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+/*
+ * ====================================================
+ * 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.
+ * ====================================================
+ */
+
+/* Modifications for 128-bit long double are
+ Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
+ and are incorporated herein by permission of the author. The author
+ reserves the right to distribute this material elsewhere under different
+ copying permissions. These modifications are distributed here under
+ the following terms:
+
+ This library is free software; you can redistribute it and/or
+ modify it under the terms of the GNU Lesser General Public
+ License as published by the Free Software Foundation; either
+ version 2.1 of the License, or (at your option) any later version.
+
+ This library is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+ Lesser General Public License for more details.
+
+ You should have received a copy of the GNU Lesser General Public
+ License along with this library; if not, write to the Free Software
+ Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */
+
+/*
+ * __ieee754_jn(n, x), __ieee754_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 "quadmath-imp.h"
+
+static const __float128
+ invsqrtpi = 5.6418958354775628694807945156077258584405E-1Q,
+ two = 2.0e0Q,
+ one = 1.0e0Q,
+ zero = 0.0Q;
+
+
+__float128
+jnq (int n, __float128 x)
+{
+ uint32_t se;
+ int32_t i, ix, sgn;
+ __float128 a, b, temp, di;
+ __float128 z, w;
+ ieee854_float128 u;
+
+
+ /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
+ * Thus, J(-n,x) = J(n,-x)
+ */
+
+ u.value = x;
+ se = u.words32.w0;
+ ix = se & 0x7fffffff;
+
+ /* if J(n,NaN) is NaN */
+ if (ix >= 0x7fff0000)
+ {
+ if ((u.words32.w0 & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3)
+ return x + x;
+ }
+
+ if (n < 0)
+ {
+ n = -n;
+ x = -x;
+ se ^= 0x80000000;
+ }
+ if (n == 0)
+ return (j0q (x));
+ if (n == 1)
+ return (j1q (x));
+ sgn = (n & 1) & (se >> 31); /* even n -- 0, odd n -- sign(x) */
+ x = fabsq (x);
+
+ if (x == 0.0Q || ix >= 0x7fff0000) /* if x is 0 or inf */
+ b = zero;
+ else if ((__float128) n <= x)
+ {
+ /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
+ if (ix >= 0x412D0000)
+ { /* x > 2**302 */
+
+ /* ??? Could use an expansion for large x here. */
+
+ /* (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
+ */
+ __float128 s;
+ __float128 c;
+ sincosq (x, &s, &c);
+ switch (n & 3)
+ {
+ case 0:
+ temp = c + s;
+ break;
+ case 1:
+ temp = -c + s;
+ break;
+ case 2:
+ temp = -c - s;
+ break;
+ case 3:
+ temp = c - s;
+ break;
+ }
+ b = invsqrtpi * temp / sqrtq (x);
+ }
+ else
+ {
+ a = j0q (x);
+ b = j1q (x);
+ for (i = 1; i < n; i++)
+ {
+ temp = b;
+ b = b * ((__float128) (i + i) / x) - a; /* avoid underflow */
+ a = temp;
+ }
+ }
+ }
+ else
+ {
+ if (ix < 0x3fc60000)
+ { /* x < 2**-57 */
+ /* x is tiny, return the first Taylor expansion of J(n,x)
+ * J(n,x) = 1/n!*(x/2)^n - ...
+ */
+ if (n >= 400) /* underflow, result < 10^-4952 */
+ b = zero;
+ else
+ {
+ temp = x * 0.5;
+ b = temp;
+ for (a = one, i = 2; i <= n; i++)
+ {
+ a *= (__float128) 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 */
+ __float128 t, v;
+ __float128 q0, q1, h, tmp;
+ int32_t k, m;
+ w = (n + n) / (__float128) x;
+ h = 2.0Q / (__float128) x;
+ q0 = w;
+ z = w + h;
+ q1 = w * z - 1.0Q;
+ k = 1;
+ while (q1 < 1.0e17Q)
+ {
+ 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
+ * __float128 1.1356523406294143949491931077970765006170e+04
+ * then recurrent value may overflow and the result is
+ * likely underflow to zero
+ */
+ tmp = n;
+ v = two / x;
+ tmp = tmp * logq (fabsq (v * tmp));
+
+ if (tmp < 1.1356523406294143949491931077970765006170e+04Q)
+ {
+ for (i = n - 1, di = (__float128) (i + i); i > 0; i--)
+ {
+ temp = b;
+ b *= di;
+ b = b / x - a;
+ a = temp;
+ di -= two;
+ }
+ }
+ else
+ {
+ for (i = n - 1, di = (__float128) (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 > 1e100Q)
+ {
+ a /= b;
+ t /= b;
+ b = one;
+ }
+ }
+ }
+ b = (t * j0q (x) / b);
+ }
+ }
+ if (sgn == 1)
+ return -b;
+ else
+ return b;
+}
+
+__float128
+ynq (int n, __float128 x)
+{
+ uint32_t se;
+ int32_t i, ix;
+ int32_t sign;
+ __float128 a, b, temp;
+ ieee854_float128 u;
+
+ u.value = x;
+ se = u.words32.w0;
+ ix = se & 0x7fffffff;
+
+ /* if Y(n,NaN) is NaN */
+ if (ix >= 0x7fff0000)
+ {
+ if ((u.words32.w0 & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3)
+ return x + x;
+ }
+ if (x <= 0.0Q)
+ {
+ if (x == 0.0Q)
+ return -HUGE_VALQ + x;
+ if (se & 0x80000000)
+ return zero / (zero * x);
+ }
+ sign = 1;
+ if (n < 0)
+ {
+ n = -n;
+ sign = 1 - ((n & 1) << 1);
+ }
+ if (n == 0)
+ return (y0q (x));
+ if (n == 1)
+ return (sign * y1q (x));
+ if (ix >= 0x7fff0000)
+ return zero;
+ if (ix >= 0x412D0000)
+ { /* x > 2**302 */
+
+ /* ??? See comment above on the possible futility of this. */
+
+ /* (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
+ */
+ __float128 s;
+ __float128 c;
+ sincosq (x, &s, &c);
+ switch (n & 3)
+ {
+ case 0:
+ temp = s - c;
+ break;
+ case 1:
+ temp = -s - c;
+ break;
+ case 2:
+ temp = -s + c;
+ break;
+ case 3:
+ temp = s + c;
+ break;
+ }
+ b = invsqrtpi * temp / sqrtq (x);
+ }
+ else
+ {
+ a = y0q (x);
+ b = y1q (x);
+ /* quit if b is -inf */
+ u.value = b;
+ se = u.words32.w0 & 0xffff0000;
+ for (i = 1; i < n && se != 0xffff0000; i++)
+ {
+ temp = b;
+ b = ((__float128) (i + i) / x) * b - a;
+ u.value = b;
+ se = u.words32.w0 & 0xffff0000;
+ a = temp;
+ }
+ }
+ if (sign > 0)
+ return b;
+ else
+ return -b;
+}