+++ /dev/null
-/* java.lang.StrictMath -- common mathematical functions, strict Java
- Copyright (C) 1998, 2001, 2002 Free Software Foundation, Inc.
-
-This file is part of GNU Classpath.
-
-GNU Classpath is free software; you can redistribute it and/or modify
-it under the terms of the GNU General Public License as published by
-the Free Software Foundation; either version 2, or (at your option)
-any later version.
-
-GNU Classpath is distributed in the hope that it will be useful, but
-WITHOUT ANY WARRANTY; without even the implied warranty of
-MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
-General Public License for more details.
-
-You should have received a copy of the GNU General Public License
-along with GNU Classpath; see the file COPYING. If not, write to the
-Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
-02111-1307 USA.
-
-Linking this library statically or dynamically with other modules is
-making a combined work based on this library. Thus, the terms and
-conditions of the GNU General Public License cover the whole
-combination.
-
-As a special exception, the copyright holders of this library give you
-permission to link this library with independent modules to produce an
-executable, regardless of the license terms of these independent
-modules, and to copy and distribute the resulting executable under
-terms of your choice, provided that you also meet, for each linked
-independent module, the terms and conditions of the license of that
-module. An independent module is a module which is not derived from
-or based on this library. If you modify this library, you may extend
-this exception to your version of the library, but you are not
-obligated to do so. If you do not wish to do so, delete this
-exception statement from your version. */
-
-/*
- * Some of the algorithms in this class are in the public domain, as part
- * of fdlibm (freely-distributable math library), available at
- * http://www.netlib.org/fdlibm/, and carry the following copyright:
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunSoft, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-
-package java.lang;
-
-import java.util.Random;
-import gnu.classpath.Configuration;
-
-/**
- * Helper class containing useful mathematical functions and constants.
- * This class mirrors {@link Math}, but is 100% portable, because it uses
- * no native methods whatsoever. Also, these algorithms are all accurate
- * to less than 1 ulp, and execute in <code>strictfp</code> mode, while
- * Math is allowed to vary in its results for some functions. Unfortunately,
- * this usually means StrictMath has less efficiency and speed, as Math can
- * use native methods.
- *
- * <p>The source of the various algorithms used is the fdlibm library, at:<br>
- * <a href="http://www.netlib.org/fdlibm/">http://www.netlib.org/fdlibm/</a>
- *
- * Note that angles are specified in radians. Conversion functions are
- * provided for your convenience.
- *
- * @author Eric Blake <ebb9@email.byu.edu>
- * @since 1.3
- */
-public final strictfp class StrictMath
-{
- /**
- * StrictMath is non-instantiable.
- */
- private StrictMath()
- {
- }
-
- /**
- * A random number generator, initialized on first use.
- *
- * @see #random()
- */
- private static Random rand;
-
- /**
- * The most accurate approximation to the mathematical constant <em>e</em>:
- * <code>2.718281828459045</code>. Used in natural log and exp.
- *
- * @see #log(double)
- * @see #exp(double)
- */
- public static final double E
- = 2.718281828459045; // Long bits 0x4005bf0z8b145769L.
-
- /**
- * The most accurate approximation to the mathematical constant <em>pi</em>:
- * <code>3.141592653589793</code>. This is the ratio of a circle's diameter
- * to its circumference.
- */
- public static final double PI
- = 3.141592653589793; // Long bits 0x400921fb54442d18L.
-
- /**
- * Take the absolute value of the argument. (Absolute value means make
- * it positive.)
- *
- * <p>Note that the the largest negative value (Integer.MIN_VALUE) cannot
- * be made positive. In this case, because of the rules of negation in
- * a computer, MIN_VALUE is what will be returned.
- * This is a <em>negative</em> value. You have been warned.
- *
- * @param i the number to take the absolute value of
- * @return the absolute value
- * @see Integer#MIN_VALUE
- */
- public static int abs(int i)
- {
- return (i < 0) ? -i : i;
- }
-
- /**
- * Take the absolute value of the argument. (Absolute value means make
- * it positive.)
- *
- * <p>Note that the the largest negative value (Long.MIN_VALUE) cannot
- * be made positive. In this case, because of the rules of negation in
- * a computer, MIN_VALUE is what will be returned.
- * This is a <em>negative</em> value. You have been warned.
- *
- * @param l the number to take the absolute value of
- * @return the absolute value
- * @see Long#MIN_VALUE
- */
- public static long abs(long l)
- {
- return (l < 0) ? -l : l;
- }
-
- /**
- * Take the absolute value of the argument. (Absolute value means make
- * it positive.)
- *
- * @param f the number to take the absolute value of
- * @return the absolute value
- */
- public static float abs(float f)
- {
- return (f <= 0) ? 0 - f : f;
- }
-
- /**
- * Take the absolute value of the argument. (Absolute value means make
- * it positive.)
- *
- * @param d the number to take the absolute value of
- * @return the absolute value
- */
- public static double abs(double d)
- {
- return (d <= 0) ? 0 - d : d;
- }
-
- /**
- * Return whichever argument is smaller.
- *
- * @param a the first number
- * @param b a second number
- * @return the smaller of the two numbers
- */
- public static int min(int a, int b)
- {
- return (a < b) ? a : b;
- }
-
- /**
- * Return whichever argument is smaller.
- *
- * @param a the first number
- * @param b a second number
- * @return the smaller of the two numbers
- */
- public static long min(long a, long b)
- {
- return (a < b) ? a : b;
- }
-
- /**
- * Return whichever argument is smaller. If either argument is NaN, the
- * result is NaN, and when comparing 0 and -0, -0 is always smaller.
- *
- * @param a the first number
- * @param b a second number
- * @return the smaller of the two numbers
- */
- public static float min(float a, float b)
- {
- // this check for NaN, from JLS 15.21.1, saves a method call
- if (a != a)
- return a;
- // no need to check if b is NaN; < will work correctly
- // recall that -0.0 == 0.0, but [+-]0.0 - [+-]0.0 behaves special
- if (a == 0 && b == 0)
- return -(-a - b);
- return (a < b) ? a : b;
- }
-
- /**
- * Return whichever argument is smaller. If either argument is NaN, the
- * result is NaN, and when comparing 0 and -0, -0 is always smaller.
- *
- * @param a the first number
- * @param b a second number
- * @return the smaller of the two numbers
- */
- public static double min(double a, double b)
- {
- // this check for NaN, from JLS 15.21.1, saves a method call
- if (a != a)
- return a;
- // no need to check if b is NaN; < will work correctly
- // recall that -0.0 == 0.0, but [+-]0.0 - [+-]0.0 behaves special
- if (a == 0 && b == 0)
- return -(-a - b);
- return (a < b) ? a : b;
- }
-
- /**
- * Return whichever argument is larger.
- *
- * @param a the first number
- * @param b a second number
- * @return the larger of the two numbers
- */
- public static int max(int a, int b)
- {
- return (a > b) ? a : b;
- }
-
- /**
- * Return whichever argument is larger.
- *
- * @param a the first number
- * @param b a second number
- * @return the larger of the two numbers
- */
- public static long max(long a, long b)
- {
- return (a > b) ? a : b;
- }
-
- /**
- * Return whichever argument is larger. If either argument is NaN, the
- * result is NaN, and when comparing 0 and -0, 0 is always larger.
- *
- * @param a the first number
- * @param b a second number
- * @return the larger of the two numbers
- */
- public static float max(float a, float b)
- {
- // this check for NaN, from JLS 15.21.1, saves a method call
- if (a != a)
- return a;
- // no need to check if b is NaN; > will work correctly
- // recall that -0.0 == 0.0, but [+-]0.0 - [+-]0.0 behaves special
- if (a == 0 && b == 0)
- return a - -b;
- return (a > b) ? a : b;
- }
-
- /**
- * Return whichever argument is larger. If either argument is NaN, the
- * result is NaN, and when comparing 0 and -0, 0 is always larger.
- *
- * @param a the first number
- * @param b a second number
- * @return the larger of the two numbers
- */
- public static double max(double a, double b)
- {
- // this check for NaN, from JLS 15.21.1, saves a method call
- if (a != a)
- return a;
- // no need to check if b is NaN; > will work correctly
- // recall that -0.0 == 0.0, but [+-]0.0 - [+-]0.0 behaves special
- if (a == 0 && b == 0)
- return a - -b;
- return (a > b) ? a : b;
- }
-
- /**
- * The trigonometric function <em>sin</em>. The sine of NaN or infinity is
- * NaN, and the sine of 0 retains its sign.
- *
- * @param a the angle (in radians)
- * @return sin(a)
- */
- public static double sin(double a)
- {
- if (a == Double.NEGATIVE_INFINITY || ! (a < Double.POSITIVE_INFINITY))
- return Double.NaN;
-
- if (abs(a) <= PI / 4)
- return sin(a, 0);
-
- // Argument reduction needed.
- double[] y = new double[2];
- int n = remPiOver2(a, y);
- switch (n & 3)
- {
- case 0:
- return sin(y[0], y[1]);
- case 1:
- return cos(y[0], y[1]);
- case 2:
- return -sin(y[0], y[1]);
- default:
- return -cos(y[0], y[1]);
- }
- }
-
- /**
- * The trigonometric function <em>cos</em>. The cosine of NaN or infinity is
- * NaN.
- *
- * @param a the angle (in radians).
- * @return cos(a).
- */
- public static double cos(double a)
- {
- if (a == Double.NEGATIVE_INFINITY || ! (a < Double.POSITIVE_INFINITY))
- return Double.NaN;
-
- if (abs(a) <= PI / 4)
- return cos(a, 0);
-
- // Argument reduction needed.
- double[] y = new double[2];
- int n = remPiOver2(a, y);
- switch (n & 3)
- {
- case 0:
- return cos(y[0], y[1]);
- case 1:
- return -sin(y[0], y[1]);
- case 2:
- return -cos(y[0], y[1]);
- default:
- return sin(y[0], y[1]);
- }
- }
-
- /**
- * The trigonometric function <em>tan</em>. The tangent of NaN or infinity
- * is NaN, and the tangent of 0 retains its sign.
- *
- * @param a the angle (in radians)
- * @return tan(a)
- */
- public static double tan(double a)
- {
- if (a == Double.NEGATIVE_INFINITY || ! (a < Double.POSITIVE_INFINITY))
- return Double.NaN;
-
- if (abs(a) <= PI / 4)
- return tan(a, 0, false);
-
- // Argument reduction needed.
- double[] y = new double[2];
- int n = remPiOver2(a, y);
- return tan(y[0], y[1], (n & 1) == 1);
- }
-
- /**
- * 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 or
- * its absolute value is beyond 1, the result is NaN; and the arcsine of
- * 0 retains its sign.
- *
- * @param x the sin to turn back into an angle
- * @return arcsin(x)
- */
- public static double asin(double x)
- {
- boolean negative = x < 0;
- if (negative)
- x = -x;
- if (! (x <= 1))
- return Double.NaN;
- if (x == 1)
- return negative ? -PI / 2 : PI / 2;
- if (x < 0.5)
- {
- if (x < 1 / TWO_27)
- return negative ? -x : x;
- double t = x * x;
- double p = t * (PS0 + t * (PS1 + t * (PS2 + t * (PS3 + t
- * (PS4 + t * PS5)))));
- double q = 1 + t * (QS1 + t * (QS2 + t * (QS3 + t * QS4)));
- return negative ? -x - x * (p / q) : x + x * (p / q);
- }
- double w = 1 - x; // 1>|x|>=0.5.
- double t = w * 0.5;
- double p = t * (PS0 + t * (PS1 + t * (PS2 + t * (PS3 + t
- * (PS4 + t * PS5)))));
- double q = 1 + t * (QS1 + t * (QS2 + t * (QS3 + t * QS4)));
- double s = sqrt(t);
- if (x >= 0.975)
- {
- w = p / q;
- t = PI / 2 - (2 * (s + s * w) - PI_L / 2);
- }
- else
- {
- w = (float) s;
- double c = (t - w * w) / (s + w);
- p = 2 * s * (p / q) - (PI_L / 2 - 2 * c);
- q = PI / 4 - 2 * w;
- t = PI / 4 - (p - q);
- }
- return negative ? -t : t;
- }
-
- /**
- * The trigonometric function <em>arccos</em>. The range of angles returned
- * is 0 to pi radians (0 to 180 degrees). If the argument is NaN or
- * its absolute value is beyond 1, the result is NaN.
- *
- * @param x the cos to turn back into an angle
- * @return arccos(x)
- */
- public static double acos(double x)
- {
- boolean negative = x < 0;
- if (negative)
- x = -x;
- if (! (x <= 1))
- return Double.NaN;
- if (x == 1)
- return negative ? PI : 0;
- if (x < 0.5)
- {
- if (x < 1 / TWO_57)
- return PI / 2;
- double z = x * x;
- double p = z * (PS0 + z * (PS1 + z * (PS2 + z * (PS3 + z
- * (PS4 + z * PS5)))));
- double q = 1 + z * (QS1 + z * (QS2 + z * (QS3 + z * QS4)));
- double r = x - (PI_L / 2 - x * (p / q));
- return negative ? PI / 2 + r : PI / 2 - r;
- }
- if (negative) // x<=-0.5.
- {
- double z = (1 + x) * 0.5;
- double p = z * (PS0 + z * (PS1 + z * (PS2 + z * (PS3 + z
- * (PS4 + z * PS5)))));
- double q = 1 + z * (QS1 + z * (QS2 + z * (QS3 + z * QS4)));
- double s = sqrt(z);
- double w = p / q * s - PI_L / 2;
- return PI - 2 * (s + w);
- }
- double z = (1 - x) * 0.5; // x>0.5.
- double s = sqrt(z);
- double df = (float) s;
- double c = (z - df * df) / (s + df);
- double p = z * (PS0 + z * (PS1 + z * (PS2 + z * (PS3 + z
- * (PS4 + z * PS5)))));
- double q = 1 + z * (QS1 + z * (QS2 + z * (QS3 + z * QS4)));
- double w = p / q * s + c;
- return 2 * (df + w);
- }
-
- /**
- * 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)
- return negative ? -PI / 2 : PI / 2;
- if (! (x >= 0.4375)) // |x|<7/16, or NaN.
- {
- if (! (x >= 1 / TWO_29)) // Small, or NaN.
- return negative ? -x : x;
- lo = hi = 0;
- }
- 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 infinte, 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);
- }
-}
projects / msp430-gcc.git / blobdiff