2011-10-12 15:27:51 +00:00
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/* Optimized by Ulrich Drepper <drepper@gmail.com>, 2011 */
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1996-03-05 21:41:30 +00:00
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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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2011-10-12 15:27:51 +00:00
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* software is freely granted, provided that this notice
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1996-03-05 21:41:30 +00:00
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* is preserved.
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* ====================================================
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*/
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/* __ieee754_cosh(x)
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2011-10-12 15:27:51 +00:00
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* Method :
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1996-03-05 21:41:30 +00:00
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* mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2
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2011-10-12 15:27:51 +00:00
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* 1. Replace x by |x| (cosh(x) = cosh(-x)).
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* 2.
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* [ exp(x) - 1 ]^2
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1996-03-05 21:41:30 +00:00
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* 0 <= x <= ln2/2 : cosh(x) := 1 + -------------------
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2011-10-12 15:27:51 +00:00
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* 2*exp(x)
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1996-03-05 21:41:30 +00:00
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*
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2011-10-12 15:27:51 +00:00
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* exp(x) + 1/exp(x)
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1996-03-05 21:41:30 +00:00
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* ln2/2 <= x <= 22 : cosh(x) := -------------------
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2011-10-12 15:27:51 +00:00
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* 2
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* 22 <= x <= lnovft : cosh(x) := exp(x)/2
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1996-03-05 21:41:30 +00:00
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* lnovft <= x <= ln2ovft: cosh(x) := exp(x/2)/2 * exp(x/2)
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* ln2ovft < x : cosh(x) := huge*huge (overflow)
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*
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* Special cases:
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* cosh(x) is |x| if x is +INF, -INF, or NaN.
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* only cosh(0)=1 is exact for finite x.
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*/
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2012-03-09 19:29:16 +00:00
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#include <math.h>
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#include <math_private.h>
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1996-03-05 21:41:30 +00:00
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2013-10-17 14:03:24 +00:00
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static const double one = 1.0, half = 0.5, huge = 1.0e300;
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1996-03-05 21:41:30 +00:00
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2011-10-12 15:27:51 +00:00
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double
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__ieee754_cosh (double x)
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{
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2013-10-17 14:03:24 +00:00
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double t, w;
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int32_t ix;
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u_int32_t lx;
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1996-03-05 21:41:30 +00:00
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2013-10-17 14:03:24 +00:00
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/* High word of |x|. */
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GET_HIGH_WORD (ix, x);
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ix &= 0x7fffffff;
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1996-03-05 21:41:30 +00:00
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2013-10-17 14:03:24 +00:00
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/* |x| in [0,22] */
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if (ix < 0x40360000)
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{
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/* |x| in [0,0.5*ln2], return 1+expm1(|x|)^2/(2*exp(|x|)) */
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if (ix < 0x3fd62e43)
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{
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Fix cosh spurious underflows from expm1 (bug 16354), inaccurate results near 0 (bug 17061).
This patch fixes bug 16354, spurious underflows from cosh when a tiny
argument is passed to expm1 and expm1 correctly underflows although
the final result of cosh should be 1. As noted in that bug, some
cases are latent because of expm1 implementations not raising
underflow (bug 16353), but all the implementations are fixed
similarly. They already contained checks for tiny arguments, but the
checks were too late to avoid underflow from expm1 (although they
would avoid underflow from subsequent squaring of the result of
expm1); they are moved before the expm1 calls.
The thresholds used for considering arguments tiny are not
particularly consistent in how they relate to the precision of the
floating-point format in question. They are, however, all sufficient
to ensure that the round-to-nearest result of cosh is indeed 1 below
the threshold (although sometimes they are smaller than necessary).
But the previous logic did not return 1, but the previously computed 1
+ expm1(abs(x)) value. And the thresholds in the ldbl-128 and
ldbl-128ibm code (0x1p-71L - I suspect 0x3f8b was intended in the code
instead of 0x3fb8 - and (roughly) 0x1p-55L) are not sufficient for
that value to be 1. So by moving the test for tiny arguments, and
consequently returning 1 directly now the expm1 value hasn't been
computed by that point, this patch also fixes bug 17061, the (large
number of ulps) inaccuracy for small arguments in those
implementations. Tests for that bug are duly added.
Tested x86_64 and x86 and ulps updated accordingly. Also tested for
mips64 and powerpc32 to validate the ldbl-128 and ldbl-128ibm changes.
[BZ #16354]
[BZ #17061]
* sysdeps/ieee754/dbl-64/e_cosh.c (__ieee754_cosh): Check for
small arguments before calling __expm1.
* sysdeps/ieee754/flt-32/e_coshf.c (__ieee754_coshf): Check for
small arguments before calling __expm1f.
* sysdeps/ieee754/ldbl-128/e_coshl.c (__ieee754_coshl): Check for
small arguments before calling __expm1l.
* sysdeps/ieee754/ldbl-128ibm/e_coshl.c (__ieee754_coshl):
Likewise.
* sysdeps/ieee754/ldbl-96/e_coshl.c (__ieee754_coshl): Likewise.
* math/auto-libm-test-in: Add more cosh tests. Do not allow
spurious underflow for some cosh tests.
* math/auto-libm-test-out: Regenerated.
* sysdeps/i386/fpu/libm-test-ulps: Update.
2014-06-23 20:20:10 +00:00
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if (ix < 0x3c800000)
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return one; /* cosh(tiny) = 1 */
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2013-10-17 14:03:24 +00:00
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t = __expm1 (fabs (x));
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w = one + t;
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return one + (t * t) / (w + w);
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1996-03-05 21:41:30 +00:00
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}
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2013-10-17 14:03:24 +00:00
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/* |x| in [0.5*ln2,22], return (exp(|x|)+1/exp(|x|)/2; */
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t = __ieee754_exp (fabs (x));
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return half * t + half / t;
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}
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1996-03-05 21:41:30 +00:00
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2013-10-17 14:03:24 +00:00
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/* |x| in [22, log(maxdouble)] return half*exp(|x|) */
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if (ix < 0x40862e42)
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return half * __ieee754_exp (fabs (x));
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/* |x| in [log(maxdouble), overflowthresold] */
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GET_LOW_WORD (lx, x);
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if (ix < 0x408633ce || ((ix == 0x408633ce) && (lx <= (u_int32_t) 0x8fb9f87d)))
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{
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w = __ieee754_exp (half * fabs (x));
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t = half * w;
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return t * w;
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}
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1996-03-05 21:41:30 +00:00
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2013-10-17 14:03:24 +00:00
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/* x is INF or NaN */
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if (ix >= 0x7ff00000)
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return x * x;
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2011-10-12 15:27:51 +00:00
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2013-10-17 14:03:24 +00:00
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/* |x| > overflowthresold, cosh(x) overflow */
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return huge * huge;
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1996-03-05 21:41:30 +00:00
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}
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2011-10-12 15:27:51 +00:00
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strong_alias (__ieee754_cosh, __cosh_finite)
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