2014-12-04 18:41:16 +01:00
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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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* 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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* Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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*
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* Permission to use, copy, modify, and distribute this software for any
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* purpose with or without fee is hereby granted, provided that the above
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* copyright notice and this permission notice appear in all copies.
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*
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* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
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* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
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* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
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* ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
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* ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
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* OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
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*/
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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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2015-01-11 23:34:27 +01:00
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#include <openlibm_math.h>
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2014-12-04 18:41:16 +01:00
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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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/* 8 * (2/sqrt(pi) - 1) */
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efx8 = 1.0270333367641005911692712249723613735048E0L,
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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 */
|
|
|
|
|
{
|
|
|
|
|
if (ix < 0x3fde8000) /* |x|<2**-33 */
|
|
|
|
|
{
|
|
|
|
|
if (ix < 0x00080000)
|
|
|
|
|
return 0.125 * (8.0 * x + efx8 * x); /*avoid underflow */
|
|
|
|
|
return x + efx * 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;
|
|
|
|
|
return x + x * y;
|
|
|
|
|
}
|
|
|
|
|
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)
|
|
|
|
|
return erx + P / Q;
|
|
|
|
|
else
|
|
|
|
|
return -erx - P / Q;
|
|
|
|
|
}
|
|
|
|
|
if (ix >= 0x4001d555) /* 6.6666259765625 */
|
|
|
|
|
{ /* inf>|x|>=6.666 */
|
|
|
|
|
if ((se & 0x8000) == 0)
|
|
|
|
|
return one - tiny;
|
|
|
|
|
else
|
|
|
|
|
return tiny - one;
|
|
|
|
|
}
|
|
|
|
|
x = fabsl (x);
|
|
|
|
|
s = one / (x * x);
|
|
|
|
|
if (ix < 0x4000b6db) /* 2.85711669921875 */
|
|
|
|
|
{
|
|
|
|
|
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
|
|
|
|
|
{ /* |x| >= 1/0.35 */
|
|
|
|
|
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))))));
|
|
|
|
|
}
|
|
|
|
|
z = x;
|
|
|
|
|
GET_LDOUBLE_WORDS (i, i0, i1, z);
|
|
|
|
|
i1 = 0;
|
|
|
|
|
SET_LDOUBLE_WORDS (z, i, i0, i1);
|
|
|
|
|
r =
|
|
|
|
|
expl (-z * z - 0.5625) * expl ((z - x) * (z + x) + R / S);
|
|
|
|
|
if ((se & 0x8000) == 0)
|
|
|
|
|
return one - r / x;
|
|
|
|
|
else
|
|
|
|
|
return r / x - one;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
long double
|
|
|
|
|
erfcl(long double x)
|
|
|
|
|
{
|
|
|
|
|
int32_t hx, ix;
|
|
|
|
|
long double R, S, P, Q, s, y, z, r;
|
|
|
|
|
u_int32_t se, i0, i1;
|
|
|
|
|
|
|
|
|
|
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 = expl (-z * z - 0.5625) *
|
|
|
|
|
expl ((z - x) * (z + x) + R / S);
|
|
|
|
|
if ((se & 0x8000) == 0)
|
|
|
|
|
return r / x;
|
|
|
|
|
else
|
|
|
|
|
return two - r / x;
|
|
|
|
|
}
|
|
|
|
|
else
|
|
|
|
|
{
|
|
|
|
|
if ((se & 0x8000) == 0)
|
|
|
|
|
return tiny * tiny;
|
|
|
|
|
else
|
|
|
|
|
return two - tiny;
|
|
|
|
|
}
|
|
|
|
|
}
|
