compute-erfc
Version:
Complementary error function.
280 lines (253 loc) • 9.91 kB
JavaScript
'use strict';
/**
* NOTE: the following copyright and license, as well as the long comment were part of the original implementation available as part of [FreeBSD]{@link https://svnweb.freebsd.org/base/release/9.3.0/lib/msun/src/s_erf.c?revision=268523&view=co}.
*
* The implementation follows the original, but has been modified for JavaScript.
*/
/**
* ===========================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc business.
* Permission to use, copy, modify, and distribute this software is freely granted, provided that this notice is preserved.
* ===========================
*/
/**
* double erfc(double x)
* x
* 2 |\
* erf(x) = ----------- | exp(-t*t)dt
* sqrt(pi) \|
* 0
*
* erfc(x) = 1 - erf(x)
* Note that
* erf(-x) = -erf(x)
* erfc(-x) = 2 - erfc(x)
*
* Method:
* 1. For |x| in [0, 0.84375)
* erf(x) = x + x*R(x^2)
* erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
* = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
* where R = P/Q where P is an odd poly of degree 8 and Q is an odd poly of degree 10.
* -57.90
* | R - (erf(x)-x)/x | <= 2
*
*
* Remark. The formula is derived by noting
* erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
* and that
* 2/sqrt(pi) = 1.128379167095512573896158903121545171688
* is close to one. The interval is chosen because the fix point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is near 0.6174), and by some experiment, 0.84375 is chosen to guarantee the error is less than one ulp for erf.
*
* 2. For |x| in [0.84375,1.25), let s = |x| - 1, and c = 0.84506291151 rounded to single (24 bits)
* erf(x) = sign(x) * (c + P1(s)/Q1(s))
* erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
* 1+(c+P1(s)/Q1(s)) if x < 0
* |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
* Remark: here we use the taylor series expansion at x=1.
* erf(1+s) = erf(1) + s*Poly(s)
* = 0.845.. + P1(s)/Q1(s)
* That is, we use rational approximation to approximate
* erf(1+s) - (c = (single)0.84506291151)
* Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] where
* P1(s) = degree 6 poly in s
* Q1(s) = degree 6 poly in s
*
* 3. For x in [1.25,1/0.35(~2.857143)),
* erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
* erf(x) = 1 - erfc(x)
* where
* R1(z) = degree 7 poly in z, (z=1/x^2)
* S1(z) = degree 8 poly in z
*
* 4. For x in [1/0.35,28]
* erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
* = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6 < x < 0
* = 2.0 - tiny if x <= -6
* erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
* erf(x) = sign(x)*(1.0 - tiny)
* where
* R2(z) = degree 6 poly in z, (z=1/x^2)
* S2(z) = degree 7 poly in z
*
* Note1:
* To compute exp(-x*x-0.5625+R/S), let s be a single precision number and s := x; then
* -x*x = -s*s + (s-x)*(s+x)
* exp(-x*x-0.5626+R/S) = exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
* Note2:
* Here 4 and 5 make use of the asymptotic series
* exp(-x*x)
* erfc(x) ~ ----------- * ( 1 + Poly(1/x^2) )
* x*sqrt(pi)
* We use rational approximation to approximate
* g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
* Here is the error bound for R1/S1 and R2/S2
* |R1/S1 - f(x)| < 2**(-62.57)
* |R2/S2 - f(x)| < 2**(-61.52)
*
* 5. For inf > x >= 28
* erf(x) = sign(x) * (1 - tiny) (raise inexact)
* erfc(x) = tiny*tiny (raise underflow) if x > 0
* = 2 - tiny if x < 0
*
* 6. Special case:
* erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
* erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
* erfc/erf(NaN) is NaN
*/
// CONSTANTS //
var INF = Number.POSITIVE_INFINITY,
NINF = Number.NEGATIVE_INFINITY,
TINY = 1e-300,
SMALL = 1.0 / (1 << 56 ), /* 2**-56; equiv is Math.pow( 2, -56 ) */
ERX = 8.45062911510467529297e-1, /* 0x3FEB0AC1, 0x60000000 */
// Coefficients for approximation to erfc on [0, 0.84375)
PP0 = 1.28379167095512558561e-1, /* 0x3FC06EBA, 0x8214DB68 */
PP1 = -3.25042107247001499370e-1, /* 0xBFD4CD7D, 0x691CB913 */
PP2 = -2.84817495755985104766e-2, /* 0xBF9D2A51, 0xDBD7194F */
PP3 = -5.77027029648944159157e-3, /* 0xBF77A291, 0x236668E4 */
PP4 = -2.37630166566501626084e-5, /* 0xBEF8EAD6, 0x120016AC */
QQ1 = 3.97917223959155352819e-1, /* 0x3FD97779, 0xCDDADC09 */
QQ2 = 6.50222499887672944485e-2, /* 0x3FB0A54C, 0x5536CEBA */
QQ3 = 5.08130628187576562776e-3, /* 0x3F74D022, 0xC4D36B0F */
QQ4 = 1.32494738004321644526e-4, /* 0x3F215DC9, 0x221C1A10 */
QQ5 = -3.96022827877536812320e-6, /* 0xBED09C43, 0x42A26120 */
// Coefficients for approximation to erfc on [0.84375, 1.25)
PA0 = -2.36211856075265944077e-3, /* 0xBF6359B8, 0xBEF77538 */
PA1 = 4.14856118683748331666e-1, /* 0x3FDA8D00, 0xAD92B34D */
PA2 = -3.72207876035701323847e-1, /* 0xBFD7D240, 0xFBB8C3F1 */
PA3 = 3.18346619901161753674e-1, /* 0x3FD45FCA, 0x805120E4 */
PA4 = -1.10894694282396677476e-1, /* 0xBFBC6398, 0x3D3E28EC */
PA5 = 3.54783043256182359371e-2, /* 0x3FA22A36, 0x599795EB */
PA6 = -2.16637559486879084300e-3, /* 0xBF61BF38, 0x0A96073F */
QA1 = 1.06420880400844228286e-1, /* 0x3FBB3E66, 0x18EEE323 */
QA2 = 5.40397917702171048937e-1, /* 0x3FE14AF0, 0x92EB6F33 */
QA3 = 7.18286544141962662868e-2, /* 0x3FB2635C, 0xD99FE9A7 */
QA4 = 1.26171219808761642112e-1, /* 0x3FC02660, 0xE763351F */
QA5 = 1.36370839120290507362e-2, /* 0x3F8BEDC2, 0x6B51DD1C */
QA6 = 1.19844998467991074170e-2, /* 0x3F888B54, 0x5735151D */
// Coefficients for approximation to erfc on [1.25, 1/0.35)
RA0 = -9.86494403484714822705e-3, /* 0xBF843412, 0x600D6435 */
RA1 = -6.93858572707181764372e-1, /* 0xBFE63416, 0xE4BA7360 */
RA2 = -1.05586262253232909814e1, /* 0xC0251E04, 0x41B0E726 */
RA3 = -6.23753324503260060396e1, /* 0xC04F300A, 0xE4CBA38D */
RA4 = -1.62396669462573470355e2, /* 0xC0644CB1, 0x84282266 */
RA5 = -1.84605092906711035994e2, /* 0xC067135C, 0xEBCCABB2 */
RA6 = -8.12874355063065934246e1, /* 0xC0545265, 0x57E4D2F2 */
RA7 = -9.81432934416914548592, /* 0xC023A0EF, 0xC69AC25C */
SA1 = 1.96512716674392571292e1, /* 0x4033A6B9, 0xBD707687 */
SA2 = 1.37657754143519042600e2, /* 0x4061350C, 0x526AE721 */
SA3 = 4.34565877475229228821e2, /* 0x407B290D, 0xD58A1A71 */
SA4 = 6.45387271733267880336e2, /* 0x40842B19, 0x21EC2868 */
SA5 = 4.29008140027567833386e2, /* 0x407AD021, 0x57700314 */
SA6 = 1.08635005541779435134e2, /* 0x405B28A3, 0xEE48AE2C */
SA7 = 6.57024977031928170135, /* 0x401A47EF, 0x8E484A93 */
SA8 = -6.04244152148580987438e-2, /* 0xBFAEEFF2, 0xEE749A62 */
// Coefficients for approximation to erfc on [1/0.35, 28]
RB0 = -9.86494292470009928597e-3, /* 0xBF843412, 0x39E86F4A */
RB1 = -7.99283237680523006574e-1, /* 0xBFE993BA, 0x70C285DE */
RB2 = -1.77579549177547519889e1, /* 0xC031C209, 0x555F995A */
RB3 = -1.60636384855821916062e2, /* 0xC064145D, 0x43C5ED98 */
RB4 = -6.37566443368389627722e2, /* 0xC083EC88, 0x1375F228 */
RB5 = -1.02509513161107724954e3, /* 0xC0900461, 0x6A2E5992 */
RB6 = -4.83519191608651397019e2, /* 0xC07E384E, 0x9BDC383F */
SB1 = 3.03380607434824582924e1, /* 0x403E568B, 0x261D5190 */
SB2 = 3.25792512996573918826e2, /* 0x40745CAE, 0x221B9F0A */
SB3 = 1.53672958608443695994e3, /* 0x409802EB, 0x189D5118 */
SB4 = 3.19985821950859553908e3, /* 0x40A8FFB7, 0x688C246A */
SB5 = 2.55305040643316442583e3, /* 0x40A3F219, 0xCEDF3BE6 */
SB6 = 4.74528541206955367215e2, /* 0x407DA874, 0xE79FE763 */
SB7 = -2.24409524465858183362e1; /* 0xC03670E2, 0x42712D62 */
// VARIABLES //
var EXP = Math.exp;
// ERFC //
/**
* FUNCTION: erfc( x )
* Evaluates the complementary error function for an input value.
*
* @param {Number} x - input value
* @returns {Number} evaluated complementary error function
*/
function erfc( x ) {
var sign = false,
tmp,
z, r, s, y, p, q;
// [1] Special cases...
// NaN:
if ( x !== x ) {
return NaN;
}
// Positive infinity:
if ( x === INF ) {
return 0;
}
// Negative infinity:
if ( x === NINF ) {
return 2;
}
// [2] Get the sign:
if ( x < 0 ) {
x = -x;
sign = true;
}
// [3] |x| < 0.84375
if ( x < 0.84375 ) {
// |x| < 2**-56
if ( x < SMALL ) {
tmp = x;
} else {
z = x * x;
r = PP0 + z*(PP1+z*(PP2+z*(PP3+z*PP4)));
s = 1 + z*(QQ1+z*(QQ2+z*(QQ3+z*(QQ4+z*QQ5))));
y = r / s;
if ( x < 0.25 ) { // |x| < 1/4
tmp = x + x*y;
} else {
tmp = 0.5 + (x*y + (x-0.5));
}
}
if ( sign ) {
return 1 + tmp;
}
return 1 - tmp;
}
// [4] 0.84375 <= |x| < 1.25
if ( x < 1.25 ) {
s = x - 1;
p = PA0 + s*(PA1+s*(PA2+s*(PA3+s*(PA4+s*(PA5+s*PA6)))));
q = 1 + s*(QA1+s*(QA2+s*(QA3+s*(QA4+s*(QA5+s*QA6)))));
if ( sign ) {
return 1 + ERX + p/q;
}
return 1 - ERX - p/q;
}
// [5] |x| < 28
if ( x < 28 ) {
s = 1 / (x*x);
// |x| < 1/0.35 ~ 2.857143
if ( x < 1/0.35 ) {
r = RA0 + s*(RA1+s*(RA2+s*(RA3+s*(RA4+s*(RA5+s*(RA6+s*RA7))))));
s = 1 + s*(SA1+s*(SA2+s*(SA3+s*(SA4+s*(SA5+s*(SA6+s*SA7))))));
} else { // |x| >= 1/0.35 ~ 2.857143
if ( sign && x > 6 ) { // x < -6
return 2 - TINY;
}
r = RB0 + s*(RB1+s*(RB2+s*(RB3+s*(RB4+s*(RB5+s*RB6)))));
s = 1 + s*(SB1+s*(SB2+s*(SB3+s*(SB4+s*(SB5+s*(SB6+s*SB7))))));
}
z = x & 0xffffffff00000000; // pseudo-single (20-bit) precision x;
r = EXP( -z*z - 0.5625 ) * EXP( (z-x)*(z+x) + r/s );
if ( sign ) {
return 2 - r/x;
}
return r/x;
}
if ( sign ) {
return 2 - TINY; // ~2
}
return TINY * TINY; // ~0
} // end FUNCTION erfc()
// EXPORTS
module.exports = erfc;