2023-11-01 04:58:29 +00:00
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/*-*- mode:c;indent-tabs-mode:t;c-basic-offset:8;tab-width:8;coding:utf-8 -*-│
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2023-12-13 07:28:11 +00:00
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│ vi: set noet ft=c ts=8 sw=8 fenc=utf-8 :vi │
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2023-11-01 04:58:29 +00:00
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╚──────────────────────────────────────────────────────────────────────────────╝
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│ │
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│ Musl Libc │
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│ Copyright © 2005-2014 Rich Felker, et al. │
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│ │
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│ Permission is hereby granted, free of charge, to any person obtaining │
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│ a copy of this software and associated documentation files (the │
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│ "Software"), to deal in the Software without restriction, including │
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│ without limitation the rights to use, copy, modify, merge, publish, │
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│ distribute, sublicense, and/or sell copies of the Software, and to │
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│ permit persons to whom the Software is furnished to do so, subject to │
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│ the following conditions: │
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│ │
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│ The above copyright notice and this permission notice shall be │
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│ included in all copies or substantial portions of the Software. │
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│ │
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│ THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, │
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│ EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF │
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│ MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. │
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│ IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY │
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│ CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, │
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│ TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE │
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│ SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. │
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│ │
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╚─────────────────────────────────────────────────────────────────────────────*/
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#include "libc/math.h"
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#include "libc/tinymath/ldshape.internal.h"
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#if LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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asm(".ident\t\"\\n\\n\
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OpenBSD libm (ISC License)\\n\
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Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>\"");
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asm(".ident\t\"\\n\\n\
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Musl libc (MIT License)\\n\
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Copyright 2005-2014 Rich Felker, et. al.\"");
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asm(".include \"libc/disclaimer.inc\"");
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// clang-format off
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/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_erfl.c */
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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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static const long double
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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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/* 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,
|
|
|
|
|
-1.765321928311155824664963633786967602934E0L,
|
|
|
|
|
-1.029403473103215800456761180695263439188E0L,
|
|
|
|
|
},
|
|
|
|
|
sc[] = {
|
|
|
|
|
8.413244363014929493035952542677768808601E-3L,
|
|
|
|
|
2.065114333816877479753334599639158060979E-1L,
|
|
|
|
|
1.639064941530797583766364412782135680148E0L,
|
|
|
|
|
4.936788463787115555582319302981666347450E0L,
|
|
|
|
|
5.005177727208955487404729933261347679090E0L,
|
|
|
|
|
/* 1.000000000000000000000000000000000000000E0 */
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
static long double erfc1(long double x)
|
|
|
|
|
{
|
|
|
|
|
long double s,P,Q;
|
|
|
|
|
|
|
|
|
|
s = fabsl(x) - 1;
|
|
|
|
|
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))))));
|
|
|
|
|
return 1 - erx - P / Q;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
static long double erfc2(uint32_t ix, long double x)
|
|
|
|
|
{
|
|
|
|
|
union ldshape u;
|
|
|
|
|
long double s,z,R,S;
|
|
|
|
|
|
|
|
|
|
if (ix < 0x3fffa000) /* 0.84375 <= |x| < 1.25 */
|
|
|
|
|
return erfc1(x);
|
|
|
|
|
|
|
|
|
|
x = fabsl(x);
|
|
|
|
|
s = 1 / (x * x);
|
|
|
|
|
if (ix < 0x4000b6db) { /* 1.25 <= |x| < 2.857 ~ 1/.35 */
|
|
|
|
|
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) { /* 2.857 <= |x| < 6.6666259765625 */
|
|
|
|
|
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 { /* 6.666 <= |x| < 107 (erfc only) */
|
|
|
|
|
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))));
|
|
|
|
|
}
|
|
|
|
|
u.f = x;
|
|
|
|
|
u.i.m &= -1ULL << 40;
|
|
|
|
|
z = u.f;
|
|
|
|
|
return expl(-z*z - 0.5625) * expl((z - x) * (z + x) + R / S) / x;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
long double erfl(long double x)
|
|
|
|
|
{
|
|
|
|
|
long double r, s, z, y;
|
|
|
|
|
union ldshape u = {x};
|
|
|
|
|
uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48;
|
|
|
|
|
int sign = u.i.se >> 15;
|
|
|
|
|
|
|
|
|
|
if (ix >= 0x7fff0000)
|
|
|
|
|
/* erf(nan)=nan, erf(+-inf)=+-1 */
|
|
|
|
|
return 1 - 2*sign + 1/x;
|
|
|
|
|
if (ix < 0x3ffed800) { /* |x| < 0.84375 */
|
|
|
|
|
if (ix < 0x3fde8000) { /* |x| < 2**-33 */
|
|
|
|
|
return 0.125 * (8 * x + efx8 * x); /* avoid underflow */
|
|
|
|
|
}
|
|
|
|
|
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 < 0x4001d555) /* |x| < 6.6666259765625 */
|
|
|
|
|
y = 1 - erfc2(ix,x);
|
|
|
|
|
else
|
|
|
|
|
y = 1 - 0x1p-16382L;
|
|
|
|
|
return sign ? -y : y;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
long double erfcl(long double x)
|
|
|
|
|
{
|
|
|
|
|
long double r, s, z, y;
|
|
|
|
|
union ldshape u = {x};
|
|
|
|
|
uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48;
|
|
|
|
|
int sign = u.i.se >> 15;
|
|
|
|
|
|
|
|
|
|
if (ix >= 0x7fff0000)
|
|
|
|
|
/* erfc(nan) = nan, erfc(+-inf) = 0,2 */
|
|
|
|
|
return 2*sign + 1/x;
|
|
|
|
|
if (ix < 0x3ffed800) { /* |x| < 0.84375 */
|
|
|
|
|
if (ix < 0x3fbe0000) /* |x| < 2**-65 */
|
|
|
|
|
return 1.0 - 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 1.0 - (x + x * y);
|
|
|
|
|
return 0.5 - (x - 0.5 + x * y);
|
|
|
|
|
}
|
|
|
|
|
if (ix < 0x4005d600) /* |x| < 107 */
|
|
|
|
|
return sign ? 2 - erfc2(ix,x) : erfc2(ix,x);
|
|
|
|
|
y = 0x1p-16382L;
|
|
|
|
|
return sign ? 2 - y : y*y;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
|
|
|
|
|
// TODO: broken implementation to make things compile
|
|
|
|
|
long double erfl(long double x)
|
|
|
|
|
{
|
|
|
|
|
return erf(x);
|
|
|
|
|
}
|
|
|
|
|
long double erfcl(long double x)
|
|
|
|
|
{
|
|
|
|
|
return erfc(x);
|
|
|
|
|
}
|
|
|
|
|
#endif /* long double is long */
|
