BoostErf.java
/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
// (C) Copyright John Maddock 2006.
// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0. (See accompanying file
// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
package org.apache.commons.numbers.gamma;
import org.apache.commons.numbers.core.DD;
/**
* Implementation of the <a href="http://mathworld.wolfram.com/Erf.html">error function</a> and
* its inverse.
*
* <p>This code has been adapted from the <a href="https://www.boost.org/">Boost</a>
* {@code c++} implementation {@code <boost/math/special_functions/erf.hpp>}.
* The erf/erfc functions and their inverses are copyright John Maddock 2006 and subject to
* the Boost Software License.
*
* <p>Additions made to support the erfcx function are original work under the Apache software
* license.
*
* @see
* <a href="https://www.boost.org/doc/libs/1_77_0/libs/math/doc/html/math_toolkit/sf_erf/error_function.html">
* Boost C++ Error functions</a>
*/
final class BoostErf {
/** 1 / sqrt(pi). Used for the scaled complementary error function erfcx. */
private static final double ONE_OVER_ROOT_PI = 0.5641895835477562869480794515607725858;
/** Threshold for the scaled complementary error function erfcx
* where the approximation {@code (1 / sqrt(pi)) / x} can be used. */
private static final double ERFCX_APPROX = 6.71e7;
/** Threshold for the erf implementation for |x| where the computation
* uses {@code erf(x)}; otherwise {@code erfc(x)} is computed. The final result is
* achieved by suitable application of symmetry. */
private static final double COMPUTE_ERF = 0.5;
/** Threshold for the scaled complementary error function erfcx for negative x
* where {@code 2 * exp(x*x)} will overflow. Value is 26.62873571375149. */
private static final double ERFCX_NEG_X_MAX = Math.sqrt(Math.log(Double.MAX_VALUE / 2));
/** Threshold for the scaled complementary error function erfcx for x
* where {@code exp(x*x) == 1; x <= t}. Value is (1 + 5/16) * 2^-27 = 9.778887033462524E-9.
* <p>Note: This is used for performance. If set to 0 then the result is computed
* using expm1(x*x) with the same final result. */
private static final double EXP_XX_1 = 0x1.5p-27;
/** Private constructor. */
private BoostErf() {
// intentionally empty.
}
// Code ported from Boost 1.77.0
//
// boost/math/special_functions/erf.hpp
// boost/math/special_functions/detail/erf_inv.hpp
//
// Original code comments, including measured deviations, are preserved.
//
// Changes to the Boost implementation:
// - Update method names to replace underscores with camel case
// - Explicitly inline the polynomial function evaluation
// using Horner's method (https://en.wikipedia.org/wiki/Horner%27s_method)
// - Support odd function for f(0.0) = -f(-0.0)
// - Support the scaled complementary error function erfcx
// Erf:
// - Change extended precision z*z to compute the square round-off
// using Dekker's method
// - Change extended precision exp(-z*z) to compute using a
// round-off addition to the standard exp result (see NUMBERS-177)
// - Change the erf threshold for z when erf(z)=1 from
// z > 5.8f to z > 5.930664
// - Change the erfc threshold for z when erfc(z)=0 from
// z < 28 to z < 27.3
// - Change rational function approximation for z > 4 to a function
// suitable for erfcx (see NUMBERS-177)
// Inverse erf:
// - Change inverse erf edge case detection to include NaN
// - Change edge case detection for integer z
//
// Note:
// Constants using the 'f' suffix are machine
// representable as a float, e.g.
// assert 0.0891314744949340820313f == 0.0891314744949340820313;
// The values are unchanged from the Boost reference.
/**
* Returns the complementary error function.
*
* @param x the value.
* @return the complementary error function.
*/
static double erfc(double x) {
return erfImp(x, true, false);
}
/**
* Returns the error function.
*
* @param x the value.
* @return the error function.
*/
static double erf(double x) {
return erfImp(x, false, false);
}
/**
* 53-bit implementation for the error function.
*
* <p>Note: The {@code scaled} flag only applies when
* {@code z >= 0.5} and {@code invert == true}.
* This functionality is used to compute erfcx(z) for positive z.
*
* @param z Point to evaluate
* @param invert true to invert the result (for the complementary error function)
* @param scaled true to compute the scaled complementary error function
* @return the error function result
*/
private static double erfImp(double z, boolean invert, boolean scaled) {
if (Double.isNaN(z)) {
return Double.NaN;
}
if (z < 0) {
// Here the scaled flag is ignored.
if (!invert) {
return -erfImp(-z, invert, false);
} else if (z < -0.5) {
return 2 - erfImp(-z, invert, false);
} else {
return 1 + erfImp(-z, false, false);
}
}
double result;
//
// Big bunch of selection statements now to pick
// which implementation to use,
// try to put most likely options first:
//
if (z < COMPUTE_ERF) {
//
// We're going to calculate erf:
//
// Here the scaled flag is ignored.
if (z < 1e-10) {
if (z == 0) {
result = z;
} else {
final double c = 0.003379167095512573896158903121545171688;
result = z * 1.125f + z * c;
}
} else {
// Maximum Deviation Found: 1.561e-17
// Expected Error Term: 1.561e-17
// Maximum Relative Change in Control Points: 1.155e-04
// Max Error found at double precision = 2.961182e-17
final double Y = 1.044948577880859375f;
final double zz = z * z;
double P;
P = -0.000322780120964605683831;
P = -0.00772758345802133288487 + P * zz;
P = -0.0509990735146777432841 + P * zz;
P = -0.338165134459360935041 + P * zz;
P = 0.0834305892146531832907 + P * zz;
double Q;
Q = 0.000370900071787748000569;
Q = 0.00858571925074406212772 + Q * zz;
Q = 0.0875222600142252549554 + Q * zz;
Q = 0.455004033050794024546 + Q * zz;
Q = 1.0 + Q * zz;
result = z * (Y + P / Q);
}
// Note: Boost threshold of 5.8f has been raised to approximately 5.93 (6073 / 1024);
// threshold of 28 has been lowered to approximately 27.3 (6989/256) where exp(-z*z) = 0.
} else if (scaled || (invert ? (z < 27.300781f) : (z < 5.9306640625f))) {
//
// We'll be calculating erfc:
//
// Here the scaled flag is used.
invert = !invert;
if (z < 1.5f) {
// Maximum Deviation Found: 3.702e-17
// Expected Error Term: 3.702e-17
// Maximum Relative Change in Control Points: 2.845e-04
// Max Error found at double precision = 4.841816e-17
final double Y = 0.405935764312744140625f;
final double zm = z - 0.5;
double P;
P = 0.00180424538297014223957;
P = 0.0195049001251218801359 + P * zm;
P = 0.0888900368967884466578 + P * zm;
P = 0.191003695796775433986 + P * zm;
P = 0.178114665841120341155 + P * zm;
P = -0.098090592216281240205 + P * zm;
double Q;
Q = 0.337511472483094676155e-5;
Q = 0.0113385233577001411017 + Q * zm;
Q = 0.12385097467900864233 + Q * zm;
Q = 0.578052804889902404909 + Q * zm;
Q = 1.42628004845511324508 + Q * zm;
Q = 1.84759070983002217845 + Q * zm;
Q = 1.0 + Q * zm;
result = Y + P / Q;
if (scaled) {
result /= z;
} else {
result *= expmxx(z) / z;
}
} else if (z < 2.5f) {
// Max Error found at double precision = 6.599585e-18
// Maximum Deviation Found: 3.909e-18
// Expected Error Term: 3.909e-18
// Maximum Relative Change in Control Points: 9.886e-05
final double Y = 0.50672817230224609375f;
final double zm = z - 1.5;
double P;
P = 0.000235839115596880717416;
P = 0.00323962406290842133584 + P * zm;
P = 0.0175679436311802092299 + P * zm;
P = 0.04394818964209516296 + P * zm;
P = 0.0386540375035707201728 + P * zm;
P = -0.0243500476207698441272 + P * zm;
double Q;
Q = 0.00410369723978904575884;
Q = 0.0563921837420478160373 + Q * zm;
Q = 0.325732924782444448493 + Q * zm;
Q = 0.982403709157920235114 + Q * zm;
Q = 1.53991494948552447182 + Q * zm;
Q = 1.0 + Q * zm;
result = Y + P / Q;
if (scaled) {
result /= z;
} else {
result *= expmxx(z) / z;
}
// Lowered Boost threshold from 4.5 to 4.0 as this is the limit
// for the Cody erfc approximation
} else if (z < 4.0f) {
// Maximum Deviation Found: 1.512e-17
// Expected Error Term: 1.512e-17
// Maximum Relative Change in Control Points: 2.222e-04
// Max Error found at double precision = 2.062515e-17
final double Y = 0.5405750274658203125f;
final double zm = z - 3.5;
double P;
P = 0.113212406648847561139e-4;
P = 0.000250269961544794627958 + P * zm;
P = 0.00212825620914618649141 + P * zm;
P = 0.00840807615555585383007 + P * zm;
P = 0.0137384425896355332126 + P * zm;
P = 0.00295276716530971662634 + P * zm;
double Q;
Q = 0.000479411269521714493907;
Q = 0.0105982906484876531489 + Q * zm;
Q = 0.0958492726301061423444 + Q * zm;
Q = 0.442597659481563127003 + Q * zm;
Q = 1.04217814166938418171 + Q * zm;
Q = 1.0 + Q * zm;
result = Y + P / Q;
if (scaled) {
result /= z;
} else {
result *= expmxx(z) / z;
}
} else {
// Rational function approximation for erfc(x > 4.0)
//
// This approximation is not the Boost implementation.
// The Boost function is suitable for [4.5 < z < 28].
//
// This function is suitable for erfcx(z) as it asymptotes
// to (1 / sqrt(pi)) / z at large z.
//
// Taken from "Rational Chebyshev approximations for the error function"
// by W. J. Cody, Math. Comp., 1969, PP. 631-638.
//
// See NUMBERS-177.
final double izz = 1 / (z * z);
double p;
p = 1.63153871373020978498e-2;
p = 3.05326634961232344035e-1 + p * izz;
p = 3.60344899949804439429e-1 + p * izz;
p = 1.25781726111229246204e-1 + p * izz;
p = 1.60837851487422766278e-2 + p * izz;
p = 6.58749161529837803157e-4 + p * izz;
double q;
q = 1;
q = 2.56852019228982242072e00 + q * izz;
q = 1.87295284992346047209e00 + q * izz;
q = 5.27905102951428412248e-1 + q * izz;
q = 6.05183413124413191178e-2 + q * izz;
q = 2.33520497626869185443e-3 + q * izz;
result = izz * p / q;
result = (ONE_OVER_ROOT_PI - result) / z;
if (!scaled) {
// exp(-z*z) can be sub-normal so
// multiply by any sub-normal after divide by z
result *= expmxx(z);
}
}
} else {
//
// Any value of z larger than 27.3 will underflow to zero:
//
result = 0;
invert = !invert;
}
if (invert) {
// Note: If 0.5 <= z < 28 and the scaled flag is true then
// invert will have been flipped to false and the
// the result is unchanged as erfcx(z)
result = 1 - result;
}
return result;
}
/**
* Returns the scaled complementary error function.
* <pre>
* erfcx(x) = exp(x^2) * erfc(x)
* </pre>
*
* @param x the value.
* @return the scaled complementary error function.
*/
static double erfcx(double x) {
if (Double.isNaN(x)) {
return Double.NaN;
}
// For |z| < 0.5 erfc is computed using erf
final double ax = Math.abs(x);
if (ax < COMPUTE_ERF) {
// Use the erf(x) result.
// (1 - erf(x)) * exp(x*x)
final double erfx = erf(x);
if (ax < EXP_XX_1) {
// No exponential required
return 1 - erfx;
}
// exp(x*x) - exp(x*x) * erf(x)
// Avoid use of exp(x*x) with expm1:
// exp(x*x) - 1 - (erf(x) * (exp(x*x) - 1)) - erf(x) + 1
// Sum small to large: |erf(x)| > expm1(x*x)
// -erf(x) * expm1(x*x) + expm1(x*x) - erf(x) + 1
// Negative x: erf(x) < 0, summed terms are positive, no cancellation occurs.
// Positive x: erf(x) > 0 so cancellation can occur.
// When terms are ordered by absolute magnitude the magnitude of the next term
// is above the round-off from adding the previous term to the sum. Thus
// cancellation is negligible compared to errors in the largest computed term (erf(x)).
final double em1 = Math.expm1(x * x);
return -erfx * em1 + em1 - erfx + 1;
}
// Handle negative arguments
if (x < 0) {
// erfcx(x) = 2*exp(x*x) - erfcx(-x)
if (x < -ERFCX_NEG_X_MAX) {
// Overflow
return Double.POSITIVE_INFINITY;
}
final double e = expxx(x);
return e - erfImp(-x, true, true) + e;
}
// Approximation for large positive x
if (x > ERFCX_APPROX) {
return ONE_OVER_ROOT_PI / x;
}
// Compute erfc scaled
return erfImp(x, true, true);
}
/**
* Returns the inverse complementary error function.
*
* @param z Value (in {@code [0, 2]}).
* @return t such that {@code z = erfc(t)}
*/
static double erfcInv(double z) {
//
// Begin by testing for domain errors, and other special cases:
//
if (z < 0 || z > 2 || Double.isNaN(z)) {
// Argument outside range [0,2] in inverse erfc function
return Double.NaN;
}
// Domain bounds must be detected as the implementation computes NaN.
// (log(q=0) creates infinity and the rational number is
// infinity / infinity)
if (z == (int) z) {
// z return
// 2 -inf
// 1 0
// 0 inf
return z == 1 ? 0 : (1 - z) * Double.POSITIVE_INFINITY;
}
//
// Normalise the input, so it's in the range [0,1], we will
// negate the result if z is outside that range. This is a simple
// application of the erfc reflection formula: erfc(-z) = 2 - erfc(z)
//
final double p;
final double q;
final double s;
if (z > 1) {
q = 2 - z;
p = 1 - q;
s = -1;
} else {
p = 1 - z;
q = z;
s = 1;
}
//
// And get the result, negating where required:
//
return s * erfInvImp(p, q);
}
/**
* Returns the inverse error function.
*
* @param z Value (in {@code [-1, 1]}).
* @return t such that {@code z = erf(t)}
*/
static double erfInv(double z) {
//
// Begin by testing for domain errors, and other special cases:
//
if (z < -1 || z > 1 || Double.isNaN(z)) {
// Argument outside range [-1, 1] in inverse erf function
return Double.NaN;
}
// Domain bounds must be detected as the implementation computes NaN.
// (log(q=0) creates infinity and the rational number is
// infinity / infinity)
if (z == (int) z) {
// z return
// -1 -inf
// -0 -0
// 0 0
// 1 inf
return z == 0 ? z : z * Double.POSITIVE_INFINITY;
}
//
// Normalise the input, so it's in the range [0,1], we will
// negate the result if z is outside that range. This is a simple
// application of the erf reflection formula: erf(-z) = -erf(z)
//
final double p;
final double q;
final double s;
if (z < 0) {
p = -z;
q = 1 - p;
s = -1;
} else {
p = z;
q = 1 - z;
s = 1;
}
//
// And get the result, negating where required:
//
return s * erfInvImp(p, q);
}
/**
* Common implementation for inverse erf and erfc functions.
*
* @param p P-value
* @param q Q-value (1-p)
* @return the inverse
*/
private static double erfInvImp(double p, double q) {
final double result;
if (p <= 0.5) {
//
// Evaluate inverse erf using the rational approximation:
//
// x = p(p+10)(Y+R(p))
//
// Where Y is a constant, and R(p) is optimised for a low
// absolute error compared to |Y|.
//
// double: Max error found: 2.001849e-18
// long double: Max error found: 1.017064e-20
// Maximum Deviation Found (actual error term at infinite precision) 8.030e-21
//
final float Y = 0.0891314744949340820313f;
double P;
P = -0.00538772965071242932965;
P = 0.00822687874676915743155 + P * p;
P = 0.0219878681111168899165 + P * p;
P = -0.0365637971411762664006 + P * p;
P = -0.0126926147662974029034 + P * p;
P = 0.0334806625409744615033 + P * p;
P = -0.00836874819741736770379 + P * p;
P = -0.000508781949658280665617 + P * p;
double Q;
Q = 0.000886216390456424707504;
Q = -0.00233393759374190016776 + Q * p;
Q = 0.0795283687341571680018 + Q * p;
Q = -0.0527396382340099713954 + Q * p;
Q = -0.71228902341542847553 + Q * p;
Q = 0.662328840472002992063 + Q * p;
Q = 1.56221558398423026363 + Q * p;
Q = -1.56574558234175846809 + Q * p;
Q = -0.970005043303290640362 + Q * p;
Q = 1.0 + Q * p;
final double g = p * (p + 10);
final double r = P / Q;
result = g * Y + g * r;
} else if (q >= 0.25) {
//
// Rational approximation for 0.5 > q >= 0.25
//
// x = sqrt(-2*log(q)) / (Y + R(q))
//
// Where Y is a constant, and R(q) is optimised for a low
// absolute error compared to Y.
//
// double : Max error found: 7.403372e-17
// long double : Max error found: 6.084616e-20
// Maximum Deviation Found (error term) 4.811e-20
//
final float Y = 2.249481201171875f;
final double xs = q - 0.25f;
double P;
P = -3.67192254707729348546;
P = 21.1294655448340526258 + P * xs;
P = 17.445385985570866523 + P * xs;
P = -44.6382324441786960818 + P * xs;
P = -18.8510648058714251895 + P * xs;
P = 17.6447298408374015486 + P * xs;
P = 8.37050328343119927838 + P * xs;
P = 0.105264680699391713268 + P * xs;
P = -0.202433508355938759655 + P * xs;
double Q;
Q = 1.72114765761200282724;
Q = -22.6436933413139721736 + Q * xs;
Q = 10.8268667355460159008 + Q * xs;
Q = 48.5609213108739935468 + Q * xs;
Q = -20.1432634680485188801 + Q * xs;
Q = -28.6608180499800029974 + Q * xs;
Q = 3.9713437953343869095 + Q * xs;
Q = 6.24264124854247537712 + Q * xs;
Q = 1.0 + Q * xs;
final double g = Math.sqrt(-2 * Math.log(q));
final double r = P / Q;
result = g / (Y + r);
} else {
//
// For q < 0.25 we have a series of rational approximations all
// of the general form:
//
// let: x = sqrt(-log(q))
//
// Then the result is given by:
//
// x(Y+R(x-B))
//
// where Y is a constant, B is the lowest value of x for which
// the approximation is valid, and R(x-B) is optimised for a low
// absolute error compared to Y.
//
// Note that almost all code will really go through the first
// or maybe second approximation. After than we're dealing with very
// small input values indeed.
//
// Limit for a double: Math.sqrt(-Math.log(Double.MIN_VALUE)) = 27.28...
// Branches for x >= 44 (supporting 80 and 128 bit long double) have been removed.
final double x = Math.sqrt(-Math.log(q));
if (x < 3) {
// Max error found: 1.089051e-20
final float Y = 0.807220458984375f;
final double xs = x - 1.125f;
double P;
P = -0.681149956853776992068e-9;
P = 0.285225331782217055858e-7 + P * xs;
P = -0.679465575181126350155e-6 + P * xs;
P = 0.00214558995388805277169 + P * xs;
P = 0.0290157910005329060432 + P * xs;
P = 0.142869534408157156766 + P * xs;
P = 0.337785538912035898924 + P * xs;
P = 0.387079738972604337464 + P * xs;
P = 0.117030156341995252019 + P * xs;
P = -0.163794047193317060787 + P * xs;
P = -0.131102781679951906451 + P * xs;
double Q;
Q = 0.01105924229346489121;
Q = 0.152264338295331783612 + Q * xs;
Q = 0.848854343457902036425 + Q * xs;
Q = 2.59301921623620271374 + Q * xs;
Q = 4.77846592945843778382 + Q * xs;
Q = 5.38168345707006855425 + Q * xs;
Q = 3.46625407242567245975 + Q * xs;
Q = 1.0 + Q * xs;
final double R = P / Q;
result = Y * x + R * x;
} else if (x < 6) {
// Max error found: 8.389174e-21
final float Y = 0.93995571136474609375f;
final double xs = x - 3;
double P;
P = 0.266339227425782031962e-11;
P = -0.230404776911882601748e-9 + P * xs;
P = 0.460469890584317994083e-5 + P * xs;
P = 0.000157544617424960554631 + P * xs;
P = 0.00187123492819559223345 + P * xs;
P = 0.00950804701325919603619 + P * xs;
P = 0.0185573306514231072324 + P * xs;
P = -0.00222426529213447927281 + P * xs;
P = -0.0350353787183177984712 + P * xs;
double Q;
Q = 0.764675292302794483503e-4;
Q = 0.00263861676657015992959 + Q * xs;
Q = 0.0341589143670947727934 + Q * xs;
Q = 0.220091105764131249824 + Q * xs;
Q = 0.762059164553623404043 + Q * xs;
Q = 1.3653349817554063097 + Q * xs;
Q = 1.0 + Q * xs;
final double R = P / Q;
result = Y * x + R * x;
} else if (x < 18) {
// Max error found: 1.481312e-19
final float Y = 0.98362827301025390625f;
final double xs = x - 6;
double P;
P = 0.99055709973310326855e-16;
P = -0.281128735628831791805e-13 + P * xs;
P = 0.462596163522878599135e-8 + P * xs;
P = 0.449696789927706453732e-6 + P * xs;
P = 0.149624783758342370182e-4 + P * xs;
P = 0.000209386317487588078668 + P * xs;
P = 0.00105628862152492910091 + P * xs;
P = -0.00112951438745580278863 + P * xs;
P = -0.0167431005076633737133 + P * xs;
double Q;
Q = 0.282243172016108031869e-6;
Q = 0.275335474764726041141e-4 + Q * xs;
Q = 0.000964011807005165528527 + Q * xs;
Q = 0.0160746087093676504695 + Q * xs;
Q = 0.138151865749083321638 + Q * xs;
Q = 0.591429344886417493481 + Q * xs;
Q = 1.0 + Q * xs;
final double R = P / Q;
result = Y * x + R * x;
} else {
// x < 44
// Max error found: 5.697761e-20
final float Y = 0.99714565277099609375f;
final double xs = x - 18;
double P;
P = -0.116765012397184275695e-17;
P = 0.145596286718675035587e-11 + P * xs;
P = 0.411632831190944208473e-9 + P * xs;
P = 0.396341011304801168516e-7 + P * xs;
P = 0.162397777342510920873e-5 + P * xs;
P = 0.254723037413027451751e-4 + P * xs;
P = -0.779190719229053954292e-5 + P * xs;
P = -0.0024978212791898131227 + P * xs;
double Q;
Q = 0.509761276599778486139e-9;
Q = 0.144437756628144157666e-6 + Q * xs;
Q = 0.145007359818232637924e-4 + Q * xs;
Q = 0.000690538265622684595676 + Q * xs;
Q = 0.0169410838120975906478 + Q * xs;
Q = 0.207123112214422517181 + Q * xs;
Q = 1.0 + Q * xs;
final double R = P / Q;
result = Y * x + R * x;
}
}
return result;
}
/**
* Compute {@code exp(x*x)} with high accuracy. This is performed using
* information in the round-off from {@code x*x}.
*
* <p>This is accurate at large x to 1 ulp.
*
* <p>At small x the accuracy cannot be improved over using exp(x*x).
* This occurs at {@code x <= 1}.
*
* <p>Warning: This has no checks for overflow. The method is never called
* when {@code x*x > log(MAX_VALUE/2)}.
*
* @param x Value
* @return exp(x*x)
*/
static double expxx(double x) {
// Note: If exp(a) overflows this can create NaN if the
// round-off b is negative or zero:
// exp(a) * exp1m(b) + exp(a)
// inf * 0 + inf or inf * -b + inf
final DD x2 = DD.ofSquare(x);
return expxx(x2.hi(), x2.lo());
}
/**
* Compute {@code exp(-x*x)} with high accuracy. This is performed using
* information in the round-off from {@code x*x}.
*
* <p>This is accurate at large x to 1 ulp until exp(-x*x) is close to
* sub-normal. For very small exp(-x*x) the adjustment is sub-normal and
* bits can be lost in the adjustment for a max observed error of {@code < 2} ulp.
*
* <p>At small x the accuracy cannot be improved over using exp(-x*x).
* This occurs at {@code x <= 1}.
*
* @param x Value
* @return exp(-x*x)
*/
static double expmxx(double x) {
final DD x2 = DD.ofSquare(x);
return expxx(-x2.hi(), -x2.lo());
}
/**
* Compute {@code exp(a+b)} with high accuracy assuming {@code a+b = a}.
*
* <p>This is accurate at large positive a to 1 ulp. If a is negative and exp(a) is
* close to sub-normal a bit of precision may be lost when adjusting result
* as the adjustment is sub-normal (max observed error {@code < 2} ulp).
* For the use case of multiplication of a number less than 1 by exp(-x*x), a = -x*x,
* the result will be sub-normal and the rounding error is lost.
*
* <p>At small |a| the accuracy cannot be improved over using exp(a) as the
* round-off is too small to create terms that can adjust the standard result by
* more than 0.5 ulp. This occurs at {@code |a| <= 1}.
*
* @param a High bits of a split number
* @param b Low bits of a split number
* @return exp(a+b)
*/
private static double expxx(double a, double b) {
// exp(a+b) = exp(a) * exp(b)
// = exp(a) * (exp(b) - 1) + exp(a)
// Assuming:
// 1. -746 < a < 710 for no under/overflow of exp(a)
// 2. a+b = a
// As b -> 0 then exp(b) -> 1; expm1(b) -> b
// The round-off b is limited to ~ 0.5 * ulp(746) ~ 5.68e-14
// and we can use an approximation for expm1 (x/1! + x^2/2! + ...)
// The second term is required for the expm1 result but the
// bits are not significant to change the following sum with exp(a)
final double ea = Math.exp(a);
// b ~ expm1(b)
return ea * b + ea;
}
}