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0bf061d3e3
Bug 16516 reports spurious underflows from erf (for all floating-point types), when the result is close to underflowing but does not actually underflow. erf (x) is about (2/sqrt(pi))*x for x close to 0, so there are subnormal arguments for which it does not underflow. The various implementations do (x + efx*x) (for efx = 2/sqrt(pi) - 1), for greater accuracy than if just using a single multiplication by an approximation to 2/sqrt(pi) (effectively, this way there are a few more bits in the approximation to 2/sqrt(pi)). This can introduce underflows when efx*x underflows even though the final result does not, so a scaled calculation with 8*efx is done in these cases - but 8 is not a big enough scale factor to avoid all such underflows. 16 is (any underflows with a scale factor of 16 would only occur when the final result underflows), so this patch changes the code to use that factor. Rather than recomputing all the values of the efx8 variable, it is removed, leaving it to the compiler's constant folding to compute 16*efx. As such scaling can also lose underflows when the final scaling down happens to be exact, appropriate checks are added to ensure underflow exceptions occur when required in such cases. Tested x86_64 and x86; no ulps updates needed. Also spot-checked for powerpc32 and mips64 to verify the changes to the ldbl-128ibm and ldbl-128 implementations. [BZ #16516] * sysdeps/ieee754/dbl-64/s_erf.c (efx8): Remove variable. (__erf): Scale by 16 instead of 8 in potentially underflowing case. Ensure exception if result actually underflows. * sysdeps/ieee754/flt-32/s_erff.c (efx8): Remove variable. (__erff): Scale by 16 instead of 8 in potentially underflowing case. Ensure exception if result actually underflows. * sysdeps/ieee754/ldbl-128/s_erfl.c: Include <float.h>. (efx8): Remove variable. (__erfl): Scale by 16 instead of 8 in potentially underflowing case. Ensure exception if result actually underflows. * sysdeps/ieee754/ldbl-128ibm/s_erfl.c: Include <float.h>. (efx8): Remove variable. (__erfl): Scale by 16 instead of 8 in potentially underflowing case. Ensure exception if result actually underflows. * sysdeps/ieee754/ldbl-96/s_erfl.c: Include <float.h>. (efx8): Remove variable. (__erfl): Scale by 16 instead of 8 in potentially underflowing case. Ensure exception if result actually underflows. * math/auto-libm-test-in: Add more tests of erf. * math/auto-libm-test-out: Regenerated.
432 lines
14 KiB
C
432 lines
14 KiB
C
/* @(#)s_erf.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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/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
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for performance improvement on pipelined processors.
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*/
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#if defined(LIBM_SCCS) && !defined(lint)
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static char rcsid[] = "$NetBSD: s_erf.c,v 1.8 1995/05/10 20:47:05 jtc Exp $";
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#endif
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/* double erf(double x)
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* double erfc(double x)
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* x
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* 2 |\
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* erf(x) = --------- | exp(-t*t)dt
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* sqrt(pi) \|
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* 0
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*
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* erfc(x) = 1-erf(x)
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* Note that
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* erf(-x) = -erf(x)
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* erfc(-x) = 2 - erfc(x)
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*
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* Method:
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* 1. For |x| in [0, 0.84375]
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* erf(x) = x + x*R(x^2)
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* erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
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* = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
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* where R = P/Q where P is an odd poly of degree 8 and
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* Q is an odd poly of degree 10.
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* -57.90
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* | R - (erf(x)-x)/x | <= 2
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*
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*
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* Remark. The formula is derived by noting
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* erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
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* and that
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* 2/sqrt(pi) = 1.128379167095512573896158903121545171688
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* is close to one. The interval is chosen because the fix
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* point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
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* near 0.6174), and by some experiment, 0.84375 is chosen to
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* guarantee the error is less than one ulp for erf.
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*
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* 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
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* c = 0.84506291151 rounded to single (24 bits)
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* erf(x) = sign(x) * (c + P1(s)/Q1(s))
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* erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
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* 1+(c+P1(s)/Q1(s)) if x < 0
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* |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
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* Remark: here we use the taylor series expansion at x=1.
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* erf(1+s) = erf(1) + s*Poly(s)
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* = 0.845.. + P1(s)/Q1(s)
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* That is, we use rational approximation to approximate
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* erf(1+s) - (c = (single)0.84506291151)
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* Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
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* where
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* P1(s) = degree 6 poly in s
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* Q1(s) = degree 6 poly in s
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*
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* 3. For x in [1.25,1/0.35(~2.857143)],
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* erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
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* erf(x) = 1 - erfc(x)
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* where
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* R1(z) = degree 7 poly in z, (z=1/x^2)
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* S1(z) = degree 8 poly in z
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*
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* 4. For x in [1/0.35,28]
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* erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
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* = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
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* = 2.0 - tiny (if x <= -6)
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* erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
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* erf(x) = sign(x)*(1.0 - tiny)
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* where
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* R2(z) = degree 6 poly in z, (z=1/x^2)
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* S2(z) = degree 7 poly in z
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*
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* Note1:
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* To compute exp(-x*x-0.5625+R/S), let s be a single
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* precision number and s := x; then
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* -x*x = -s*s + (s-x)*(s+x)
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* exp(-x*x-0.5626+R/S) =
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* exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
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* Note2:
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* Here 4 and 5 make use of the asymptotic series
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* exp(-x*x)
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* erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
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* x*sqrt(pi)
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* We use rational approximation to approximate
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* g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
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* Here is the error bound for R1/S1 and R2/S2
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* |R1/S1 - f(x)| < 2**(-62.57)
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* |R2/S2 - f(x)| < 2**(-61.52)
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*
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* 5. For inf > x >= 28
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* erf(x) = sign(x) *(1 - tiny) (raise inexact)
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* erfc(x) = tiny*tiny (raise underflow) if x > 0
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* = 2 - tiny if x<0
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*
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* 7. Special case:
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* erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
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* erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
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* erfc/erf(NaN) is NaN
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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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tiny = 1e-300,
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half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
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one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
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two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
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/* c = (float)0.84506291151 */
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erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
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/*
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* Coefficients for approximation to erf on [0,0.84375]
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*/
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efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
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pp[] = { 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
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-3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
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-2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
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-5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
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-2.37630166566501626084e-05 }, /* 0xBEF8EAD6, 0x120016AC */
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qq[] = { 0.0, 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
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6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
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5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
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1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
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-3.96022827877536812320e-06 }, /* 0xBED09C43, 0x42A26120 */
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/*
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* Coefficients for approximation to erf in [0.84375,1.25]
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*/
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pa[] = { -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
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4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
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-3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
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3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
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-1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
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3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
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-2.16637559486879084300e-03 }, /* 0xBF61BF38, 0x0A96073F */
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qa[] = { 0.0, 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
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5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
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7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
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1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
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1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
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1.19844998467991074170e-02 }, /* 0x3F888B54, 0x5735151D */
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/*
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* Coefficients for approximation to erfc in [1.25,1/0.35]
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*/
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ra[] = { -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
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-6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
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-1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
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-6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
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-1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
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-1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
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-8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
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-9.81432934416914548592e+00 }, /* 0xC023A0EF, 0xC69AC25C */
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sa[] = { 0.0, 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
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1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
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4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
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6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
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4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
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1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
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6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
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-6.04244152148580987438e-02 }, /* 0xBFAEEFF2, 0xEE749A62 */
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/*
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* Coefficients for approximation to erfc in [1/.35,28]
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*/
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rb[] = { -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
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-7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
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-1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
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-1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
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-6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
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-1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
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-4.83519191608651397019e+02 }, /* 0xC07E384E, 0x9BDC383F */
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sb[] = { 0.0, 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
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3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
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1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
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3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
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2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
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4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
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-2.24409524465858183362e+01 }; /* 0xC03670E2, 0x42712D62 */
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double
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__erf (double x)
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{
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int32_t hx, ix, i;
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double R, S, P, Q, s, y, z, r;
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GET_HIGH_WORD (hx, x);
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ix = hx & 0x7fffffff;
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if (ix >= 0x7ff00000) /* erf(nan)=nan */
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{
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i = ((u_int32_t) hx >> 31) << 1;
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return (double) (1 - i) + one / x; /* erf(+-inf)=+-1 */
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}
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if (ix < 0x3feb0000) /* |x|<0.84375 */
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{
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double r1, r2, s1, s2, s3, z2, z4;
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if (ix < 0x3e300000) /* |x|<2**-28 */
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{
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if (ix < 0x00800000)
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{
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/* Avoid spurious underflow. */
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double ret = 0.0625 * (16.0 * x + (16.0 * efx) * x);
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if (fabs (ret) < DBL_MIN)
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{
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double force_underflow = ret * ret;
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math_force_eval (force_underflow);
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}
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return ret;
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}
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return x + efx * x;
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}
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z = x * x;
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r1 = pp[0] + z * pp[1]; z2 = z * z;
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r2 = pp[2] + z * pp[3]; z4 = z2 * z2;
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s1 = one + z * qq[1];
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s2 = qq[2] + z * qq[3];
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s3 = qq[4] + z * qq[5];
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r = r1 + z2 * r2 + z4 * pp[4];
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s = s1 + z2 * s2 + z4 * s3;
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y = r / s;
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return x + x * y;
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}
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if (ix < 0x3ff40000) /* 0.84375 <= |x| < 1.25 */
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{
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double s2, s4, s6, P1, P2, P3, P4, Q1, Q2, Q3, Q4;
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s = fabs (x) - one;
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P1 = pa[0] + s * pa[1]; s2 = s * s;
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Q1 = one + s * qa[1]; s4 = s2 * s2;
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P2 = pa[2] + s * pa[3]; s6 = s4 * s2;
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Q2 = qa[2] + s * qa[3];
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P3 = pa[4] + s * pa[5];
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Q3 = qa[4] + s * qa[5];
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P4 = pa[6];
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Q4 = qa[6];
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P = P1 + s2 * P2 + s4 * P3 + s6 * P4;
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Q = Q1 + s2 * Q2 + s4 * Q3 + s6 * Q4;
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if (hx >= 0)
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return erx + P / Q;
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else
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return -erx - P / Q;
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}
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if (ix >= 0x40180000) /* inf>|x|>=6 */
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{
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if (hx >= 0)
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return one - tiny;
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else
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return tiny - one;
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}
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x = fabs (x);
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s = one / (x * x);
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if (ix < 0x4006DB6E) /* |x| < 1/0.35 */
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{
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double R1, R2, R3, R4, S1, S2, S3, S4, s2, s4, s6, s8;
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R1 = ra[0] + s * ra[1]; s2 = s * s;
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S1 = one + s * sa[1]; s4 = s2 * s2;
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R2 = ra[2] + s * ra[3]; s6 = s4 * s2;
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S2 = sa[2] + s * sa[3]; s8 = s4 * s4;
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R3 = ra[4] + s * ra[5];
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S3 = sa[4] + s * sa[5];
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R4 = ra[6] + s * ra[7];
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S4 = sa[6] + s * sa[7];
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R = R1 + s2 * R2 + s4 * R3 + s6 * R4;
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S = S1 + s2 * S2 + s4 * S3 + s6 * S4 + s8 * sa[8];
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}
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else /* |x| >= 1/0.35 */
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{
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double R1, R2, R3, S1, S2, S3, S4, s2, s4, s6;
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R1 = rb[0] + s * rb[1]; s2 = s * s;
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S1 = one + s * sb[1]; s4 = s2 * s2;
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R2 = rb[2] + s * rb[3]; s6 = s4 * s2;
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S2 = sb[2] + s * sb[3];
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R3 = rb[4] + s * rb[5];
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S3 = sb[4] + s * sb[5];
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S4 = sb[6] + s * sb[7];
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R = R1 + s2 * R2 + s4 * R3 + s6 * rb[6];
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S = S1 + s2 * S2 + s4 * S3 + s6 * S4;
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}
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z = x;
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SET_LOW_WORD (z, 0);
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r = __ieee754_exp (-z * z - 0.5625) *
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__ieee754_exp ((z - x) * (z + x) + R / S);
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if (hx >= 0)
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return one - r / x;
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else
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return r / x - one;
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}
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weak_alias (__erf, erf)
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#ifdef NO_LONG_DOUBLE
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strong_alias (__erf, __erfl)
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weak_alias (__erf, erfl)
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#endif
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double
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__erfc (double x)
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{
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int32_t hx, ix;
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double R, S, P, Q, s, y, z, r;
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GET_HIGH_WORD (hx, x);
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ix = hx & 0x7fffffff;
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if (ix >= 0x7ff00000) /* erfc(nan)=nan */
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{ /* erfc(+-inf)=0,2 */
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return (double) (((u_int32_t) hx >> 31) << 1) + one / x;
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}
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if (ix < 0x3feb0000) /* |x|<0.84375 */
|
|
{
|
|
double r1, r2, s1, s2, s3, z2, z4;
|
|
if (ix < 0x3c700000) /* |x|<2**-56 */
|
|
return one - x;
|
|
z = x * x;
|
|
r1 = pp[0] + z * pp[1]; z2 = z * z;
|
|
r2 = pp[2] + z * pp[3]; z4 = z2 * z2;
|
|
s1 = one + z * qq[1];
|
|
s2 = qq[2] + z * qq[3];
|
|
s3 = qq[4] + z * qq[5];
|
|
r = r1 + z2 * r2 + z4 * pp[4];
|
|
s = s1 + z2 * s2 + z4 * s3;
|
|
y = r / s;
|
|
if (hx < 0x3fd00000) /* x<1/4 */
|
|
{
|
|
return one - (x + x * y);
|
|
}
|
|
else
|
|
{
|
|
r = x * y;
|
|
r += (x - half);
|
|
return half - r;
|
|
}
|
|
}
|
|
if (ix < 0x3ff40000) /* 0.84375 <= |x| < 1.25 */
|
|
{
|
|
double s2, s4, s6, P1, P2, P3, P4, Q1, Q2, Q3, Q4;
|
|
s = fabs (x) - one;
|
|
P1 = pa[0] + s * pa[1]; s2 = s * s;
|
|
Q1 = one + s * qa[1]; s4 = s2 * s2;
|
|
P2 = pa[2] + s * pa[3]; s6 = s4 * s2;
|
|
Q2 = qa[2] + s * qa[3];
|
|
P3 = pa[4] + s * pa[5];
|
|
Q3 = qa[4] + s * qa[5];
|
|
P4 = pa[6];
|
|
Q4 = qa[6];
|
|
P = P1 + s2 * P2 + s4 * P3 + s6 * P4;
|
|
Q = Q1 + s2 * Q2 + s4 * Q3 + s6 * Q4;
|
|
if (hx >= 0)
|
|
{
|
|
z = one - erx; return z - P / Q;
|
|
}
|
|
else
|
|
{
|
|
z = erx + P / Q; return one + z;
|
|
}
|
|
}
|
|
if (ix < 0x403c0000) /* |x|<28 */
|
|
{
|
|
x = fabs (x);
|
|
s = one / (x * x);
|
|
if (ix < 0x4006DB6D) /* |x| < 1/.35 ~ 2.857143*/
|
|
{
|
|
double R1, R2, R3, R4, S1, S2, S3, S4, s2, s4, s6, s8;
|
|
R1 = ra[0] + s * ra[1]; s2 = s * s;
|
|
S1 = one + s * sa[1]; s4 = s2 * s2;
|
|
R2 = ra[2] + s * ra[3]; s6 = s4 * s2;
|
|
S2 = sa[2] + s * sa[3]; s8 = s4 * s4;
|
|
R3 = ra[4] + s * ra[5];
|
|
S3 = sa[4] + s * sa[5];
|
|
R4 = ra[6] + s * ra[7];
|
|
S4 = sa[6] + s * sa[7];
|
|
R = R1 + s2 * R2 + s4 * R3 + s6 * R4;
|
|
S = S1 + s2 * S2 + s4 * S3 + s6 * S4 + s8 * sa[8];
|
|
}
|
|
else /* |x| >= 1/.35 ~ 2.857143 */
|
|
{
|
|
double R1, R2, R3, S1, S2, S3, S4, s2, s4, s6;
|
|
if (hx < 0 && ix >= 0x40180000)
|
|
return two - tiny; /* x < -6 */
|
|
R1 = rb[0] + s * rb[1]; s2 = s * s;
|
|
S1 = one + s * sb[1]; s4 = s2 * s2;
|
|
R2 = rb[2] + s * rb[3]; s6 = s4 * s2;
|
|
S2 = sb[2] + s * sb[3];
|
|
R3 = rb[4] + s * rb[5];
|
|
S3 = sb[4] + s * sb[5];
|
|
S4 = sb[6] + s * sb[7];
|
|
R = R1 + s2 * R2 + s4 * R3 + s6 * rb[6];
|
|
S = S1 + s2 * S2 + s4 * S3 + s6 * S4;
|
|
}
|
|
z = x;
|
|
SET_LOW_WORD (z, 0);
|
|
r = __ieee754_exp (-z * z - 0.5625) *
|
|
__ieee754_exp ((z - x) * (z + x) + R / S);
|
|
if (hx > 0)
|
|
{
|
|
#if FLT_EVAL_METHOD != 0
|
|
volatile
|
|
#endif
|
|
double ret = r / x;
|
|
if (ret == 0)
|
|
__set_errno (ERANGE);
|
|
return ret;
|
|
}
|
|
else
|
|
return two - r / x;
|
|
}
|
|
else
|
|
{
|
|
if (hx > 0)
|
|
{
|
|
__set_errno (ERANGE);
|
|
return tiny * tiny;
|
|
}
|
|
else
|
|
return two - tiny;
|
|
}
|
|
}
|
|
weak_alias (__erfc, erfc)
|
|
#ifdef NO_LONG_DOUBLE
|
|
strong_alias (__erfc, __erfcl)
|
|
weak_alias (__erfc, erfcl)
|
|
#endif
|