*i)/x - a;
+					a = temp;
+				}
+			} else {
+				for (i=nm1; i>0; i--) {
+					temp = b;
+					b = b*(2.0*i)/x - a;
+					a = temp;
+					/* scale b to avoid spurious overflow */
+					if (b > 0x1p500) {
+						a /= b;
+						t /= b;
+						b  = 1.0;
+					}
+				}
+			}
+			z = j0(x);
+			w = j1(x);
+			if (fabs(z) >= fabs(w))
+				b = t*z/b;
+			else
+				b = t*w/a;
+		}
+	}
+	return sign ? -b : b;
+}
+
+
+double yn(int n, double x)
+{
+	uint32_t ix, lx, ib;
+	int nm1, sign, i;
+	double a, b, temp;
+
+	EXTRACT_WORDS(ix, lx, x);
+	sign = ix>>31;
+	ix &= 0x7fffffff;
+
+	if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */
+		return x;
+	if (sign && (ix|lx)!=0) /* x < 0 */
+		return 0/0.0;
+	if (ix == 0x7ff00000)
+		return 0.0;
+
+	if (n == 0)
+		return y0(x);
+	if (n < 0) {
+		nm1 = -(n+1);
+		sign = n&1;
+	} else {
+		nm1 = n-1;
+		sign = 0;
+	}
+	if (nm1 == 0)
+		return sign ? -y1(x) : y1(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(nm1&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;
+		default:
+		case 3: temp =  sin(x)-cos(x); break;
+		}
+		b = invsqrtpi*temp/sqrt(x);
+	} else {
+		a = y0(x);
+		b = y1(x);
+		/* quit if b is -inf */
+		GET_HIGH_WORD(ib, b);
+		for (i=0; i<nm1 && ib!=0xfff00000; ){
+			i++;
+			temp = b;
+			b = (2.0*i/x)*b - a;
+			GET_HIGH_WORD(ib, b);
+			a = temp;
+		}
+	}
+	return sign ? -b : b;
+}