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Diffstat (limited to 'libjava/java/lang/StrictMath.java')
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diff --git a/libjava/java/lang/StrictMath.java b/libjava/java/lang/StrictMath.java deleted file mode 100644 index 32bd354..0000000 --- a/libjava/java/lang/StrictMath.java +++ /dev/null @@ -1,1844 +0,0 @@ -/* java.lang.StrictMath -- common mathematical functions, strict Java - Copyright (C) 1998, 2001, 2002, 2003 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., 51 Franklin Street, Fifth Floor, Boston, MA -02110-1301 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 gnu.classpath.Configuration; - -import java.util.Random; - -/** - * 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 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); - } -} |