source: azure_iot_hub/trunk/musl-1.1.18/src/math/jnf.c@ 389

/* origin: FreeBSD /usr/src/lib/msun/src/e_jnf.c */
/*
 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

#define _GNU_SOURCE
#include "libm.h"

float jnf(int n, float x)
{
        uint32_t ix;
        int nm1, sign, i;
        float a, b, temp;

        GET_FLOAT_WORD(ix, x);
        sign = ix>>31;
        ix &= 0x7fffffff;
        if (ix > 0x7f800000) /* nan */
                return x;

        /* J(-n,x) = J(n,-x), use |n|-1 to avoid overflow in -n */
        if (n == 0)
                return j0f(x);
        if (n < 0) {
                nm1 = -(n+1);
                x = -x;
                sign ^= 1;
        } else
                nm1 = n-1;
        if (nm1 == 0)
                return j1f(x);

        sign &= n;  /* even n: 0, odd n: signbit(x) */
        x = fabsf(x);
        if (ix == 0 || ix == 0x7f800000)  /* if x is 0 or inf */
                b = 0.0f;
        else if (nm1 < x) {
                /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
                a = j0f(x);
                b = j1f(x);
                for (i=0; i<nm1; ){
                        i++;
                        temp = b;
                        b = b*(2.0f*i/x) - a;
                        a = temp;
                }
        } else {
                if (ix < 0x35800000) { /* x < 2**-20 */
                        /* x is tiny, return the first Taylor expansion of J(n,x)
                         * J(n,x) = 1/n!*(x/2)^n  - ...
                         */
                        if (nm1 > 8)  /* underflow */
                                nm1 = 8;
                        temp = 0.5f * x;
                        b = temp;
                        a = 1.0f;
                        for (i=2; i<=nm1+1; i++) {
                                a *= (float)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 */
                        float t,q0,q1,w,h,z,tmp,nf;
                        int k;

                        nf = nm1+1.0f;
                        w = 2*nf/x;
                        h = 2/x;
                        z = w+h;
                        q0 = w;
                        q1 = w*z - 1.0f;
                        k = 1;
                        while (q1 < 1.0e4f) {
                                k += 1;
                                z += h;
                                tmp = z*q1 - q0;
                                q0 = q1;
                                q1 = tmp;
                        }
                        for (t=0.0f, i=k; i>=0; i--)
                                t = 1.0f/(2*(i+nf)/x-t);
                        a = t;
                        b = 1.0f;
                        /*  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*logf(fabsf(w));
                        if (tmp < 88.721679688f) {
                                for (i=nm1; i>0; i--) {
                                        temp = b;
                                        b = 2.0f*i*b/x - a;
                                        a = temp;
                                }
                        } else {
                                for (i=nm1; i>0; i--){
                                        temp = b;
                                        b = 2.0f*i*b/x - a;
                                        a = temp;
                                        /* scale b to avoid spurious overflow */
                                        if (b > 0x1p60f) {
                                                a /= b;
                                                t /= b;
                                                b = 1.0f;
                                        }
                                }
                        }
                        z = j0f(x);
                        w = j1f(x);
                        if (fabsf(z) >= fabsf(w))
                                b = t*z/b;
                        else
                                b = t*w/a;
                }
        }
        return sign ? -b : b;
}

float ynf(int n, float x)
{
        uint32_t ix, ib;
        int nm1, sign, i;
        float a, b, temp;

        GET_FLOAT_WORD(ix, x);
        sign = ix>>31;
        ix &= 0x7fffffff;
        if (ix > 0x7f800000) /* nan */
                return x;
        if (sign && ix != 0) /* x < 0 */
                return 0/0.0f;
        if (ix == 0x7f800000)
                return 0.0f;

        if (n == 0)
                return y0f(x);
        if (n < 0) {
                nm1 = -(n+1);
                sign = n&1;
        } else {
                nm1 = n-1;
                sign = 0;
        }
        if (nm1 == 0)
                return sign ? -y1f(x) : y1f(x);

        a = y0f(x);
        b = y1f(x);
        /* quit if b is -inf */
        GET_FLOAT_WORD(ib,b);
        for (i = 0; i < nm1 && ib != 0xff800000; ) {
                i++;
                temp = b;
                b = (2.0f*i/x)*b - a;
                GET_FLOAT_WORD(ib, b);
                a = temp;
        }
        return sign ? -b : b;
}