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

fusl/src/math/jn.c

+/* 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;
+}