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be25493251
This patch fixes bugs 16561 and 16562, bad results of yn in overflow cases in non-default rounding modes, both because an intermediate overflow in the recurrence does not get detected if the result is not an infinity and because an overflowing result may occur in the wrong sign. The fix is to set FE_TONEAREST mode internally for the parts of the function where such overflows can occur (which includes the call to y1 - where yn is used to compute a Bessel function of order -1, negating the result of y1 isn't correct for overflowing results in directed rounding modes) and then compute an overflowing value in the original rounding mode if the to-nearest result was an infinity. Tested x86_64 and x86 and ulps updated accordingly. Also tested for mips64 and powerpc32 to test the ldbl-128 and ldbl-128ibm changes. (The tests for these bugs were added in my previous y1 patch, so the only thing this patch has to do with the testsuite is enable yn testing in all rounding modes.) [BZ #16561] [BZ #16562] * sysdeps/ieee754/dbl-64/e_jn.c: Include <float.h>. (__ieee754_yn): Set FE_TONEAREST mode internally and then recompute overflowing results in original rounding mode. * sysdeps/ieee754/flt-32/e_jnf.c: Include <float.h>. (__ieee754_ynf): Set FE_TONEAREST mode internally and then recompute overflowing results in original rounding mode. * sysdeps/ieee754/ldbl-128/e_jnl.c: Include <float.h>. (__ieee754_ynl): Set FE_TONEAREST mode internally and then recompute overflowing results in original rounding mode. * sysdeps/ieee754/ldbl-128ibm/e_jnl.c: Include <float.h>. (__ieee754_ynl): Set FE_TONEAREST mode internally and then recompute overflowing results in original rounding mode. * sysdeps/ieee754/ldbl-96/e_jnl.c: Include <float.h>. (__ieee754_ynl): Set FE_TONEAREST mode internally and then recompute overflowing results in original rounding mode. * sysdeps/i386/fpu/fenv_private.h [!__SSE2_MATH__] (libc_feholdsetround_ctx): New macro. * math/libm-test.inc (yn_test): Use ALL_RM_TEST. * sysdeps/i386/fpu/libm-test-ulps: Update. * sysdeps/x86_64/fpu/libm-test-ulps : Likewise.
336 lines
8.2 KiB
C
336 lines
8.2 KiB
C
/* @(#)e_jn.c 5.1 93/09/24 */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/*
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* __ieee754_jn(n, x), __ieee754_yn(n, x)
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* floating point Bessel's function of the 1st and 2nd kind
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* of order n
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*
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* Special cases:
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* y0(0)=y1(0)=yn(n,0) = -inf with overflow signal;
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* y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
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* Note 2. About jn(n,x), yn(n,x)
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* For n=0, j0(x) is called,
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* for n=1, j1(x) is called,
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* for n<x, forward recursion us used starting
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* from values of j0(x) and j1(x).
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* for n>x, a continued fraction approximation to
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* j(n,x)/j(n-1,x) is evaluated and then backward
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* recursion is used starting from a supposed value
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* for j(n,x). The resulting value of j(0,x) is
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* compared with the actual value to correct the
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* supposed value of j(n,x).
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*
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* yn(n,x) is similar in all respects, except
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* that forward recursion is used for all
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* values of n>1.
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*
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*/
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#include <errno.h>
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#include <float.h>
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#include <math.h>
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#include <math_private.h>
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static const double
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invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
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two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
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one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
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static const double zero = 0.00000000000000000000e+00;
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double
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__ieee754_jn (int n, double x)
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{
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int32_t i, hx, ix, lx, sgn;
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double a, b, temp, di;
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double z, w;
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/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
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* Thus, J(-n,x) = J(n,-x)
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*/
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EXTRACT_WORDS (hx, lx, x);
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ix = 0x7fffffff & hx;
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/* if J(n,NaN) is NaN */
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if (__glibc_unlikely ((ix | ((u_int32_t) (lx | -lx)) >> 31) > 0x7ff00000))
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return x + x;
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if (n < 0)
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{
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n = -n;
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x = -x;
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hx ^= 0x80000000;
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}
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if (n == 0)
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return (__ieee754_j0 (x));
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if (n == 1)
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return (__ieee754_j1 (x));
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sgn = (n & 1) & (hx >> 31); /* even n -- 0, odd n -- sign(x) */
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x = fabs (x);
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if (__glibc_unlikely ((ix | lx) == 0 || ix >= 0x7ff00000))
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/* if x is 0 or inf */
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b = zero;
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else if ((double) n <= x)
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{
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/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
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if (ix >= 0x52D00000) /* x > 2**302 */
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{ /* (x >> n**2)
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* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Let s=sin(x), c=cos(x),
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
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*
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* n sin(xn)*sqt2 cos(xn)*sqt2
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* ----------------------------------
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* 0 s-c c+s
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* 1 -s-c -c+s
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* 2 -s+c -c-s
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* 3 s+c c-s
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*/
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double s;
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double c;
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__sincos (x, &s, &c);
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switch (n & 3)
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{
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case 0: temp = c + s; break;
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case 1: temp = -c + s; break;
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case 2: temp = -c - s; break;
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case 3: temp = c - s; break;
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}
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b = invsqrtpi * temp / __ieee754_sqrt (x);
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}
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else
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{
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a = __ieee754_j0 (x);
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b = __ieee754_j1 (x);
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for (i = 1; i < n; i++)
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{
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temp = b;
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b = b * ((double) (i + i) / x) - a; /* avoid underflow */
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a = temp;
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}
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}
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}
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else
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{
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if (ix < 0x3e100000) /* x < 2**-29 */
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{ /* x is tiny, return the first Taylor expansion of J(n,x)
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* J(n,x) = 1/n!*(x/2)^n - ...
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*/
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if (n > 33) /* underflow */
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b = zero;
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else
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{
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temp = x * 0.5; b = temp;
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for (a = one, i = 2; i <= n; i++)
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{
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a *= (double) i; /* a = n! */
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b *= temp; /* b = (x/2)^n */
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}
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b = b / a;
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}
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}
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else
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{
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/* use backward recurrence */
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/* x x^2 x^2
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* J(n,x)/J(n-1,x) = ---- ------ ------ .....
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* 2n - 2(n+1) - 2(n+2)
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*
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* 1 1 1
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* (for large x) = ---- ------ ------ .....
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* 2n 2(n+1) 2(n+2)
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* -- - ------ - ------ -
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* x x x
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*
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* Let w = 2n/x and h=2/x, then the above quotient
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* is equal to the continued fraction:
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* 1
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* = -----------------------
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* 1
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* w - -----------------
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* 1
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* w+h - ---------
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* w+2h - ...
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*
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* To determine how many terms needed, let
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* Q(0) = w, Q(1) = w(w+h) - 1,
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* Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
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* When Q(k) > 1e4 good for single
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* When Q(k) > 1e9 good for double
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* When Q(k) > 1e17 good for quadruple
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*/
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/* determine k */
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double t, v;
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double q0, q1, h, tmp; int32_t k, m;
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w = (n + n) / (double) x; h = 2.0 / (double) x;
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q0 = w; z = w + h; q1 = w * z - 1.0; k = 1;
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while (q1 < 1.0e9)
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{
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k += 1; z += h;
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tmp = z * q1 - q0;
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q0 = q1;
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q1 = tmp;
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}
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m = n + n;
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for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
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t = one / (i / x - t);
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a = t;
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b = one;
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/* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
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* Hence, if n*(log(2n/x)) > ...
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* single 8.8722839355e+01
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* double 7.09782712893383973096e+02
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* long double 1.1356523406294143949491931077970765006170e+04
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* then recurrent value may overflow and the result is
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* likely underflow to zero
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*/
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tmp = n;
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v = two / x;
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tmp = tmp * __ieee754_log (fabs (v * tmp));
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if (tmp < 7.09782712893383973096e+02)
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{
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for (i = n - 1, di = (double) (i + i); i > 0; i--)
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{
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temp = b;
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b *= di;
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b = b / x - a;
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a = temp;
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di -= two;
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}
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}
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else
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{
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for (i = n - 1, di = (double) (i + i); i > 0; i--)
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{
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temp = b;
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b *= di;
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b = b / x - a;
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a = temp;
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di -= two;
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/* scale b to avoid spurious overflow */
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if (b > 1e100)
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{
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a /= b;
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t /= b;
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b = one;
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}
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}
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}
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/* j0() and j1() suffer enormous loss of precision at and
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* near zero; however, we know that their zero points never
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* coincide, so just choose the one further away from zero.
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*/
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z = __ieee754_j0 (x);
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w = __ieee754_j1 (x);
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if (fabs (z) >= fabs (w))
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b = (t * z / b);
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else
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b = (t * w / a);
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}
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}
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if (sgn == 1)
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return -b;
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else
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return b;
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}
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strong_alias (__ieee754_jn, __jn_finite)
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double
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__ieee754_yn (int n, double x)
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{
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int32_t i, hx, ix, lx;
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int32_t sign;
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double a, b, temp, ret;
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EXTRACT_WORDS (hx, lx, x);
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ix = 0x7fffffff & hx;
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/* if Y(n,NaN) is NaN */
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if (__glibc_unlikely ((ix | ((u_int32_t) (lx | -lx)) >> 31) > 0x7ff00000))
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return x + x;
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if (__glibc_unlikely ((ix | lx) == 0))
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return -HUGE_VAL + x;
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/* -inf and overflow exception. */;
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if (__glibc_unlikely (hx < 0))
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return zero / (zero * x);
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sign = 1;
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if (n < 0)
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{
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n = -n;
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sign = 1 - ((n & 1) << 1);
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}
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if (n == 0)
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return (__ieee754_y0 (x));
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{
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SET_RESTORE_ROUND (FE_TONEAREST);
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if (n == 1)
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{
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ret = sign * __ieee754_y1 (x);
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goto out;
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}
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if (__glibc_unlikely (ix == 0x7ff00000))
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return zero;
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if (ix >= 0x52D00000) /* x > 2**302 */
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{ /* (x >> n**2)
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* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Let s=sin(x), c=cos(x),
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
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*
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* n sin(xn)*sqt2 cos(xn)*sqt2
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* ----------------------------------
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* 0 s-c c+s
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* 1 -s-c -c+s
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* 2 -s+c -c-s
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* 3 s+c c-s
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*/
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double c;
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double s;
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__sincos (x, &s, &c);
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switch (n & 3)
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{
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case 0: temp = s - c; break;
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case 1: temp = -s - c; break;
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case 2: temp = -s + c; break;
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case 3: temp = s + c; break;
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}
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b = invsqrtpi * temp / __ieee754_sqrt (x);
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}
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else
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{
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u_int32_t high;
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a = __ieee754_y0 (x);
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b = __ieee754_y1 (x);
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/* quit if b is -inf */
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GET_HIGH_WORD (high, b);
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for (i = 1; i < n && high != 0xfff00000; i++)
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{
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temp = b;
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b = ((double) (i + i) / x) * b - a;
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GET_HIGH_WORD (high, b);
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a = temp;
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}
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/* If B is +-Inf, set up errno accordingly. */
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if (!__finite (b))
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__set_errno (ERANGE);
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}
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if (sign > 0)
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ret = b;
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else
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ret = -b;
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}
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out:
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if (__isinf (ret))
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ret = __copysign (DBL_MAX, ret) * DBL_MAX;
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return ret;
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}
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strong_alias (__ieee754_yn, __yn_finite)
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