first commit of the new libm!

thanks to the hard work of Szabolcs Nagy (nsz), identifying the best
(from correctness and license standpoint) implementations from freebsd
and openbsd and cleaning them up! musl should now fully support c99
float and long double math functions, and has near-complete complex
math support. tgmath should also work (fully on gcc-compatible
compilers, and mostly on any c99 compiler).

based largely on commit 0376d44a890fea261506f1fc63833e7a686dca19 from
nsz's libm git repo, with some additions (dummy versions of a few
missing long double complex functions, etc.) by me.

various cleanups still need to be made, including re-adding (if
they're correct) some asm functions that were dropped.

1 files changed, 282 insertions, 0 deletions

diff --git a/src/math/jn.c b/src/math/jn.c
new file mode 100644
index 00000000..082a17bc
--- /dev/null
+++ b/src/math/jn.c
@@ -0,0 +1,282 @@
+/* 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;
+}