777bb1d438
* java/lang/StrictMath.java: Typo fix. * java/lang/Math.java: Typo fix. 2003-08-26 Stephen Crawley <crawley@dstc.edu.au> * java/lang/ThreadGroup.java (removeThread): null the 'group' field of the removed Thread. 2003-08-26 Mark Wielaard <mark@klomp.org> Reported by David Holmes <dholmes@dltech.com.au>. * java/lang/InheritableThreadLocal.java (threadMap): Wrap inside Collections.synchronizedMap. * java/lang/ThreadLocal.java (valueMap): Likewise. From-SVN: r70828
1844 lines
60 KiB
Java
1844 lines
60 KiB
Java
/* java.lang.StrictMath -- common mathematical functions, strict Java
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Copyright (C) 1998, 2001, 2002, 2003 Free Software Foundation, Inc.
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This file is part of GNU Classpath.
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GNU Classpath is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2, or (at your option)
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any later version.
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GNU Classpath is distributed in the hope that it will be useful, but
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WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with GNU Classpath; see the file COPYING. If not, write to the
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Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
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02111-1307 USA.
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Linking this library statically or dynamically with other modules is
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making a combined work based on this library. Thus, the terms and
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conditions of the GNU General Public License cover the whole
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combination.
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As a special exception, the copyright holders of this library give you
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permission to link this library with independent modules to produce an
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executable, regardless of the license terms of these independent
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modules, and to copy and distribute the resulting executable under
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terms of your choice, provided that you also meet, for each linked
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independent module, the terms and conditions of the license of that
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module. An independent module is a module which is not derived from
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or based on this library. If you modify this library, you may extend
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this exception to your version of the library, but you are not
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obligated to do so. If you do not wish to do so, delete this
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exception statement from your version. */
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/*
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* Some of the algorithms in this class are in the public domain, as part
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* of fdlibm (freely-distributable math library), available at
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* http://www.netlib.org/fdlibm/, and carry the following copyright:
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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 SunSoft, 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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package java.lang;
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import java.util.Random;
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import gnu.classpath.Configuration;
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/**
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* Helper class containing useful mathematical functions and constants.
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* This class mirrors {@link Math}, but is 100% portable, because it uses
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* no native methods whatsoever. Also, these algorithms are all accurate
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* to less than 1 ulp, and execute in <code>strictfp</code> mode, while
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* Math is allowed to vary in its results for some functions. Unfortunately,
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* this usually means StrictMath has less efficiency and speed, as Math can
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* use native methods.
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*
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* <p>The source of the various algorithms used is the fdlibm library, at:<br>
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* <a href="http://www.netlib.org/fdlibm/">http://www.netlib.org/fdlibm/</a>
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*
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* Note that angles are specified in radians. Conversion functions are
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* provided for your convenience.
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*
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* @author Eric Blake <ebb9@email.byu.edu>
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* @since 1.3
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*/
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public final strictfp class StrictMath
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{
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/**
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* StrictMath is non-instantiable.
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*/
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private StrictMath()
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{
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}
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/**
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* A random number generator, initialized on first use.
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*
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* @see #random()
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*/
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private static Random rand;
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/**
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* The most accurate approximation to the mathematical constant <em>e</em>:
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* <code>2.718281828459045</code>. Used in natural log and exp.
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*
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* @see #log(double)
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* @see #exp(double)
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*/
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public static final double E
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= 2.718281828459045; // Long bits 0x4005bf0z8b145769L.
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/**
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* The most accurate approximation to the mathematical constant <em>pi</em>:
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* <code>3.141592653589793</code>. This is the ratio of a circle's diameter
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* to its circumference.
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*/
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public static final double PI
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= 3.141592653589793; // Long bits 0x400921fb54442d18L.
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/**
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* Take the absolute value of the argument. (Absolute value means make
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* it positive.)
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*
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* <p>Note that the the largest negative value (Integer.MIN_VALUE) cannot
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* be made positive. In this case, because of the rules of negation in
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* a computer, MIN_VALUE is what will be returned.
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* This is a <em>negative</em> value. You have been warned.
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*
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* @param i the number to take the absolute value of
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* @return the absolute value
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* @see Integer#MIN_VALUE
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*/
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public static int abs(int i)
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{
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return (i < 0) ? -i : i;
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}
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/**
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* Take the absolute value of the argument. (Absolute value means make
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* it positive.)
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*
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* <p>Note that the the largest negative value (Long.MIN_VALUE) cannot
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* be made positive. In this case, because of the rules of negation in
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* a computer, MIN_VALUE is what will be returned.
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* This is a <em>negative</em> value. You have been warned.
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*
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* @param l the number to take the absolute value of
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* @return the absolute value
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* @see Long#MIN_VALUE
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*/
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public static long abs(long l)
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{
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return (l < 0) ? -l : l;
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}
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/**
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* Take the absolute value of the argument. (Absolute value means make
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* it positive.)
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*
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* @param f the number to take the absolute value of
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* @return the absolute value
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*/
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public static float abs(float f)
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{
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return (f <= 0) ? 0 - f : f;
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}
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/**
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* Take the absolute value of the argument. (Absolute value means make
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* it positive.)
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*
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* @param d the number to take the absolute value of
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* @return the absolute value
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*/
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public static double abs(double d)
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{
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return (d <= 0) ? 0 - d : d;
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}
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/**
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* Return whichever argument is smaller.
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*
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* @param a the first number
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* @param b a second number
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* @return the smaller of the two numbers
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*/
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public static int min(int a, int b)
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{
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return (a < b) ? a : b;
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}
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/**
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* Return whichever argument is smaller.
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*
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* @param a the first number
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* @param b a second number
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* @return the smaller of the two numbers
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*/
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public static long min(long a, long b)
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{
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return (a < b) ? a : b;
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}
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/**
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* Return whichever argument is smaller. If either argument is NaN, the
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* result is NaN, and when comparing 0 and -0, -0 is always smaller.
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*
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* @param a the first number
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* @param b a second number
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* @return the smaller of the two numbers
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*/
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public static float min(float a, float b)
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{
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// this check for NaN, from JLS 15.21.1, saves a method call
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if (a != a)
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return a;
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// no need to check if b is NaN; < will work correctly
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// recall that -0.0 == 0.0, but [+-]0.0 - [+-]0.0 behaves special
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if (a == 0 && b == 0)
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return -(-a - b);
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return (a < b) ? a : b;
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}
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/**
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* Return whichever argument is smaller. If either argument is NaN, the
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* result is NaN, and when comparing 0 and -0, -0 is always smaller.
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*
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* @param a the first number
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* @param b a second number
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* @return the smaller of the two numbers
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*/
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public static double min(double a, double b)
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{
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// this check for NaN, from JLS 15.21.1, saves a method call
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if (a != a)
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return a;
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// no need to check if b is NaN; < will work correctly
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// recall that -0.0 == 0.0, but [+-]0.0 - [+-]0.0 behaves special
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if (a == 0 && b == 0)
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return -(-a - b);
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return (a < b) ? a : b;
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}
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/**
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* Return whichever argument is larger.
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*
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* @param a the first number
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* @param b a second number
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* @return the larger of the two numbers
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*/
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public static int max(int a, int b)
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{
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return (a > b) ? a : b;
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}
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/**
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* Return whichever argument is larger.
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*
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* @param a the first number
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* @param b a second number
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* @return the larger of the two numbers
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*/
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public static long max(long a, long b)
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{
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return (a > b) ? a : b;
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}
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/**
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* Return whichever argument is larger. If either argument is NaN, the
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* result is NaN, and when comparing 0 and -0, 0 is always larger.
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*
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* @param a the first number
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* @param b a second number
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* @return the larger of the two numbers
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*/
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public static float max(float a, float b)
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{
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// this check for NaN, from JLS 15.21.1, saves a method call
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if (a != a)
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return a;
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// no need to check if b is NaN; > will work correctly
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// recall that -0.0 == 0.0, but [+-]0.0 - [+-]0.0 behaves special
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if (a == 0 && b == 0)
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return a - -b;
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return (a > b) ? a : b;
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}
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/**
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* Return whichever argument is larger. If either argument is NaN, the
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* result is NaN, and when comparing 0 and -0, 0 is always larger.
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*
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* @param a the first number
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* @param b a second number
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* @return the larger of the two numbers
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*/
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public static double max(double a, double b)
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{
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// this check for NaN, from JLS 15.21.1, saves a method call
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if (a != a)
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return a;
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// no need to check if b is NaN; > will work correctly
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// recall that -0.0 == 0.0, but [+-]0.0 - [+-]0.0 behaves special
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if (a == 0 && b == 0)
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return a - -b;
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return (a > b) ? a : b;
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}
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/**
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* The trigonometric function <em>sin</em>. The sine of NaN or infinity is
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* NaN, and the sine of 0 retains its sign.
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*
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* @param a the angle (in radians)
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* @return sin(a)
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*/
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public static double sin(double a)
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{
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if (a == Double.NEGATIVE_INFINITY || ! (a < Double.POSITIVE_INFINITY))
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return Double.NaN;
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if (abs(a) <= PI / 4)
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return sin(a, 0);
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// Argument reduction needed.
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double[] y = new double[2];
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int n = remPiOver2(a, y);
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switch (n & 3)
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{
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case 0:
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return sin(y[0], y[1]);
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case 1:
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return cos(y[0], y[1]);
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case 2:
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return -sin(y[0], y[1]);
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default:
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return -cos(y[0], y[1]);
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}
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}
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/**
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* The trigonometric function <em>cos</em>. The cosine of NaN or infinity is
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* NaN.
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*
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* @param a the angle (in radians).
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* @return cos(a).
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*/
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public static double cos(double a)
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{
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if (a == Double.NEGATIVE_INFINITY || ! (a < Double.POSITIVE_INFINITY))
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return Double.NaN;
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if (abs(a) <= PI / 4)
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return cos(a, 0);
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// Argument reduction needed.
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double[] y = new double[2];
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int n = remPiOver2(a, y);
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switch (n & 3)
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{
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case 0:
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return cos(y[0], y[1]);
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case 1:
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return -sin(y[0], y[1]);
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case 2:
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return -cos(y[0], y[1]);
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default:
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return sin(y[0], y[1]);
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}
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}
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/**
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* The trigonometric function <em>tan</em>. The tangent of NaN or infinity
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* is NaN, and the tangent of 0 retains its sign.
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*
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* @param a the angle (in radians)
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* @return tan(a)
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*/
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public static double tan(double a)
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{
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if (a == Double.NEGATIVE_INFINITY || ! (a < Double.POSITIVE_INFINITY))
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return Double.NaN;
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if (abs(a) <= PI / 4)
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return tan(a, 0, false);
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// Argument reduction needed.
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double[] y = new double[2];
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int n = remPiOver2(a, y);
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return tan(y[0], y[1], (n & 1) == 1);
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}
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/**
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* The trigonometric function <em>arcsin</em>. The range of angles returned
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* is -pi/2 to pi/2 radians (-90 to 90 degrees). If the argument is NaN or
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* its absolute value is beyond 1, the result is NaN; and the arcsine of
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* 0 retains its sign.
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*
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* @param x the sin to turn back into an angle
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* @return arcsin(x)
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*/
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public static double asin(double x)
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{
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boolean negative = x < 0;
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if (negative)
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x = -x;
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if (! (x <= 1))
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return Double.NaN;
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if (x == 1)
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return negative ? -PI / 2 : PI / 2;
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if (x < 0.5)
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{
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if (x < 1 / TWO_27)
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return negative ? -x : x;
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double t = x * x;
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double p = t * (PS0 + t * (PS1 + t * (PS2 + t * (PS3 + t
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* (PS4 + t * PS5)))));
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double q = 1 + t * (QS1 + t * (QS2 + t * (QS3 + t * QS4)));
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return negative ? -x - x * (p / q) : x + x * (p / q);
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}
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double w = 1 - x; // 1>|x|>=0.5.
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double t = w * 0.5;
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double p = t * (PS0 + t * (PS1 + t * (PS2 + t * (PS3 + t
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* (PS4 + t * PS5)))));
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double q = 1 + t * (QS1 + t * (QS2 + t * (QS3 + t * QS4)));
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double s = sqrt(t);
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if (x >= 0.975)
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{
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w = p / q;
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t = PI / 2 - (2 * (s + s * w) - PI_L / 2);
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}
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else
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{
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w = (float) s;
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double c = (t - w * w) / (s + w);
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p = 2 * s * (p / q) - (PI_L / 2 - 2 * c);
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q = PI / 4 - 2 * w;
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t = PI / 4 - (p - q);
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}
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return negative ? -t : t;
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}
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|
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/**
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* The trigonometric function <em>arccos</em>. The range of angles returned
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* is 0 to pi radians (0 to 180 degrees). If the argument is NaN or
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* its absolute value is beyond 1, the result is NaN.
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*
|
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* @param x the cos to turn back into an angle
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* @return arccos(x)
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*/
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public static double acos(double x)
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{
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boolean negative = x < 0;
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if (negative)
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x = -x;
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if (! (x <= 1))
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return Double.NaN;
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if (x == 1)
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return negative ? PI : 0;
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if (x < 0.5)
|
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{
|
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if (x < 1 / TWO_57)
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return PI / 2;
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double z = x * x;
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double p = z * (PS0 + z * (PS1 + z * (PS2 + z * (PS3 + z
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* (PS4 + z * PS5)))));
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double q = 1 + z * (QS1 + z * (QS2 + z * (QS3 + z * QS4)));
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double r = x - (PI_L / 2 - x * (p / q));
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return negative ? PI / 2 + r : PI / 2 - r;
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}
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if (negative) // x<=-0.5.
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{
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double z = (1 + x) * 0.5;
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double p = z * (PS0 + z * (PS1 + z * (PS2 + z * (PS3 + z
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* (PS4 + z * PS5)))));
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double q = 1 + z * (QS1 + z * (QS2 + z * (QS3 + z * QS4)));
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double s = sqrt(z);
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double w = p / q * s - PI_L / 2;
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return PI - 2 * (s + w);
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}
|
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double z = (1 - x) * 0.5; // x>0.5.
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double s = sqrt(z);
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double df = (float) s;
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double c = (z - df * df) / (s + df);
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double p = z * (PS0 + z * (PS1 + z * (PS2 + z * (PS3 + z
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* (PS4 + z * PS5)))));
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double q = 1 + z * (QS1 + z * (QS2 + z * (QS3 + z * QS4)));
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double w = p / q * s + c;
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return 2 * (df + w);
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}
|
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|
|
/**
|
|
* The trigonometric function <em>arcsin</em>. The range of angles returned
|
|
* is -pi/2 to pi/2 radians (-90 to 90 degrees). If the argument is NaN, the
|
|
* result is NaN; and the arctangent of 0 retains its sign.
|
|
*
|
|
* @param x the tan to turn back into an angle
|
|
* @return arcsin(x)
|
|
* @see #atan2(double, double)
|
|
*/
|
|
public static double atan(double x)
|
|
{
|
|
double lo;
|
|
double hi;
|
|
boolean negative = x < 0;
|
|
if (negative)
|
|
x = -x;
|
|
if (x >= TWO_66)
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|
return negative ? -PI / 2 : PI / 2;
|
|
if (! (x >= 0.4375)) // |x|<7/16, or NaN.
|
|
{
|
|
if (! (x >= 1 / TWO_29)) // Small, or NaN.
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|
return negative ? -x : x;
|
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lo = hi = 0;
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|
}
|
|
else if (x < 1.1875)
|
|
{
|
|
if (x < 0.6875) // 7/16<=|x|<11/16.
|
|
{
|
|
x = (2 * x - 1) / (2 + x);
|
|
hi = ATAN_0_5H;
|
|
lo = ATAN_0_5L;
|
|
}
|
|
else // 11/16<=|x|<19/16.
|
|
{
|
|
x = (x - 1) / (x + 1);
|
|
hi = PI / 4;
|
|
lo = PI_L / 4;
|
|
}
|
|
}
|
|
else if (x < 2.4375) // 19/16<=|x|<39/16.
|
|
{
|
|
x = (x - 1.5) / (1 + 1.5 * x);
|
|
hi = ATAN_1_5H;
|
|
lo = ATAN_1_5L;
|
|
}
|
|
else // 39/16<=|x|<2**66.
|
|
{
|
|
x = -1 / x;
|
|
hi = PI / 2;
|
|
lo = PI_L / 2;
|
|
}
|
|
|
|
// Break sum from i=0 to 10 ATi*z**(i+1) into odd and even poly.
|
|
double z = x * x;
|
|
double w = z * z;
|
|
double s1 = z * (AT0 + w * (AT2 + w * (AT4 + w * (AT6 + w
|
|
* (AT8 + w * AT10)))));
|
|
double s2 = w * (AT1 + w * (AT3 + w * (AT5 + w * (AT7 + w * AT9))));
|
|
if (hi == 0)
|
|
return negative ? x * (s1 + s2) - x : x - x * (s1 + s2);
|
|
z = hi - ((x * (s1 + s2) - lo) - x);
|
|
return negative ? -z : z;
|
|
}
|
|
|
|
/**
|
|
* A special version of the trigonometric function <em>arctan</em>, for
|
|
* converting rectangular coordinates <em>(x, y)</em> to polar
|
|
* <em>(r, theta)</em>. This computes the arctangent of x/y in the range
|
|
* of -pi to pi radians (-180 to 180 degrees). Special cases:<ul>
|
|
* <li>If either argument is NaN, the result is NaN.</li>
|
|
* <li>If the first argument is positive zero and the second argument is
|
|
* positive, or the first argument is positive and finite and the second
|
|
* argument is positive infinity, then the result is positive zero.</li>
|
|
* <li>If the first argument is negative zero and the second argument is
|
|
* positive, or the first argument is negative and finite and the second
|
|
* argument is positive infinity, then the result is negative zero.</li>
|
|
* <li>If the first argument is positive zero and the second argument is
|
|
* negative, or the first argument is positive and finite and the second
|
|
* argument is negative infinity, then the result is the double value
|
|
* closest to pi.</li>
|
|
* <li>If the first argument is negative zero and the second argument is
|
|
* negative, or the first argument is negative and finite and the second
|
|
* argument is negative infinity, then the result is the double value
|
|
* closest to -pi.</li>
|
|
* <li>If the first argument is positive and the second argument is
|
|
* positive zero or negative zero, or the first argument is positive
|
|
* infinity and the second argument is finite, then the result is the
|
|
* double value closest to pi/2.</li>
|
|
* <li>If the first argument is negative and the second argument is
|
|
* positive zero or negative zero, or the first argument is negative
|
|
* infinity and the second argument is finite, then the result is the
|
|
* double value closest to -pi/2.</li>
|
|
* <li>If both arguments are positive infinity, then the result is the
|
|
* double value closest to pi/4.</li>
|
|
* <li>If the first argument is positive infinity and the second argument
|
|
* is negative infinity, then the result is the double value closest to
|
|
* 3*pi/4.</li>
|
|
* <li>If the first argument is negative infinity and the second argument
|
|
* is positive infinity, then the result is the double value closest to
|
|
* -pi/4.</li>
|
|
* <li>If both arguments are negative infinity, then the result is the
|
|
* double value closest to -3*pi/4.</li>
|
|
*
|
|
* </ul><p>This returns theta, the angle of the point. To get r, albeit
|
|
* slightly inaccurately, use sqrt(x*x+y*y).
|
|
*
|
|
* @param y the y position
|
|
* @param x the x position
|
|
* @return <em>theta</em> in the conversion of (x, y) to (r, theta)
|
|
* @see #atan(double)
|
|
*/
|
|
public static double atan2(double y, double x)
|
|
{
|
|
if (x != x || y != y)
|
|
return Double.NaN;
|
|
if (x == 1)
|
|
return atan(y);
|
|
if (x == Double.POSITIVE_INFINITY)
|
|
{
|
|
if (y == Double.POSITIVE_INFINITY)
|
|
return PI / 4;
|
|
if (y == Double.NEGATIVE_INFINITY)
|
|
return -PI / 4;
|
|
return 0 * y;
|
|
}
|
|
if (x == Double.NEGATIVE_INFINITY)
|
|
{
|
|
if (y == Double.POSITIVE_INFINITY)
|
|
return 3 * PI / 4;
|
|
if (y == Double.NEGATIVE_INFINITY)
|
|
return -3 * PI / 4;
|
|
return (1 / (0 * y) == Double.POSITIVE_INFINITY) ? PI : -PI;
|
|
}
|
|
if (y == 0)
|
|
{
|
|
if (1 / (0 * x) == Double.POSITIVE_INFINITY)
|
|
return y;
|
|
return (1 / y == Double.POSITIVE_INFINITY) ? PI : -PI;
|
|
}
|
|
if (y == Double.POSITIVE_INFINITY || y == Double.NEGATIVE_INFINITY
|
|
|| x == 0)
|
|
return y < 0 ? -PI / 2 : PI / 2;
|
|
|
|
double z = abs(y / x); // Safe to do y/x.
|
|
if (z > TWO_60)
|
|
z = PI / 2 + 0.5 * PI_L;
|
|
else if (x < 0 && z < 1 / TWO_60)
|
|
z = 0;
|
|
else
|
|
z = atan(z);
|
|
if (x > 0)
|
|
return y > 0 ? z : -z;
|
|
return y > 0 ? PI - (z - PI_L) : z - PI_L - PI;
|
|
}
|
|
|
|
/**
|
|
* Take <em>e</em><sup>a</sup>. The opposite of <code>log()</code>. If the
|
|
* argument is NaN, the result is NaN; if the argument is positive infinity,
|
|
* the result is positive infinity; and if the argument is negative
|
|
* infinity, the result is positive zero.
|
|
*
|
|
* @param x the number to raise to the power
|
|
* @return the number raised to the power of <em>e</em>
|
|
* @see #log(double)
|
|
* @see #pow(double, double)
|
|
*/
|
|
public static double exp(double x)
|
|
{
|
|
if (x != x)
|
|
return x;
|
|
if (x > EXP_LIMIT_H)
|
|
return Double.POSITIVE_INFINITY;
|
|
if (x < EXP_LIMIT_L)
|
|
return 0;
|
|
|
|
// Argument reduction.
|
|
double hi;
|
|
double lo;
|
|
int k;
|
|
double t = abs(x);
|
|
if (t > 0.5 * LN2)
|
|
{
|
|
if (t < 1.5 * LN2)
|
|
{
|
|
hi = t - LN2_H;
|
|
lo = LN2_L;
|
|
k = 1;
|
|
}
|
|
else
|
|
{
|
|
k = (int) (INV_LN2 * t + 0.5);
|
|
hi = t - k * LN2_H;
|
|
lo = k * LN2_L;
|
|
}
|
|
if (x < 0)
|
|
{
|
|
hi = -hi;
|
|
lo = -lo;
|
|
k = -k;
|
|
}
|
|
x = hi - lo;
|
|
}
|
|
else if (t < 1 / TWO_28)
|
|
return 1;
|
|
else
|
|
lo = hi = k = 0;
|
|
|
|
// Now x is in primary range.
|
|
t = x * x;
|
|
double c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
|
|
if (k == 0)
|
|
return 1 - (x * c / (c - 2) - x);
|
|
double y = 1 - (lo - x * c / (2 - c) - hi);
|
|
return scale(y, k);
|
|
}
|
|
|
|
/**
|
|
* Take ln(a) (the natural log). The opposite of <code>exp()</code>. If the
|
|
* argument is NaN or negative, the result is NaN; if the argument is
|
|
* positive infinity, the result is positive infinity; and if the argument
|
|
* is either zero, the result is negative infinity.
|
|
*
|
|
* <p>Note that the way to get log<sub>b</sub>(a) is to do this:
|
|
* <code>ln(a) / ln(b)</code>.
|
|
*
|
|
* @param x the number to take the natural log of
|
|
* @return the natural log of <code>a</code>
|
|
* @see #exp(double)
|
|
*/
|
|
public static double log(double x)
|
|
{
|
|
if (x == 0)
|
|
return Double.NEGATIVE_INFINITY;
|
|
if (x < 0)
|
|
return Double.NaN;
|
|
if (! (x < Double.POSITIVE_INFINITY))
|
|
return x;
|
|
|
|
// Normalize x.
|
|
long bits = Double.doubleToLongBits(x);
|
|
int exp = (int) (bits >> 52);
|
|
if (exp == 0) // Subnormal x.
|
|
{
|
|
x *= TWO_54;
|
|
bits = Double.doubleToLongBits(x);
|
|
exp = (int) (bits >> 52) - 54;
|
|
}
|
|
exp -= 1023; // Unbias exponent.
|
|
bits = (bits & 0x000fffffffffffffL) | 0x3ff0000000000000L;
|
|
x = Double.longBitsToDouble(bits);
|
|
if (x >= SQRT_2)
|
|
{
|
|
x *= 0.5;
|
|
exp++;
|
|
}
|
|
x--;
|
|
if (abs(x) < 1 / TWO_20)
|
|
{
|
|
if (x == 0)
|
|
return exp * LN2_H + exp * LN2_L;
|
|
double r = x * x * (0.5 - 1 / 3.0 * x);
|
|
if (exp == 0)
|
|
return x - r;
|
|
return exp * LN2_H - ((r - exp * LN2_L) - x);
|
|
}
|
|
double s = x / (2 + x);
|
|
double z = s * s;
|
|
double w = z * z;
|
|
double t1 = w * (LG2 + w * (LG4 + w * LG6));
|
|
double t2 = z * (LG1 + w * (LG3 + w * (LG5 + w * LG7)));
|
|
double r = t2 + t1;
|
|
if (bits >= 0x3ff6174a00000000L && bits < 0x3ff6b85200000000L)
|
|
{
|
|
double h = 0.5 * x * x; // Need more accuracy for x near sqrt(2).
|
|
if (exp == 0)
|
|
return x - (h - s * (h + r));
|
|
return exp * LN2_H - ((h - (s * (h + r) + exp * LN2_L)) - x);
|
|
}
|
|
if (exp == 0)
|
|
return x - s * (x - r);
|
|
return exp * LN2_H - ((s * (x - r) - exp * LN2_L) - x);
|
|
}
|
|
|
|
/**
|
|
* Take a square root. If the argument is NaN or negative, the result is
|
|
* NaN; if the argument is positive infinity, the result is positive
|
|
* infinity; and if the result is either zero, the result is the same.
|
|
*
|
|
* <p>For other roots, use pow(x, 1/rootNumber).
|
|
*
|
|
* @param x the numeric argument
|
|
* @return the square root of the argument
|
|
* @see #pow(double, double)
|
|
*/
|
|
public static double sqrt(double x)
|
|
{
|
|
if (x < 0)
|
|
return Double.NaN;
|
|
if (x == 0 || ! (x < Double.POSITIVE_INFINITY))
|
|
return x;
|
|
|
|
// Normalize x.
|
|
long bits = Double.doubleToLongBits(x);
|
|
int exp = (int) (bits >> 52);
|
|
if (exp == 0) // Subnormal x.
|
|
{
|
|
x *= TWO_54;
|
|
bits = Double.doubleToLongBits(x);
|
|
exp = (int) (bits >> 52) - 54;
|
|
}
|
|
exp -= 1023; // Unbias exponent.
|
|
bits = (bits & 0x000fffffffffffffL) | 0x0010000000000000L;
|
|
if ((exp & 1) == 1) // Odd exp, double x to make it even.
|
|
bits <<= 1;
|
|
exp >>= 1;
|
|
|
|
// Generate sqrt(x) bit by bit.
|
|
bits <<= 1;
|
|
long q = 0;
|
|
long s = 0;
|
|
long r = 0x0020000000000000L; // Move r right to left.
|
|
while (r != 0)
|
|
{
|
|
long t = s + r;
|
|
if (t <= bits)
|
|
{
|
|
s = t + r;
|
|
bits -= t;
|
|
q += r;
|
|
}
|
|
bits <<= 1;
|
|
r >>= 1;
|
|
}
|
|
|
|
// Use floating add to round correctly.
|
|
if (bits != 0)
|
|
q += q & 1;
|
|
return Double.longBitsToDouble((q >> 1) + ((exp + 1022L) << 52));
|
|
}
|
|
|
|
/**
|
|
* Raise a number to a power. Special cases:<ul>
|
|
* <li>If the second argument is positive or negative zero, then the result
|
|
* is 1.0.</li>
|
|
* <li>If the second argument is 1.0, then the result is the same as the
|
|
* first argument.</li>
|
|
* <li>If the second argument is NaN, then the result is NaN.</li>
|
|
* <li>If the first argument is NaN and the second argument is nonzero,
|
|
* then the result is NaN.</li>
|
|
* <li>If the absolute value of the first argument is greater than 1 and
|
|
* the second argument is positive infinity, or the absolute value of the
|
|
* first argument is less than 1 and the second argument is negative
|
|
* infinity, then the result is positive infinity.</li>
|
|
* <li>If the absolute value of the first argument is greater than 1 and
|
|
* the second argument is negative infinity, or the absolute value of the
|
|
* first argument is less than 1 and the second argument is positive
|
|
* infinity, then the result is positive zero.</li>
|
|
* <li>If the absolute value of the first argument equals 1 and the second
|
|
* argument is infinite, then the result is NaN.</li>
|
|
* <li>If the first argument is positive zero and the second argument is
|
|
* greater than zero, or the first argument is positive infinity and the
|
|
* second argument is less than zero, then the result is positive zero.</li>
|
|
* <li>If the first argument is positive zero and the second argument is
|
|
* less than zero, or the first argument is positive infinity and the
|
|
* second argument is greater than zero, then the result is positive
|
|
* infinity.</li>
|
|
* <li>If the first argument is negative zero and the second argument is
|
|
* greater than zero but not a finite odd integer, or the first argument is
|
|
* negative infinity and the second argument is less than zero but not a
|
|
* finite odd integer, then the result is positive zero.</li>
|
|
* <li>If the first argument is negative zero and the second argument is a
|
|
* positive finite odd integer, or the first argument is negative infinity
|
|
* and the second argument is a negative finite odd integer, then the result
|
|
* is negative zero.</li>
|
|
* <li>If the first argument is negative zero and the second argument is
|
|
* less than zero but not a finite odd integer, or the first argument is
|
|
* negative infinity and the second argument is greater than zero but not a
|
|
* finite odd integer, then the result is positive infinity.</li>
|
|
* <li>If the first argument is negative zero and the second argument is a
|
|
* negative finite odd integer, or the first argument is negative infinity
|
|
* and the second argument is a positive finite odd integer, then the result
|
|
* is negative infinity.</li>
|
|
* <li>If the first argument is less than zero and the second argument is a
|
|
* finite even integer, then the result is equal to the result of raising
|
|
* the absolute value of the first argument to the power of the second
|
|
* argument.</li>
|
|
* <li>If the first argument is less than zero and the second argument is a
|
|
* finite odd integer, then the result is equal to the negative of the
|
|
* result of raising the absolute value of the first argument to the power
|
|
* of the second argument.</li>
|
|
* <li>If the first argument is finite and less than zero and the second
|
|
* argument is finite and not an integer, then the result is NaN.</li>
|
|
* <li>If both arguments are integers, then the result is exactly equal to
|
|
* the mathematical result of raising the first argument to the power of
|
|
* the second argument if that result can in fact be represented exactly as
|
|
* a double value.</li>
|
|
*
|
|
* </ul><p>(In the foregoing descriptions, a floating-point value is
|
|
* considered to be an integer if and only if it is a fixed point of the
|
|
* method {@link #ceil(double)} or, equivalently, a fixed point of the
|
|
* method {@link #floor(double)}. A value is a fixed point of a one-argument
|
|
* method if and only if the result of applying the method to the value is
|
|
* equal to the value.)
|
|
*
|
|
* @param x the number to raise
|
|
* @param y the power to raise it to
|
|
* @return x<sup>y</sup>
|
|
*/
|
|
public static double pow(double x, double y)
|
|
{
|
|
// Special cases first.
|
|
if (y == 0)
|
|
return 1;
|
|
if (y == 1)
|
|
return x;
|
|
if (y == -1)
|
|
return 1 / x;
|
|
if (x != x || y != y)
|
|
return Double.NaN;
|
|
|
|
// When x < 0, yisint tells if y is not an integer (0), even(1),
|
|
// or odd (2).
|
|
int yisint = 0;
|
|
if (x < 0 && floor(y) == y)
|
|
yisint = (y % 2 == 0) ? 2 : 1;
|
|
double ax = abs(x);
|
|
double ay = abs(y);
|
|
|
|
// More special cases, of y.
|
|
if (ay == Double.POSITIVE_INFINITY)
|
|
{
|
|
if (ax == 1)
|
|
return Double.NaN;
|
|
if (ax > 1)
|
|
return y > 0 ? y : 0;
|
|
return y < 0 ? -y : 0;
|
|
}
|
|
if (y == 2)
|
|
return x * x;
|
|
if (y == 0.5)
|
|
return sqrt(x);
|
|
|
|
// More special cases, of x.
|
|
if (x == 0 || ax == Double.POSITIVE_INFINITY || ax == 1)
|
|
{
|
|
if (y < 0)
|
|
ax = 1 / ax;
|
|
if (x < 0)
|
|
{
|
|
if (x == -1 && yisint == 0)
|
|
ax = Double.NaN;
|
|
else if (yisint == 1)
|
|
ax = -ax;
|
|
}
|
|
return ax;
|
|
}
|
|
if (x < 0 && yisint == 0)
|
|
return Double.NaN;
|
|
|
|
// Now we can start!
|
|
double t;
|
|
double t1;
|
|
double t2;
|
|
double u;
|
|
double v;
|
|
double w;
|
|
if (ay > TWO_31)
|
|
{
|
|
if (ay > TWO_64) // Automatic over/underflow.
|
|
return ((ax < 1) ? y < 0 : y > 0) ? Double.POSITIVE_INFINITY : 0;
|
|
// Over/underflow if x is not close to one.
|
|
if (ax < 0.9999995231628418)
|
|
return y < 0 ? Double.POSITIVE_INFINITY : 0;
|
|
if (ax >= 1.0000009536743164)
|
|
return y > 0 ? Double.POSITIVE_INFINITY : 0;
|
|
// Now |1-x| is <= 2**-20, sufficient to compute
|
|
// log(x) by x-x^2/2+x^3/3-x^4/4.
|
|
t = x - 1;
|
|
w = t * t * (0.5 - t * (1 / 3.0 - t * 0.25));
|
|
u = INV_LN2_H * t;
|
|
v = t * INV_LN2_L - w * INV_LN2;
|
|
t1 = (float) (u + v);
|
|
t2 = v - (t1 - u);
|
|
}
|
|
else
|
|
{
|
|
long bits = Double.doubleToLongBits(ax);
|
|
int exp = (int) (bits >> 52);
|
|
if (exp == 0) // Subnormal x.
|
|
{
|
|
ax *= TWO_54;
|
|
bits = Double.doubleToLongBits(ax);
|
|
exp = (int) (bits >> 52) - 54;
|
|
}
|
|
exp -= 1023; // Unbias exponent.
|
|
ax = Double.longBitsToDouble((bits & 0x000fffffffffffffL)
|
|
| 0x3ff0000000000000L);
|
|
boolean k;
|
|
if (ax < SQRT_1_5) // |x|<sqrt(3/2).
|
|
k = false;
|
|
else if (ax < SQRT_3) // |x|<sqrt(3).
|
|
k = true;
|
|
else
|
|
{
|
|
k = false;
|
|
ax *= 0.5;
|
|
exp++;
|
|
}
|
|
|
|
// Compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5).
|
|
u = ax - (k ? 1.5 : 1);
|
|
v = 1 / (ax + (k ? 1.5 : 1));
|
|
double s = u * v;
|
|
double s_h = (float) s;
|
|
double t_h = (float) (ax + (k ? 1.5 : 1));
|
|
double t_l = ax - (t_h - (k ? 1.5 : 1));
|
|
double s_l = v * ((u - s_h * t_h) - s_h * t_l);
|
|
// Compute log(ax).
|
|
double s2 = s * s;
|
|
double r = s_l * (s_h + s) + s2 * s2
|
|
* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
|
|
s2 = s_h * s_h;
|
|
t_h = (float) (3.0 + s2 + r);
|
|
t_l = r - (t_h - 3.0 - s2);
|
|
// u+v = s*(1+...).
|
|
u = s_h * t_h;
|
|
v = s_l * t_h + t_l * s;
|
|
// 2/(3log2)*(s+...).
|
|
double p_h = (float) (u + v);
|
|
double p_l = v - (p_h - u);
|
|
double z_h = CP_H * p_h;
|
|
double z_l = CP_L * p_h + p_l * CP + (k ? DP_L : 0);
|
|
// log2(ax) = (s+..)*2/(3*log2) = exp + dp_h + z_h + z_l.
|
|
t = exp;
|
|
t1 = (float) (z_h + z_l + (k ? DP_H : 0) + t);
|
|
t2 = z_l - (t1 - t - (k ? DP_H : 0) - z_h);
|
|
}
|
|
|
|
// Split up y into y1+y2 and compute (y1+y2)*(t1+t2).
|
|
boolean negative = x < 0 && yisint == 1;
|
|
double y1 = (float) y;
|
|
double p_l = (y - y1) * t1 + y * t2;
|
|
double p_h = y1 * t1;
|
|
double z = p_l + p_h;
|
|
if (z >= 1024) // Detect overflow.
|
|
{
|
|
if (z > 1024 || p_l + OVT > z - p_h)
|
|
return negative ? Double.NEGATIVE_INFINITY
|
|
: Double.POSITIVE_INFINITY;
|
|
}
|
|
else if (z <= -1075) // Detect underflow.
|
|
{
|
|
if (z < -1075 || p_l <= z - p_h)
|
|
return negative ? -0.0 : 0;
|
|
}
|
|
|
|
// Compute 2**(p_h+p_l).
|
|
int n = round((float) z);
|
|
p_h -= n;
|
|
t = (float) (p_l + p_h);
|
|
u = t * LN2_H;
|
|
v = (p_l - (t - p_h)) * LN2 + t * LN2_L;
|
|
z = u + v;
|
|
w = v - (z - u);
|
|
t = z * z;
|
|
t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
|
|
double r = (z * t1) / (t1 - 2) - (w + z * w);
|
|
z = scale(1 - (r - z), n);
|
|
return negative ? -z : z;
|
|
}
|
|
|
|
/**
|
|
* Get the IEEE 754 floating point remainder on two numbers. This is the
|
|
* value of <code>x - y * <em>n</em></code>, where <em>n</em> is the closest
|
|
* double to <code>x / y</code> (ties go to the even n); for a zero
|
|
* remainder, the sign is that of <code>x</code>. If either argument is NaN,
|
|
* the first argument is infinite, or the second argument is zero, the result
|
|
* is NaN; if x is finite but y is infinite, the result is x.
|
|
*
|
|
* @param x the dividend (the top half)
|
|
* @param y the divisor (the bottom half)
|
|
* @return the IEEE 754-defined floating point remainder of x/y
|
|
* @see #rint(double)
|
|
*/
|
|
public static double IEEEremainder(double x, double y)
|
|
{
|
|
// Purge off exception values.
|
|
if (x == Double.NEGATIVE_INFINITY || ! (x < Double.POSITIVE_INFINITY)
|
|
|| y == 0 || y != y)
|
|
return Double.NaN;
|
|
|
|
boolean negative = x < 0;
|
|
x = abs(x);
|
|
y = abs(y);
|
|
if (x == y || x == 0)
|
|
return 0 * x; // Get correct sign.
|
|
|
|
// Achieve x < 2y, then take first shot at remainder.
|
|
if (y < TWO_1023)
|
|
x %= y + y;
|
|
|
|
// Now adjust x to get correct precision.
|
|
if (y < 4 / TWO_1023)
|
|
{
|
|
if (x + x > y)
|
|
{
|
|
x -= y;
|
|
if (x + x >= y)
|
|
x -= y;
|
|
}
|
|
}
|
|
else
|
|
{
|
|
y *= 0.5;
|
|
if (x > y)
|
|
{
|
|
x -= y;
|
|
if (x >= y)
|
|
x -= y;
|
|
}
|
|
}
|
|
return negative ? -x : x;
|
|
}
|
|
|
|
/**
|
|
* Take the nearest integer that is that is greater than or equal to the
|
|
* argument. If the argument is NaN, infinite, or zero, the result is the
|
|
* same; if the argument is between -1 and 0, the result is negative zero.
|
|
* Note that <code>Math.ceil(x) == -Math.floor(-x)</code>.
|
|
*
|
|
* @param a the value to act upon
|
|
* @return the nearest integer >= <code>a</code>
|
|
*/
|
|
public static double ceil(double a)
|
|
{
|
|
return -floor(-a);
|
|
}
|
|
|
|
/**
|
|
* Take the nearest integer that is that is less than or equal to the
|
|
* argument. If the argument is NaN, infinite, or zero, the result is the
|
|
* same. Note that <code>Math.ceil(x) == -Math.floor(-x)</code>.
|
|
*
|
|
* @param a the value to act upon
|
|
* @return the nearest integer <= <code>a</code>
|
|
*/
|
|
public static double floor(double a)
|
|
{
|
|
double x = abs(a);
|
|
if (! (x < TWO_52) || (long) a == a)
|
|
return a; // No fraction bits; includes NaN and infinity.
|
|
if (x < 1)
|
|
return a >= 0 ? 0 * a : -1; // Worry about signed zero.
|
|
return a < 0 ? (long) a - 1.0 : (long) a; // Cast to long truncates.
|
|
}
|
|
|
|
/**
|
|
* Take the nearest integer to the argument. If it is exactly between
|
|
* two integers, the even integer is taken. If the argument is NaN,
|
|
* infinite, or zero, the result is the same.
|
|
*
|
|
* @param a the value to act upon
|
|
* @return the nearest integer to <code>a</code>
|
|
*/
|
|
public static double rint(double a)
|
|
{
|
|
double x = abs(a);
|
|
if (! (x < TWO_52))
|
|
return a; // No fraction bits; includes NaN and infinity.
|
|
if (x <= 0.5)
|
|
return 0 * a; // Worry about signed zero.
|
|
if (x % 2 <= 0.5)
|
|
return (long) a; // Catch round down to even.
|
|
return (long) (a + (a < 0 ? -0.5 : 0.5)); // Cast to long truncates.
|
|
}
|
|
|
|
/**
|
|
* Take the nearest integer to the argument. This is equivalent to
|
|
* <code>(int) Math.floor(f + 0.5f)</code>. If the argument is NaN, the
|
|
* result is 0; otherwise if the argument is outside the range of int, the
|
|
* result will be Integer.MIN_VALUE or Integer.MAX_VALUE, as appropriate.
|
|
*
|
|
* @param f the argument to round
|
|
* @return the nearest integer to the argument
|
|
* @see Integer#MIN_VALUE
|
|
* @see Integer#MAX_VALUE
|
|
*/
|
|
public static int round(float f)
|
|
{
|
|
return (int) floor(f + 0.5f);
|
|
}
|
|
|
|
/**
|
|
* Take the nearest long to the argument. This is equivalent to
|
|
* <code>(long) Math.floor(d + 0.5)</code>. If the argument is NaN, the
|
|
* result is 0; otherwise if the argument is outside the range of long, the
|
|
* result will be Long.MIN_VALUE or Long.MAX_VALUE, as appropriate.
|
|
*
|
|
* @param d the argument to round
|
|
* @return the nearest long to the argument
|
|
* @see Long#MIN_VALUE
|
|
* @see Long#MAX_VALUE
|
|
*/
|
|
public static long round(double d)
|
|
{
|
|
return (long) floor(d + 0.5);
|
|
}
|
|
|
|
/**
|
|
* Get a random number. This behaves like Random.nextDouble(), seeded by
|
|
* System.currentTimeMillis() when first called. In other words, the number
|
|
* is from a pseudorandom sequence, and lies in the range [+0.0, 1.0).
|
|
* This random sequence is only used by this method, and is threadsafe,
|
|
* although you may want your own random number generator if it is shared
|
|
* among threads.
|
|
*
|
|
* @return a random number
|
|
* @see Random#nextDouble()
|
|
* @see System#currentTimeMillis()
|
|
*/
|
|
public static synchronized double random()
|
|
{
|
|
if (rand == null)
|
|
rand = new Random();
|
|
return rand.nextDouble();
|
|
}
|
|
|
|
/**
|
|
* Convert from degrees to radians. The formula for this is
|
|
* radians = degrees * (pi/180); however it is not always exact given the
|
|
* limitations of floating point numbers.
|
|
*
|
|
* @param degrees an angle in degrees
|
|
* @return the angle in radians
|
|
*/
|
|
public static double toRadians(double degrees)
|
|
{
|
|
return degrees * (PI / 180);
|
|
}
|
|
|
|
/**
|
|
* Convert from radians to degrees. The formula for this is
|
|
* degrees = radians * (180/pi); however it is not always exact given the
|
|
* limitations of floating point numbers.
|
|
*
|
|
* @param rads an angle in radians
|
|
* @return the angle in degrees
|
|
*/
|
|
public static double toDegrees(double rads)
|
|
{
|
|
return rads * (180 / PI);
|
|
}
|
|
|
|
/**
|
|
* Constants for scaling and comparing doubles by powers of 2. The compiler
|
|
* must automatically inline constructs like (1/TWO_54), so we don't list
|
|
* negative powers of two here.
|
|
*/
|
|
private static final double
|
|
TWO_16 = 0x10000, // Long bits 0x40f0000000000000L.
|
|
TWO_20 = 0x100000, // Long bits 0x4130000000000000L.
|
|
TWO_24 = 0x1000000, // Long bits 0x4170000000000000L.
|
|
TWO_27 = 0x8000000, // Long bits 0x41a0000000000000L.
|
|
TWO_28 = 0x10000000, // Long bits 0x41b0000000000000L.
|
|
TWO_29 = 0x20000000, // Long bits 0x41c0000000000000L.
|
|
TWO_31 = 0x80000000L, // Long bits 0x41e0000000000000L.
|
|
TWO_49 = 0x2000000000000L, // Long bits 0x4300000000000000L.
|
|
TWO_52 = 0x10000000000000L, // Long bits 0x4330000000000000L.
|
|
TWO_54 = 0x40000000000000L, // Long bits 0x4350000000000000L.
|
|
TWO_57 = 0x200000000000000L, // Long bits 0x4380000000000000L.
|
|
TWO_60 = 0x1000000000000000L, // Long bits 0x43b0000000000000L.
|
|
TWO_64 = 1.8446744073709552e19, // Long bits 0x43f0000000000000L.
|
|
TWO_66 = 7.378697629483821e19, // Long bits 0x4410000000000000L.
|
|
TWO_1023 = 8.98846567431158e307; // Long bits 0x7fe0000000000000L.
|
|
|
|
/**
|
|
* Super precision for 2/pi in 24-bit chunks, for use in
|
|
* {@link #remPiOver2()}.
|
|
*/
|
|
private static final int TWO_OVER_PI[] = {
|
|
0xa2f983, 0x6e4e44, 0x1529fc, 0x2757d1, 0xf534dd, 0xc0db62,
|
|
0x95993c, 0x439041, 0xfe5163, 0xabdebb, 0xc561b7, 0x246e3a,
|
|
0x424dd2, 0xe00649, 0x2eea09, 0xd1921c, 0xfe1deb, 0x1cb129,
|
|
0xa73ee8, 0x8235f5, 0x2ebb44, 0x84e99c, 0x7026b4, 0x5f7e41,
|
|
0x3991d6, 0x398353, 0x39f49c, 0x845f8b, 0xbdf928, 0x3b1ff8,
|
|
0x97ffde, 0x05980f, 0xef2f11, 0x8b5a0a, 0x6d1f6d, 0x367ecf,
|
|
0x27cb09, 0xb74f46, 0x3f669e, 0x5fea2d, 0x7527ba, 0xc7ebe5,
|
|
0xf17b3d, 0x0739f7, 0x8a5292, 0xea6bfb, 0x5fb11f, 0x8d5d08,
|
|
0x560330, 0x46fc7b, 0x6babf0, 0xcfbc20, 0x9af436, 0x1da9e3,
|
|
0x91615e, 0xe61b08, 0x659985, 0x5f14a0, 0x68408d, 0xffd880,
|
|
0x4d7327, 0x310606, 0x1556ca, 0x73a8c9, 0x60e27b, 0xc08c6b,
|
|
};
|
|
|
|
/**
|
|
* Super precision for pi/2 in 24-bit chunks, for use in
|
|
* {@link #remPiOver2()}.
|
|
*/
|
|
private static final double PI_OVER_TWO[] = {
|
|
1.570796251296997, // Long bits 0x3ff921fb40000000L.
|
|
7.549789415861596e-8, // Long bits 0x3e74442d00000000L.
|
|
5.390302529957765e-15, // Long bits 0x3cf8469880000000L.
|
|
3.282003415807913e-22, // Long bits 0x3b78cc5160000000L.
|
|
1.270655753080676e-29, // Long bits 0x39f01b8380000000L.
|
|
1.2293330898111133e-36, // Long bits 0x387a252040000000L.
|
|
2.7337005381646456e-44, // Long bits 0x36e3822280000000L.
|
|
2.1674168387780482e-51, // Long bits 0x3569f31d00000000L.
|
|
};
|
|
|
|
/**
|
|
* More constants related to pi, used in {@link #remPiOver2()} and
|
|
* elsewhere.
|
|
*/
|
|
private static final double
|
|
PI_L = 1.2246467991473532e-16, // Long bits 0x3ca1a62633145c07L.
|
|
PIO2_1 = 1.5707963267341256, // Long bits 0x3ff921fb54400000L.
|
|
PIO2_1L = 6.077100506506192e-11, // Long bits 0x3dd0b4611a626331L.
|
|
PIO2_2 = 6.077100506303966e-11, // Long bits 0x3dd0b4611a600000L.
|
|
PIO2_2L = 2.0222662487959506e-21, // Long bits 0x3ba3198a2e037073L.
|
|
PIO2_3 = 2.0222662487111665e-21, // Long bits 0x3ba3198a2e000000L.
|
|
PIO2_3L = 8.4784276603689e-32; // Long bits 0x397b839a252049c1L.
|
|
|
|
/**
|
|
* Natural log and square root constants, for calculation of
|
|
* {@link #exp(double)}, {@link #log(double)} and
|
|
* {@link #power(double, double)}. CP is 2/(3*ln(2)).
|
|
*/
|
|
private static final double
|
|
SQRT_1_5 = 1.224744871391589, // Long bits 0x3ff3988e1409212eL.
|
|
SQRT_2 = 1.4142135623730951, // Long bits 0x3ff6a09e667f3bcdL.
|
|
SQRT_3 = 1.7320508075688772, // Long bits 0x3ffbb67ae8584caaL.
|
|
EXP_LIMIT_H = 709.782712893384, // Long bits 0x40862e42fefa39efL.
|
|
EXP_LIMIT_L = -745.1332191019411, // Long bits 0xc0874910d52d3051L.
|
|
CP = 0.9617966939259756, // Long bits 0x3feec709dc3a03fdL.
|
|
CP_H = 0.9617967009544373, // Long bits 0x3feec709e0000000L.
|
|
CP_L = -7.028461650952758e-9, // Long bits 0xbe3e2fe0145b01f5L.
|
|
LN2 = 0.6931471805599453, // Long bits 0x3fe62e42fefa39efL.
|
|
LN2_H = 0.6931471803691238, // Long bits 0x3fe62e42fee00000L.
|
|
LN2_L = 1.9082149292705877e-10, // Long bits 0x3dea39ef35793c76L.
|
|
INV_LN2 = 1.4426950408889634, // Long bits 0x3ff71547652b82feL.
|
|
INV_LN2_H = 1.4426950216293335, // Long bits 0x3ff7154760000000L.
|
|
INV_LN2_L = 1.9259629911266175e-8; // Long bits 0x3e54ae0bf85ddf44L.
|
|
|
|
/**
|
|
* Constants for computing {@link #log(double)}.
|
|
*/
|
|
private static final double
|
|
LG1 = 0.6666666666666735, // Long bits 0x3fe5555555555593L.
|
|
LG2 = 0.3999999999940942, // Long bits 0x3fd999999997fa04L.
|
|
LG3 = 0.2857142874366239, // Long bits 0x3fd2492494229359L.
|
|
LG4 = 0.22222198432149784, // Long bits 0x3fcc71c51d8e78afL.
|
|
LG5 = 0.1818357216161805, // Long bits 0x3fc7466496cb03deL.
|
|
LG6 = 0.15313837699209373, // Long bits 0x3fc39a09d078c69fL.
|
|
LG7 = 0.14798198605116586; // Long bits 0x3fc2f112df3e5244L.
|
|
|
|
/**
|
|
* Constants for computing {@link #pow(double, double)}. L and P are
|
|
* coefficients for series; OVT is -(1024-log2(ovfl+.5ulp)); and DP is ???.
|
|
* The P coefficients also calculate {@link #exp(double)}.
|
|
*/
|
|
private static final double
|
|
L1 = 0.5999999999999946, // Long bits 0x3fe3333333333303L.
|
|
L2 = 0.4285714285785502, // Long bits 0x3fdb6db6db6fabffL.
|
|
L3 = 0.33333332981837743, // Long bits 0x3fd55555518f264dL.
|
|
L4 = 0.272728123808534, // Long bits 0x3fd17460a91d4101L.
|
|
L5 = 0.23066074577556175, // Long bits 0x3fcd864a93c9db65L.
|
|
L6 = 0.20697501780033842, // Long bits 0x3fca7e284a454eefL.
|
|
P1 = 0.16666666666666602, // Long bits 0x3fc555555555553eL.
|
|
P2 = -2.7777777777015593e-3, // Long bits 0xbf66c16c16bebd93L.
|
|
P3 = 6.613756321437934e-5, // Long bits 0x3f11566aaf25de2cL.
|
|
P4 = -1.6533902205465252e-6, // Long bits 0xbebbbd41c5d26bf1L.
|
|
P5 = 4.1381367970572385e-8, // Long bits 0x3e66376972bea4d0L.
|
|
DP_H = 0.5849624872207642, // Long bits 0x3fe2b80340000000L.
|
|
DP_L = 1.350039202129749e-8, // Long bits 0x3e4cfdeb43cfd006L.
|
|
OVT = 8.008566259537294e-17; // Long bits 0x3c971547652b82feL.
|
|
|
|
/**
|
|
* Coefficients for computing {@link #sin(double)}.
|
|
*/
|
|
private static final double
|
|
S1 = -0.16666666666666632, // Long bits 0xbfc5555555555549L.
|
|
S2 = 8.33333333332249e-3, // Long bits 0x3f8111111110f8a6L.
|
|
S3 = -1.984126982985795e-4, // Long bits 0xbf2a01a019c161d5L.
|
|
S4 = 2.7557313707070068e-6, // Long bits 0x3ec71de357b1fe7dL.
|
|
S5 = -2.5050760253406863e-8, // Long bits 0xbe5ae5e68a2b9cebL.
|
|
S6 = 1.58969099521155e-10; // Long bits 0x3de5d93a5acfd57cL.
|
|
|
|
/**
|
|
* Coefficients for computing {@link #cos(double)}.
|
|
*/
|
|
private static final double
|
|
C1 = 0.0416666666666666, // Long bits 0x3fa555555555554cL.
|
|
C2 = -1.388888888887411e-3, // Long bits 0xbf56c16c16c15177L.
|
|
C3 = 2.480158728947673e-5, // Long bits 0x3efa01a019cb1590L.
|
|
C4 = -2.7557314351390663e-7, // Long bits 0xbe927e4f809c52adL.
|
|
C5 = 2.087572321298175e-9, // Long bits 0x3e21ee9ebdb4b1c4L.
|
|
C6 = -1.1359647557788195e-11; // Long bits 0xbda8fae9be8838d4L.
|
|
|
|
/**
|
|
* Coefficients for computing {@link #tan(double)}.
|
|
*/
|
|
private static final double
|
|
T0 = 0.3333333333333341, // Long bits 0x3fd5555555555563L.
|
|
T1 = 0.13333333333320124, // Long bits 0x3fc111111110fe7aL.
|
|
T2 = 0.05396825397622605, // Long bits 0x3faba1ba1bb341feL.
|
|
T3 = 0.021869488294859542, // Long bits 0x3f9664f48406d637L.
|
|
T4 = 8.8632398235993e-3, // Long bits 0x3f8226e3e96e8493L.
|
|
T5 = 3.5920791075913124e-3, // Long bits 0x3f6d6d22c9560328L.
|
|
T6 = 1.4562094543252903e-3, // Long bits 0x3f57dbc8fee08315L.
|
|
T7 = 5.880412408202641e-4, // Long bits 0x3f4344d8f2f26501L.
|
|
T8 = 2.464631348184699e-4, // Long bits 0x3f3026f71a8d1068L.
|
|
T9 = 7.817944429395571e-5, // Long bits 0x3f147e88a03792a6L.
|
|
T10 = 7.140724913826082e-5, // Long bits 0x3f12b80f32f0a7e9L.
|
|
T11 = -1.8558637485527546e-5, // Long bits 0xbef375cbdb605373L.
|
|
T12 = 2.590730518636337e-5; // Long bits 0x3efb2a7074bf7ad4L.
|
|
|
|
/**
|
|
* Coefficients for computing {@link #asin(double)} and
|
|
* {@link #acos(double)}.
|
|
*/
|
|
private static final double
|
|
PS0 = 0.16666666666666666, // Long bits 0x3fc5555555555555L.
|
|
PS1 = -0.3255658186224009, // Long bits 0xbfd4d61203eb6f7dL.
|
|
PS2 = 0.20121253213486293, // Long bits 0x3fc9c1550e884455L.
|
|
PS3 = -0.04005553450067941, // Long bits 0xbfa48228b5688f3bL.
|
|
PS4 = 7.915349942898145e-4, // Long bits 0x3f49efe07501b288L.
|
|
PS5 = 3.479331075960212e-5, // Long bits 0x3f023de10dfdf709L.
|
|
QS1 = -2.403394911734414, // Long bits 0xc0033a271c8a2d4bL.
|
|
QS2 = 2.0209457602335057, // Long bits 0x40002ae59c598ac8L.
|
|
QS3 = -0.6882839716054533, // Long bits 0xbfe6066c1b8d0159L.
|
|
QS4 = 0.07703815055590194; // Long bits 0x3fb3b8c5b12e9282L.
|
|
|
|
/**
|
|
* Coefficients for computing {@link #atan(double)}.
|
|
*/
|
|
private static final double
|
|
ATAN_0_5H = 0.4636476090008061, // Long bits 0x3fddac670561bb4fL.
|
|
ATAN_0_5L = 2.2698777452961687e-17, // Long bits 0x3c7a2b7f222f65e2L.
|
|
ATAN_1_5H = 0.982793723247329, // Long bits 0x3fef730bd281f69bL.
|
|
ATAN_1_5L = 1.3903311031230998e-17, // Long bits 0x3c7007887af0cbbdL.
|
|
AT0 = 0.3333333333333293, // Long bits 0x3fd555555555550dL.
|
|
AT1 = -0.19999999999876483, // Long bits 0xbfc999999998ebc4L.
|
|
AT2 = 0.14285714272503466, // Long bits 0x3fc24924920083ffL.
|
|
AT3 = -0.11111110405462356, // Long bits 0xbfbc71c6fe231671L.
|
|
AT4 = 0.09090887133436507, // Long bits 0x3fb745cdc54c206eL.
|
|
AT5 = -0.0769187620504483, // Long bits 0xbfb3b0f2af749a6dL.
|
|
AT6 = 0.06661073137387531, // Long bits 0x3fb10d66a0d03d51L.
|
|
AT7 = -0.058335701337905735, // Long bits 0xbfadde2d52defd9aL.
|
|
AT8 = 0.049768779946159324, // Long bits 0x3fa97b4b24760debL.
|
|
AT9 = -0.036531572744216916, // Long bits 0xbfa2b4442c6a6c2fL.
|
|
AT10 = 0.016285820115365782; // Long bits 0x3f90ad3ae322da11L.
|
|
|
|
/**
|
|
* Helper function for reducing an angle to a multiple of pi/2 within
|
|
* [-pi/4, pi/4].
|
|
*
|
|
* @param x the angle; not infinity or NaN, and outside pi/4
|
|
* @param y an array of 2 doubles modified to hold the remander x % pi/2
|
|
* @return the quadrant of the result, mod 4: 0: [-pi/4, pi/4],
|
|
* 1: [pi/4, 3*pi/4], 2: [3*pi/4, 5*pi/4], 3: [-3*pi/4, -pi/4]
|
|
*/
|
|
private static int remPiOver2(double x, double[] y)
|
|
{
|
|
boolean negative = x < 0;
|
|
x = abs(x);
|
|
double z;
|
|
int n;
|
|
if (Configuration.DEBUG && (x <= PI / 4 || x != x
|
|
|| x == Double.POSITIVE_INFINITY))
|
|
throw new InternalError("Assertion failure");
|
|
if (x < 3 * PI / 4) // If |x| is small.
|
|
{
|
|
z = x - PIO2_1;
|
|
if ((float) x != (float) (PI / 2)) // 33+53 bit pi is good enough.
|
|
{
|
|
y[0] = z - PIO2_1L;
|
|
y[1] = z - y[0] - PIO2_1L;
|
|
}
|
|
else // Near pi/2, use 33+33+53 bit pi.
|
|
{
|
|
z -= PIO2_2;
|
|
y[0] = z - PIO2_2L;
|
|
y[1] = z - y[0] - PIO2_2L;
|
|
}
|
|
n = 1;
|
|
}
|
|
else if (x <= TWO_20 * PI / 2) // Medium size.
|
|
{
|
|
n = (int) (2 / PI * x + 0.5);
|
|
z = x - n * PIO2_1;
|
|
double w = n * PIO2_1L; // First round good to 85 bits.
|
|
y[0] = z - w;
|
|
if (n >= 32 || (float) x == (float) (w))
|
|
{
|
|
if (x / y[0] >= TWO_16) // Second iteration, good to 118 bits.
|
|
{
|
|
double t = z;
|
|
w = n * PIO2_2;
|
|
z = t - w;
|
|
w = n * PIO2_2L - (t - z - w);
|
|
y[0] = z - w;
|
|
if (x / y[0] >= TWO_49) // Third iteration, 151 bits accuracy.
|
|
{
|
|
t = z;
|
|
w = n * PIO2_3;
|
|
z = t - w;
|
|
w = n * PIO2_3L - (t - z - w);
|
|
y[0] = z - w;
|
|
}
|
|
}
|
|
}
|
|
y[1] = z - y[0] - w;
|
|
}
|
|
else
|
|
{
|
|
// All other (large) arguments.
|
|
int e0 = (int) (Double.doubleToLongBits(x) >> 52) - 1046;
|
|
z = scale(x, -e0); // e0 = ilogb(z) - 23.
|
|
double[] tx = new double[3];
|
|
for (int i = 0; i < 2; i++)
|
|
{
|
|
tx[i] = (int) z;
|
|
z = (z - tx[i]) * TWO_24;
|
|
}
|
|
tx[2] = z;
|
|
int nx = 2;
|
|
while (tx[nx] == 0)
|
|
nx--;
|
|
n = remPiOver2(tx, y, e0, nx);
|
|
}
|
|
if (negative)
|
|
{
|
|
y[0] = -y[0];
|
|
y[1] = -y[1];
|
|
return -n;
|
|
}
|
|
return n;
|
|
}
|
|
|
|
/**
|
|
* Helper function for reducing an angle to a multiple of pi/2 within
|
|
* [-pi/4, pi/4].
|
|
*
|
|
* @param x the positive angle, broken into 24-bit chunks
|
|
* @param y an array of 2 doubles modified to hold the remander x % pi/2
|
|
* @param e0 the exponent of x[0]
|
|
* @param nx the last index used in x
|
|
* @return the quadrant of the result, mod 4: 0: [-pi/4, pi/4],
|
|
* 1: [pi/4, 3*pi/4], 2: [3*pi/4, 5*pi/4], 3: [-3*pi/4, -pi/4]
|
|
*/
|
|
private static int remPiOver2(double[] x, double[] y, int e0, int nx)
|
|
{
|
|
int i;
|
|
int ih;
|
|
int n;
|
|
double fw;
|
|
double z;
|
|
int[] iq = new int[20];
|
|
double[] f = new double[20];
|
|
double[] q = new double[20];
|
|
boolean recompute = false;
|
|
|
|
// Initialize jk, jz, jv, q0; note that 3>q0.
|
|
int jk = 4;
|
|
int jz = jk;
|
|
int jv = max((e0 - 3) / 24, 0);
|
|
int q0 = e0 - 24 * (jv + 1);
|
|
|
|
// Set up f[0] to f[nx+jk] where f[nx+jk] = TWO_OVER_PI[jv+jk].
|
|
int j = jv - nx;
|
|
int m = nx + jk;
|
|
for (i = 0; i <= m; i++, j++)
|
|
f[i] = (j < 0) ? 0 : TWO_OVER_PI[j];
|
|
|
|
// Compute q[0],q[1],...q[jk].
|
|
for (i = 0; i <= jk; i++)
|
|
{
|
|
for (j = 0, fw = 0; j <= nx; j++)
|
|
fw += x[j] * f[nx + i - j];
|
|
q[i] = fw;
|
|
}
|
|
|
|
do
|
|
{
|
|
// Distill q[] into iq[] reversingly.
|
|
for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--)
|
|
{
|
|
fw = (int) (1 / TWO_24 * z);
|
|
iq[i] = (int) (z - TWO_24 * fw);
|
|
z = q[j - 1] + fw;
|
|
}
|
|
|
|
// Compute n.
|
|
z = scale(z, q0);
|
|
z -= 8 * floor(z * 0.125); // Trim off integer >= 8.
|
|
n = (int) z;
|
|
z -= n;
|
|
ih = 0;
|
|
if (q0 > 0) // Need iq[jz-1] to determine n.
|
|
{
|
|
i = iq[jz - 1] >> (24 - q0);
|
|
n += i;
|
|
iq[jz - 1] -= i << (24 - q0);
|
|
ih = iq[jz - 1] >> (23 - q0);
|
|
}
|
|
else if (q0 == 0)
|
|
ih = iq[jz - 1] >> 23;
|
|
else if (z >= 0.5)
|
|
ih = 2;
|
|
|
|
if (ih > 0) // If q > 0.5.
|
|
{
|
|
n += 1;
|
|
int carry = 0;
|
|
for (i = 0; i < jz; i++) // Compute 1-q.
|
|
{
|
|
j = iq[i];
|
|
if (carry == 0)
|
|
{
|
|
if (j != 0)
|
|
{
|
|
carry = 1;
|
|
iq[i] = 0x1000000 - j;
|
|
}
|
|
}
|
|
else
|
|
iq[i] = 0xffffff - j;
|
|
}
|
|
switch (q0)
|
|
{
|
|
case 1: // Rare case: chance is 1 in 12 for non-default.
|
|
iq[jz - 1] &= 0x7fffff;
|
|
break;
|
|
case 2:
|
|
iq[jz - 1] &= 0x3fffff;
|
|
}
|
|
if (ih == 2)
|
|
{
|
|
z = 1 - z;
|
|
if (carry != 0)
|
|
z -= scale(1, q0);
|
|
}
|
|
}
|
|
|
|
// Check if recomputation is needed.
|
|
if (z == 0)
|
|
{
|
|
j = 0;
|
|
for (i = jz - 1; i >= jk; i--)
|
|
j |= iq[i];
|
|
if (j == 0) // Need recomputation.
|
|
{
|
|
int k;
|
|
for (k = 1; iq[jk - k] == 0; k++); // k = no. of terms needed.
|
|
|
|
for (i = jz + 1; i <= jz + k; i++) // Add q[jz+1] to q[jz+k].
|
|
{
|
|
f[nx + i] = TWO_OVER_PI[jv + i];
|
|
for (j = 0, fw = 0; j <= nx; j++)
|
|
fw += x[j] * f[nx + i - j];
|
|
q[i] = fw;
|
|
}
|
|
jz += k;
|
|
recompute = true;
|
|
}
|
|
}
|
|
}
|
|
while (recompute);
|
|
|
|
// Chop off zero terms.
|
|
if (z == 0)
|
|
{
|
|
jz--;
|
|
q0 -= 24;
|
|
while (iq[jz] == 0)
|
|
{
|
|
jz--;
|
|
q0 -= 24;
|
|
}
|
|
}
|
|
else // Break z into 24-bit if necessary.
|
|
{
|
|
z = scale(z, -q0);
|
|
if (z >= TWO_24)
|
|
{
|
|
fw = (int) (1 / TWO_24 * z);
|
|
iq[jz] = (int) (z - TWO_24 * fw);
|
|
jz++;
|
|
q0 += 24;
|
|
iq[jz] = (int) fw;
|
|
}
|
|
else
|
|
iq[jz] = (int) z;
|
|
}
|
|
|
|
// Convert integer "bit" chunk to floating-point value.
|
|
fw = scale(1, q0);
|
|
for (i = jz; i >= 0; i--)
|
|
{
|
|
q[i] = fw * iq[i];
|
|
fw *= 1 / TWO_24;
|
|
}
|
|
|
|
// Compute PI_OVER_TWO[0,...,jk]*q[jz,...,0].
|
|
double[] fq = new double[20];
|
|
for (i = jz; i >= 0; i--)
|
|
{
|
|
fw = 0;
|
|
for (int k = 0; k <= jk && k <= jz - i; k++)
|
|
fw += PI_OVER_TWO[k] * q[i + k];
|
|
fq[jz - i] = fw;
|
|
}
|
|
|
|
// Compress fq[] into y[].
|
|
fw = 0;
|
|
for (i = jz; i >= 0; i--)
|
|
fw += fq[i];
|
|
y[0] = (ih == 0) ? fw : -fw;
|
|
fw = fq[0] - fw;
|
|
for (i = 1; i <= jz; i++)
|
|
fw += fq[i];
|
|
y[1] = (ih == 0) ? fw : -fw;
|
|
return n;
|
|
}
|
|
|
|
/**
|
|
* Helper method for scaling a double by a power of 2.
|
|
*
|
|
* @param x the double
|
|
* @param n the scale; |n| < 2048
|
|
* @return x * 2**n
|
|
*/
|
|
private static double scale(double x, int n)
|
|
{
|
|
if (Configuration.DEBUG && abs(n) >= 2048)
|
|
throw new InternalError("Assertion failure");
|
|
if (x == 0 || x == Double.NEGATIVE_INFINITY
|
|
|| ! (x < Double.POSITIVE_INFINITY) || n == 0)
|
|
return x;
|
|
long bits = Double.doubleToLongBits(x);
|
|
int exp = (int) (bits >> 52) & 0x7ff;
|
|
if (exp == 0) // Subnormal x.
|
|
{
|
|
x *= TWO_54;
|
|
exp = ((int) (Double.doubleToLongBits(x) >> 52) & 0x7ff) - 54;
|
|
}
|
|
exp += n;
|
|
if (exp > 0x7fe) // Overflow.
|
|
return Double.POSITIVE_INFINITY * x;
|
|
if (exp > 0) // Normal.
|
|
return Double.longBitsToDouble((bits & 0x800fffffffffffffL)
|
|
| ((long) exp << 52));
|
|
if (exp <= -54)
|
|
return 0 * x; // Underflow.
|
|
exp += 54; // Subnormal result.
|
|
x = Double.longBitsToDouble((bits & 0x800fffffffffffffL)
|
|
| ((long) exp << 52));
|
|
return x * (1 / TWO_54);
|
|
}
|
|
|
|
/**
|
|
* Helper trig function; computes sin in range [-pi/4, pi/4].
|
|
*
|
|
* @param x angle within about pi/4
|
|
* @param y tail of x, created by remPiOver2
|
|
* @return sin(x+y)
|
|
*/
|
|
private static double sin(double x, double y)
|
|
{
|
|
if (Configuration.DEBUG && abs(x + y) > 0.7854)
|
|
throw new InternalError("Assertion failure");
|
|
if (abs(x) < 1 / TWO_27)
|
|
return x; // If |x| ~< 2**-27, already know answer.
|
|
|
|
double z = x * x;
|
|
double v = z * x;
|
|
double r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6)));
|
|
if (y == 0)
|
|
return x + v * (S1 + z * r);
|
|
return x - ((z * (0.5 * y - v * r) - y) - v * S1);
|
|
}
|
|
|
|
/**
|
|
* Helper trig function; computes cos in range [-pi/4, pi/4].
|
|
*
|
|
* @param x angle within about pi/4
|
|
* @param y tail of x, created by remPiOver2
|
|
* @return cos(x+y)
|
|
*/
|
|
private static double cos(double x, double y)
|
|
{
|
|
if (Configuration.DEBUG && abs(x + y) > 0.7854)
|
|
throw new InternalError("Assertion failure");
|
|
x = abs(x);
|
|
if (x < 1 / TWO_27)
|
|
return 1; // If |x| ~< 2**-27, already know answer.
|
|
|
|
double z = x * x;
|
|
double r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
|
|
|
|
if (x < 0.3)
|
|
return 1 - (0.5 * z - (z * r - x * y));
|
|
|
|
double qx = (x > 0.78125) ? 0.28125 : (x * 0.25);
|
|
return 1 - qx - ((0.5 * z - qx) - (z * r - x * y));
|
|
}
|
|
|
|
/**
|
|
* Helper trig function; computes tan in range [-pi/4, pi/4].
|
|
*
|
|
* @param x angle within about pi/4
|
|
* @param y tail of x, created by remPiOver2
|
|
* @param invert true iff -1/tan should be returned instead
|
|
* @return tan(x+y)
|
|
*/
|
|
private static double tan(double x, double y, boolean invert)
|
|
{
|
|
// PI/2 is irrational, so no double is a perfect multiple of it.
|
|
if (Configuration.DEBUG && (abs(x + y) > 0.7854 || (x == 0 && invert)))
|
|
throw new InternalError("Assertion failure");
|
|
boolean negative = x < 0;
|
|
if (negative)
|
|
{
|
|
x = -x;
|
|
y = -y;
|
|
}
|
|
if (x < 1 / TWO_28) // If |x| ~< 2**-28, already know answer.
|
|
return (negative ? -1 : 1) * (invert ? -1 / x : x);
|
|
|
|
double z;
|
|
double w;
|
|
boolean large = x >= 0.6744;
|
|
if (large)
|
|
{
|
|
z = PI / 4 - x;
|
|
w = PI_L / 4 - y;
|
|
x = z + w;
|
|
y = 0;
|
|
}
|
|
z = x * x;
|
|
w = z * z;
|
|
// Break x**5*(T1+x**2*T2+...) into
|
|
// x**5(T1+x**4*T3+...+x**20*T11)
|
|
// + x**5(x**2*(T2+x**4*T4+...+x**22*T12)).
|
|
double r = T1 + w * (T3 + w * (T5 + w * (T7 + w * (T9 + w * T11))));
|
|
double v = z * (T2 + w * (T4 + w * (T6 + w * (T8 + w * (T10 + w * T12)))));
|
|
double s = z * x;
|
|
r = y + z * (s * (r + v) + y);
|
|
r += T0 * s;
|
|
w = x + r;
|
|
if (large)
|
|
{
|
|
v = invert ? -1 : 1;
|
|
return (negative ? -1 : 1) * (v - 2 * (x - (w * w / (w + v) - r)));
|
|
}
|
|
if (! invert)
|
|
return w;
|
|
|
|
// Compute -1.0/(x+r) accurately.
|
|
z = (float) w;
|
|
v = r - (z - x);
|
|
double a = -1 / w;
|
|
double t = (float) a;
|
|
return t + a * (1 + t * z + t * v);
|
|
}
|
|
}
|