pioneer/src/win32/WinMath.cpp

// Copyright © 2008-2021 Pioneer Developers. See AUTHORS.txt for details
// Licensed under the terms of the GPL v3. See licenses/GPL-3.txt

// Visit http://www.johndcook.com/stand_alone_code.html for the source of this code and more like it.

#include "Win32Setup.h"

#include "Pi.h"
#include "libs.h"
#include <cmath>
#include <iostream>
#include <sstream>
#include <stdexcept>

#include "WinMath.h"

// Note that the functions Gamma and LogGamma are mutually dependent.

double Gamma(
	double x // We require x > 0
)
{
	if (x <= 0.0) {
		std::stringstream os;
		os << "Invalid input argument " << x << ". Argument must be positive.";
		throw std::invalid_argument(os.str());
	}

	// Split the function domain into three intervals:
	// (0, 0.001), [0.001, 12), and (12, infinity)

	///////////////////////////////////////////////////////////////////////////
	// First interval: (0, 0.001)
	//
	// For small x, 1/Gamma(x) has power series x + gamma x^2  - ...
	// So in this range, 1/Gamma(x) = x + gamma x^2 with error on the order of x^3.
	// The relative error over this interval is less than 6e-7.

	const double gamma = 0.577215664901532860606512090; // Euler's gamma constant

	if (x < 0.001)
		return 1.0 / (x * (1.0 + gamma * x));

	///////////////////////////////////////////////////////////////////////////
	// Second interval: [0.001, 12)

	if (x < 12.0) {
		// The algorithm directly approximates gamma over (1,2) and uses
		// reduction identities to reduce other arguments to this interval.

		double y = x;
		int n = 0;
		bool arg_was_less_than_one = (y < 1.0);

		// Add or subtract integers as necessary to bring y into (1,2)
		// Will correct for this below
		if (arg_was_less_than_one) {
			y += 1.0;
		} else {
			n = static_cast<int>(floor(y)) - 1; // will use n later
			y -= n;
		}

		// numerator coefficients for approximation over the interval (1,2)
		static const double p[] = {
			-1.71618513886549492533811E+0,
			2.47656508055759199108314E+1,
			-3.79804256470945635097577E+2,
			6.29331155312818442661052E+2,
			8.66966202790413211295064E+2,
			-3.14512729688483675254357E+4,
			-3.61444134186911729807069E+4,
			6.64561438202405440627855E+4
		};

		// denominator coefficients for approximation over the interval (1,2)
		static const double q[] = {
			-3.08402300119738975254353E+1,
			3.15350626979604161529144E+2,
			-1.01515636749021914166146E+3,
			-3.10777167157231109440444E+3,
			2.25381184209801510330112E+4,
			4.75584627752788110767815E+3,
			-1.34659959864969306392456E+5,
			-1.15132259675553483497211E+5
		};

		double num = 0.0;
		double den = 1.0;
		int i;

		double z = y - 1;
		for (i = 0; i < 8; i++) {
			num = (num + p[i]) * z;
			den = den * z + q[i];
		}
		double result = num / den + 1.0;

		// Apply correction if argument was not initially in (1,2)
		if (arg_was_less_than_one) {
			// Use identity gamma(z) = gamma(z+1)/z
			// The variable "result" now holds gamma of the original y + 1
			// Thus we use y-1 to get back the orginal y.
			result /= (y - 1.0);
		} else {
			// Use the identity gamma(z+n) = z*(z+1)* ... *(z+n-1)*gamma(z)
			for (i = 0; i < n; i++)
				result *= y++;
		}

		return result;
	}

	///////////////////////////////////////////////////////////////////////////
	// Third interval: [12, infinity)

	if (x > 171.624) {
		// Correct answer too large to display. Force +infinity.
		double temp = DBL_MAX;
		return temp * 2.0;
	}

	return exp(LogGamma(x));
}

double LogGamma(
	double x // x must be positive
)
{
	if (x <= 0.0) {
		std::stringstream os;
		os << "Invalid input argument " << x << ". Argument must be positive.";
		throw std::invalid_argument(os.str());
	}

	if (x < 12.0) {
		return log(fabs(Gamma(x)));
	}

	// Abramowitz and Stegun 6.1.41
	// Asymptotic series should be good to at least 11 or 12 figures
	// For error analysis, see Whittiker and Watson
	// A Course in Modern Analysis (1927), page 252

	static const double c[8] = {
		1.0 / 12.0,
		-1.0 / 360.0,
		1.0 / 1260.0,
		-1.0 / 1680.0,
		1.0 / 1188.0,
		-691.0 / 360360.0,
		1.0 / 156.0,
		-3617.0 / 122400.0
	};
	double z = 1.0 / (x * x);
	double sum = c[7];
	for (int i = 6; i >= 0; i--) {
		sum *= z;
		sum += c[i];
	}
	double series = sum / x;

	static const double halfLogTwoPi = 0.91893853320467274178032973640562;
	double logGamma = (x - 0.5) * log(x) - x + halfLogTwoPi + series;
	return logGamma;
}

// Can delete these functions and the #include for <iostream> if not testing.

void TestGamma()
{
	struct TestCase {
		double input;
		double expected;
	};

	TestCase test[] = {
		// Test near branches in code for (0, 0.001), [0.001, 12), (12, infinity)
		{ 1e-20, 1e+20 },
		{ 2.19824158876e-16, 4.5490905327e+15 }, // 0.99*DBL_EPSILON
		{ 2.24265050974e-16, 4.45900953205e+15 }, // 1.01*DBL_EPSILON
		{ 0.00099, 1009.52477271 },
		{ 0.00100, 999.423772485 },
		{ 0.00101, 989.522792258 },
		{ 6.1, 142.451944066 },
		{ 11.999, 39819417.4793 },
		{ 12, 39916800.0 },
		{ 12.001, 40014424.1571 },
		{ 15.2, 149037380723.0 }
	};

	size_t numTests = sizeof(test) / sizeof(TestCase);

	double worst_absolute_error = 0.0;
	double worst_relative_error = 0.0;
	size_t worst_absolute_error_case = 0;
	size_t worst_relative_error_case = 0;

	for (size_t t = 0; t < numTests; t++) {
		double computed = Gamma(test[t].input);
		double absolute_error = fabs(computed - test[t].expected);
		double relative_error = absolute_error / test[t].expected;

		if (absolute_error > worst_absolute_error) {
			worst_absolute_error = absolute_error;
			worst_absolute_error_case = t;
		}

		if (relative_error > worst_relative_error) {
			worst_relative_error = absolute_error;
			worst_relative_error_case = t;
		}
	}

	size_t t = worst_absolute_error_case;
	double x = test[t].input;
	double y = test[t].expected;
	std::cout << "Worst absolute error: "
			  << fabs(Gamma(x) - y)
			  << "\nGamma( "
			  << x
			  << ") computed as "
			  << Gamma(x)
			  << " but exact value is "
			  << y
			  << "\n";

	t = worst_relative_error_case;
	x = test[t].input;
	y = test[t].expected;
	std::cout << "Worst relative error: "
			  << (Gamma(x) - y) / y
			  << "\nGamma( "
			  << x
			  << ") computed as "
			  << Gamma(x)
			  << " but exact value is "
			  << y
			  << "\n";
}

void TestLogGamma()
{
	struct TestCase {
		double input;
		double expected;
	};

	TestCase test[] = {
		{ 1e-12, 27.6310211159 },
		{ 0.9999, 5.77297915613e-05 },
		{ 1.0001, -5.77133422205e-05 },
		{ 3.1, 0.787375083274 },
		{ 6.3, 5.30734288962 },
		{ 11.9999, 17.5020635801 },
		{ 12, 17.5023078459 },
		{ 12.0001, 17.5025521125 },
		{ 27.4, 62.5755868211 }
	};

	size_t numTests = sizeof(test) / sizeof(TestCase);

	double worst_absolute_error = 0.0;
	double worst_relative_error = 0.0;
	size_t worst_absolute_error_case = 0;
	size_t worst_relative_error_case = 0;

	for (size_t t = 0; t < numTests; t++) {
		double computed = LogGamma(test[t].input);
		double absolute_error = fabs(computed - test[t].expected);
		double relative_error = absolute_error / test[t].expected;

		if (absolute_error > worst_absolute_error) {
			worst_absolute_error = absolute_error;
			worst_absolute_error_case = t;
		}

		if (relative_error > worst_relative_error) {
			worst_relative_error = absolute_error;
			worst_relative_error_case = t;
		}
	}

	size_t t = worst_absolute_error_case;
	double x = test[t].input;
	double y = test[t].expected;
	std::cout << "Worst absolute error: "
			  << fabs(LogGamma(x) - y)
			  << "\nGamma( "
			  << x
			  << ") computed as "
			  << LogGamma(x)
			  << " but exact value is "
			  << y
			  << "\n";

	t = worst_relative_error_case;
	x = test[t].input;
	y = test[t].expected;
	std::cout << "Worst relative error: "
			  << (LogGamma(x) - y) / y
			  << "\nGamma( "
			  << x
			  << ") computed as "
			  << LogGamma(x)
			  << " but exact value is "
			  << y
			  << "\n";
}

#if defined(_MSC_VER) && (_MSC_VER <= 1700)
// http://social.msdn.microsoft.com/Forums/en-US/Vsexpressvc/thread/25c923af-a824-40f8-8fd4-e5574bc147af/
double asinh(double value)
{
	double returned;
	if (value > 0.0)
		returned = log(value + sqrt(value * value + 1.0));
	else
		returned = -log(-value + sqrt(value * value + 1.0));
	return returned;
}

// http://stackoverflow.com/questions/15539116/atanh-arc-hyperbolic-tangent-function-missing-in-ms-visual-c
double atanh(double x) //implements: return (log(1+x) - log(1-x))/2
{
	return (LogOnePlusX(x) - LogOnePlusX(-x)) / 2.0;
}
#endif