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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.
456 lines
14 KiB
C
456 lines
14 KiB
C
/*
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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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/* Long double expansions are
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Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
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and are incorporated herein by permission of the author. The author
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reserves the right to distribute this material elsewhere under different
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copying permissions. These modifications are distributed here under
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the following terms:
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This library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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This library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with this library; if not, see
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<http://www.gnu.org/licenses/>. */
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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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* 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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* 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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* Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
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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(z)/S1(z))
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* z=1/x^2
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* erf(x) = 1 - erfc(x)
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*
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* 4. For x in [1/0.35,107]
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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(z)/S2(z))
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* if -6.666<x<0
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* = 2.0 - tiny (if x <= -6.666)
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* z=1/x^2
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* erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6.666, else
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* erf(x) = sign(x)*(1.0 - tiny)
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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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*
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* 5. For inf > x >= 107
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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 long double
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tiny = 1e-4931L,
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half = 0.5L,
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one = 1.0L,
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two = 2.0L,
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/* c = (float)0.84506291151 */
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erx = 0.845062911510467529296875L,
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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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/* 2/sqrt(pi) - 1 */
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efx = 1.2837916709551257389615890312154517168810E-1L,
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pp[6] = {
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1.122751350964552113068262337278335028553E6L,
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-2.808533301997696164408397079650699163276E6L,
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-3.314325479115357458197119660818768924100E5L,
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-6.848684465326256109712135497895525446398E4L,
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-2.657817695110739185591505062971929859314E3L,
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-1.655310302737837556654146291646499062882E2L,
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},
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qq[6] = {
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8.745588372054466262548908189000448124232E6L,
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3.746038264792471129367533128637019611485E6L,
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7.066358783162407559861156173539693900031E5L,
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7.448928604824620999413120955705448117056E4L,
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4.511583986730994111992253980546131408924E3L,
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1.368902937933296323345610240009071254014E2L,
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/* 1.000000000000000000000000000000000000000E0 */
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},
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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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/* erf(x+1) = 0.845062911510467529296875 + pa(x)/qa(x)
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-0.15625 <= x <= +.25
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Peak relative error 8.5e-22 */
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pa[8] = {
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-1.076952146179812072156734957705102256059E0L,
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1.884814957770385593365179835059971587220E2L,
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-5.339153975012804282890066622962070115606E1L,
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4.435910679869176625928504532109635632618E1L,
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1.683219516032328828278557309642929135179E1L,
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-2.360236618396952560064259585299045804293E0L,
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1.852230047861891953244413872297940938041E0L,
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9.394994446747752308256773044667843200719E-2L,
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},
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qa[7] = {
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4.559263722294508998149925774781887811255E2L,
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3.289248982200800575749795055149780689738E2L,
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2.846070965875643009598627918383314457912E2L,
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1.398715859064535039433275722017479994465E2L,
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6.060190733759793706299079050985358190726E1L,
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2.078695677795422351040502569964299664233E1L,
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4.641271134150895940966798357442234498546E0L,
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/* 1.000000000000000000000000000000000000000E0 */
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},
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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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/* erfc(1/x) = x exp (-1/x^2 - 0.5625 + ra(x^2)/sa(x^2))
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1/2.85711669921875 < 1/x < 1/1.25
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Peak relative error 3.1e-21 */
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ra[] = {
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1.363566591833846324191000679620738857234E-1L,
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1.018203167219873573808450274314658434507E1L,
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1.862359362334248675526472871224778045594E2L,
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1.411622588180721285284945138667933330348E3L,
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5.088538459741511988784440103218342840478E3L,
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8.928251553922176506858267311750789273656E3L,
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7.264436000148052545243018622742770549982E3L,
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2.387492459664548651671894725748959751119E3L,
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2.220916652813908085449221282808458466556E2L,
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},
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sa[] = {
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-1.382234625202480685182526402169222331847E1L,
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-3.315638835627950255832519203687435946482E2L,
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-2.949124863912936259747237164260785326692E3L,
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-1.246622099070875940506391433635999693661E4L,
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-2.673079795851665428695842853070996219632E4L,
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-2.880269786660559337358397106518918220991E4L,
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-1.450600228493968044773354186390390823713E4L,
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-2.874539731125893533960680525192064277816E3L,
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-1.402241261419067750237395034116942296027E2L,
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/* 1.000000000000000000000000000000000000000E0 */
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},
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/*
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* Coefficients for approximation to erfc in [1/.35,107]
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*/
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/* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rb(x^2)/sb(x^2))
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1/6.6666259765625 < 1/x < 1/2.85711669921875
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Peak relative error 4.2e-22 */
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rb[] = {
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-4.869587348270494309550558460786501252369E-5L,
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-4.030199390527997378549161722412466959403E-3L,
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-9.434425866377037610206443566288917589122E-2L,
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-9.319032754357658601200655161585539404155E-1L,
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-4.273788174307459947350256581445442062291E0L,
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-8.842289940696150508373541814064198259278E0L,
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-7.069215249419887403187988144752613025255E0L,
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-1.401228723639514787920274427443330704764E0L,
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},
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sb[] = {
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4.936254964107175160157544545879293019085E-3L,
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1.583457624037795744377163924895349412015E-1L,
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1.850647991850328356622940552450636420484E0L,
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9.927611557279019463768050710008450625415E0L,
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2.531667257649436709617165336779212114570E1L,
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2.869752886406743386458304052862814690045E1L,
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1.182059497870819562441683560749192539345E1L,
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/* 1.000000000000000000000000000000000000000E0 */
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},
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/* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rc(x^2)/sc(x^2))
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1/107 <= 1/x <= 1/6.6666259765625
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Peak relative error 1.1e-21 */
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rc[] = {
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-8.299617545269701963973537248996670806850E-5L,
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-6.243845685115818513578933902532056244108E-3L,
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-1.141667210620380223113693474478394397230E-1L,
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-7.521343797212024245375240432734425789409E-1L,
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-1.765321928311155824664963633786967602934E0L,
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-1.029403473103215800456761180695263439188E0L,
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},
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sc[] = {
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8.413244363014929493035952542677768808601E-3L,
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2.065114333816877479753334599639158060979E-1L,
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1.639064941530797583766364412782135680148E0L,
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4.936788463787115555582319302981666347450E0L,
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5.005177727208955487404729933261347679090E0L,
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/* 1.000000000000000000000000000000000000000E0 */
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};
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long double
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__erfl (long double x)
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{
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long double R, S, P, Q, s, y, z, r;
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int32_t ix, i;
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u_int32_t se, i0, i1;
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GET_LDOUBLE_WORDS (se, i0, i1, x);
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ix = se & 0x7fff;
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if (ix >= 0x7fff)
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{ /* erf(nan)=nan */
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i = ((se & 0xffff) >> 15) << 1;
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return (long double) (1 - i) + one / x; /* erf(+-inf)=+-1 */
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}
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ix = (ix << 16) | (i0 >> 16);
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if (ix < 0x3ffed800) /* |x|<0.84375 */
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{
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if (ix < 0x3fde8000) /* |x|<2**-33 */
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{
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if (ix < 0x00080000)
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{
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/* Avoid spurious underflow. */
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long double ret = 0.0625 * (16.0 * x + (16.0 * efx) * x);
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if (fabsl (ret) < LDBL_MIN)
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{
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long 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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r = pp[0] + z * (pp[1]
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+ z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
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s = qq[0] + z * (qq[1]
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+ z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
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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 < 0x3fffa000) /* 1.25 */
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{ /* 0.84375 <= |x| < 1.25 */
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s = fabsl (x) - one;
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P = pa[0] + s * (pa[1] + s * (pa[2]
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+ s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
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Q = qa[0] + s * (qa[1] + s * (qa[2]
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+ s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
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if ((se & 0x8000) == 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 >= 0x4001d555) /* 6.6666259765625 */
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{ /* inf>|x|>=6.666 */
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if ((se & 0x8000) == 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 = fabsl (x);
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s = one / (x * x);
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if (ix < 0x4000b6db) /* 2.85711669921875 */
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{
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R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
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s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
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S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
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s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
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}
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else
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{ /* |x| >= 1/0.35 */
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R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
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s * (rb[5] + s * (rb[6] + s * rb[7]))))));
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S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
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s * (sb[5] + s * (sb[6] + s))))));
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}
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z = x;
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GET_LDOUBLE_WORDS (i, i0, i1, z);
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i1 = 0;
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SET_LDOUBLE_WORDS (z, i, i0, i1);
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r =
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__ieee754_expl (-z * z - 0.5625) * __ieee754_expl ((z - x) * (z + x) +
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R / S);
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if ((se & 0x8000) == 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 (__erfl, erfl)
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long double
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__erfcl (long double x)
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{
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int32_t hx, ix;
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long double R, S, P, Q, s, y, z, r;
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u_int32_t se, i0, i1;
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|
|
GET_LDOUBLE_WORDS (se, i0, i1, x);
|
|
ix = se & 0x7fff;
|
|
if (ix >= 0x7fff)
|
|
{ /* erfc(nan)=nan */
|
|
/* erfc(+-inf)=0,2 */
|
|
return (long double) (((se & 0xffff) >> 15) << 1) + one / x;
|
|
}
|
|
|
|
ix = (ix << 16) | (i0 >> 16);
|
|
if (ix < 0x3ffed800) /* |x|<0.84375 */
|
|
{
|
|
if (ix < 0x3fbe0000) /* |x|<2**-65 */
|
|
return one - x;
|
|
z = x * x;
|
|
r = pp[0] + z * (pp[1]
|
|
+ z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
|
|
s = qq[0] + z * (qq[1]
|
|
+ z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
|
|
y = r / s;
|
|
if (ix < 0x3ffd8000) /* x<1/4 */
|
|
{
|
|
return one - (x + x * y);
|
|
}
|
|
else
|
|
{
|
|
r = x * y;
|
|
r += (x - half);
|
|
return half - r;
|
|
}
|
|
}
|
|
if (ix < 0x3fffa000) /* 1.25 */
|
|
{ /* 0.84375 <= |x| < 1.25 */
|
|
s = fabsl (x) - one;
|
|
P = pa[0] + s * (pa[1] + s * (pa[2]
|
|
+ s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
|
|
Q = qa[0] + s * (qa[1] + s * (qa[2]
|
|
+ s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
|
|
if ((se & 0x8000) == 0)
|
|
{
|
|
z = one - erx;
|
|
return z - P / Q;
|
|
}
|
|
else
|
|
{
|
|
z = erx + P / Q;
|
|
return one + z;
|
|
}
|
|
}
|
|
if (ix < 0x4005d600) /* 107 */
|
|
{ /* |x|<107 */
|
|
x = fabsl (x);
|
|
s = one / (x * x);
|
|
if (ix < 0x4000b6db) /* 2.85711669921875 */
|
|
{ /* |x| < 1/.35 ~ 2.857143 */
|
|
R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
|
|
s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
|
|
S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
|
|
s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
|
|
}
|
|
else if (ix < 0x4001d555) /* 6.6666259765625 */
|
|
{ /* 6.666 > |x| >= 1/.35 ~ 2.857143 */
|
|
R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
|
|
s * (rb[5] + s * (rb[6] + s * rb[7]))))));
|
|
S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
|
|
s * (sb[5] + s * (sb[6] + s))))));
|
|
}
|
|
else
|
|
{ /* |x| >= 6.666 */
|
|
if (se & 0x8000)
|
|
return two - tiny; /* x < -6.666 */
|
|
|
|
R = rc[0] + s * (rc[1] + s * (rc[2] + s * (rc[3] +
|
|
s * (rc[4] + s * rc[5]))));
|
|
S = sc[0] + s * (sc[1] + s * (sc[2] + s * (sc[3] +
|
|
s * (sc[4] + s))));
|
|
}
|
|
z = x;
|
|
GET_LDOUBLE_WORDS (hx, i0, i1, z);
|
|
i1 = 0;
|
|
i0 &= 0xffffff00;
|
|
SET_LDOUBLE_WORDS (z, hx, i0, i1);
|
|
r = __ieee754_expl (-z * z - 0.5625) *
|
|
__ieee754_expl ((z - x) * (z + x) + R / S);
|
|
if ((se & 0x8000) == 0)
|
|
{
|
|
long double ret = r / x;
|
|
if (ret == 0)
|
|
__set_errno (ERANGE);
|
|
return ret;
|
|
}
|
|
else
|
|
return two - r / x;
|
|
}
|
|
else
|
|
{
|
|
if ((se & 0x8000) == 0)
|
|
{
|
|
__set_errno (ERANGE);
|
|
return tiny * tiny;
|
|
}
|
|
else
|
|
return two - tiny;
|
|
}
|
|
}
|
|
|
|
weak_alias (__erfcl, erfcl)
|