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JDK 11 java.base.jmod - Base Module

JDK 11 java.base.jmod is the JMOD file for JDK 11 Base module.

JDK 11 Base module compiled class files are stored in \fyicenter\jdk-11.0.1\jmods\java.base.jmod.

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JDK 11 Base module source code files are stored in \fyicenter\jdk-11.0.1\lib\src.zip\java.base.

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**⏎ java/lang/Math.java**

/* * Copyright (c) 1994, 2017, Oracle and/or its affiliates. All rights reserved. * ORACLE PROPRIETARY/CONFIDENTIAL. Use is subject to license terms. * * * * * * * * * * * * * * * * * * * * */ package java.lang; import java.math.BigDecimal; import java.util.Random; import jdk.internal.math.FloatConsts; import jdk.internal.math.DoubleConsts; import jdk.internal.HotSpotIntrinsicCandidate; /** * The class {@code Math} contains methods for performing basic * numeric operations such as the elementary exponential, logarithm, * square root, and trigonometric functions. * * <p>Unlike some of the numeric methods of class * {@code StrictMath}, all implementations of the equivalent * functions of class {@code Math} are not defined to return the * bit-for-bit same results. This relaxation permits * better-performing implementations where strict reproducibility is * not required. * * <p>By default many of the {@code Math} methods simply call * the equivalent method in {@code StrictMath} for their * implementation. Code generators are encouraged to use * platform-specific native libraries or microprocessor instructions, * where available, to provide higher-performance implementations of * {@code Math} methods. Such higher-performance * implementations still must conform to the specification for * {@code Math}. * * <p>The quality of implementation specifications concern two * properties, accuracy of the returned result and monotonicity of the * method. Accuracy of the floating-point {@code Math} methods is * measured in terms of <i>ulps</i>, units in the last place. For a * given floating-point format, an {@linkplain #ulp(double) ulp} of a * specific real number value is the distance between the two * floating-point values bracketing that numerical value. When * discussing the accuracy of a method as a whole rather than at a * specific argument, the number of ulps cited is for the worst-case * error at any argument. If a method always has an error less than * 0.5 ulps, the method always returns the floating-point number * nearest the exact result; such a method is <i>correctly * rounded</i>. A correctly rounded method is generally the best a * floating-point approximation can be; however, it is impractical for * many floating-point methods to be correctly rounded. Instead, for * the {@code Math} class, a larger error bound of 1 or 2 ulps is * allowed for certain methods. Informally, with a 1 ulp error bound, * when the exact result is a representable number, the exact result * should be returned as the computed result; otherwise, either of the * two floating-point values which bracket the exact result may be * returned. For exact results large in magnitude, one of the * endpoints of the bracket may be infinite. Besides accuracy at * individual arguments, maintaining proper relations between the * method at different arguments is also important. Therefore, most * methods with more than 0.5 ulp errors are required to be * <i>semi-monotonic</i>: whenever the mathematical function is * non-decreasing, so is the floating-point approximation, likewise, * whenever the mathematical function is non-increasing, so is the * floating-point approximation. Not all approximations that have 1 * ulp accuracy will automatically meet the monotonicity requirements. * * <p> * The platform uses signed two's complement integer arithmetic with * int and long primitive types. The developer should choose * the primitive type to ensure that arithmetic operations consistently * produce correct results, which in some cases means the operations * will not overflow the range of values of the computation. * The best practice is to choose the primitive type and algorithm to avoid * overflow. In cases where the size is {@code int} or {@code long} and * overflow errors need to be detected, the methods {@code addExact}, * {@code subtractExact}, {@code multiplyExact}, and {@code toIntExact} * throw an {@code ArithmeticException} when the results overflow. * For other arithmetic operations such as divide, absolute value, * increment by one, decrement by one, and negation, overflow occurs only with * a specific minimum or maximum value and should be checked against * the minimum or maximum as appropriate. * * @author unascribed * @author Joseph D. Darcy * @since 1.0 */ public final class Math { /** * Don't let anyone instantiate this class. */ private Math() {} /** * The {@code double} value that is closer than any other to * <i>e</i>, the base of the natural logarithms. */ public static final double E = 2.7182818284590452354; /** * The {@code double} value that is closer than any other to * <i>pi</i>, the ratio of the circumference of a circle to its * diameter. */ public static final double PI = 3.14159265358979323846; /** * Constant by which to multiply an angular value in degrees to obtain an * angular value in radians. */ private static final double DEGREES_TO_RADIANS = 0.017453292519943295; /** * Constant by which to multiply an angular value in radians to obtain an * angular value in degrees. */ private static final double RADIANS_TO_DEGREES = 57.29577951308232; /** * Returns the trigonometric sine of an angle. Special cases: * <ul><li>If the argument is NaN or an infinity, then the * result is NaN. * <li>If the argument is zero, then the result is a zero with the * same sign as the argument.</ul> * * <p>The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. * * @param a an angle, in radians. * @return the sine of the argument. */ @HotSpotIntrinsicCandidate public static double sin(double a) { return StrictMath.sin(a); // default impl. delegates to StrictMath } /** * Returns the trigonometric cosine of an angle. Special cases: * <ul><li>If the argument is NaN or an infinity, then the * result is NaN.</ul> * * <p>The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. * * @param a an angle, in radians. * @return the cosine of the argument. */ @HotSpotIntrinsicCandidate public static double cos(double a) { return StrictMath.cos(a); // default impl. delegates to StrictMath } /** * Returns the trigonometric tangent of an angle. Special cases: * <ul><li>If the argument is NaN or an infinity, then the result * is NaN. * <li>If the argument is zero, then the result is a zero with the * same sign as the argument.</ul> * * <p>The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. * * @param a an angle, in radians. * @return the tangent of the argument. */ @HotSpotIntrinsicCandidate public static double tan(double a) { return StrictMath.tan(a); // default impl. delegates to StrictMath } /** * Returns the arc sine of a value; the returned angle is in the * range -<i>pi</i>/2 through <i>pi</i>/2. Special cases: * <ul><li>If the argument is NaN or its absolute value is greater * than 1, then the result is NaN. * <li>If the argument is zero, then the result is a zero with the * same sign as the argument.</ul> * * <p>The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. * * @param a the value whose arc sine is to be returned. * @return the arc sine of the argument. */ public static double asin(double a) { return StrictMath.asin(a); // default impl. delegates to StrictMath } /** * Returns the arc cosine of a value; the returned angle is in the * range 0.0 through <i>pi</i>. Special case: * <ul><li>If the argument is NaN or its absolute value is greater * than 1, then the result is NaN.</ul> * * <p>The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. * * @param a the value whose arc cosine is to be returned. * @return the arc cosine of the argument. */ public static double acos(double a) { return StrictMath.acos(a); // default impl. delegates to StrictMath } /** * Returns the arc tangent of a value; the returned angle is in the * range -<i>pi</i>/2 through <i>pi</i>/2. Special cases: * <ul><li>If the argument is NaN, then the result is NaN. * <li>If the argument is zero, then the result is a zero with the * same sign as the argument.</ul> * * <p>The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. * * @param a the value whose arc tangent is to be returned. * @return the arc tangent of the argument. */ public static double atan(double a) { return StrictMath.atan(a); // default impl. delegates to StrictMath } /** * Converts an angle measured in degrees to an approximately * equivalent angle measured in radians. The conversion from * degrees to radians is generally inexact. * * @param angdeg an angle, in degrees * @return the measurement of the angle {@code angdeg} * in radians. * @since 1.2 */ public static double toRadians(double angdeg) { return angdeg * DEGREES_TO_RADIANS; } /** * Converts an angle measured in radians to an approximately * equivalent angle measured in degrees. The conversion from * radians to degrees is generally inexact; users should * <i>not</i> expect {@code cos(toRadians(90.0))} to exactly * equal {@code 0.0}. * * @param angrad an angle, in radians * @return the measurement of the angle {@code angrad} * in degrees. * @since 1.2 */ public static double toDegrees(double angrad) { return angrad * RADIANS_TO_DEGREES; } /** * Returns Euler's number <i>e</i> raised to the power of a * {@code double} value. Special cases: * <ul><li>If the argument is NaN, the result is NaN. * <li>If the argument is positive infinity, then the result is * positive infinity. * <li>If the argument is negative infinity, then the result is * positive zero.</ul> * * <p>The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. * * @param a the exponent to raise <i>e</i> to. * @return the value <i>e</i><sup>{@code a}</sup>, * where <i>e</i> is the base of the natural logarithms. */ @HotSpotIntrinsicCandidate public static double exp(double a) { return StrictMath.exp(a); // default impl. delegates to StrictMath } /** * Returns the natural logarithm (base <i>e</i>) of a {@code double} * value. Special cases: * <ul><li>If the argument is NaN or less than zero, then the result * is NaN. * <li>If the argument is positive infinity, then the result is * positive infinity. * <li>If the argument is positive zero or negative zero, then the * result is negative infinity.</ul> * * <p>The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. * * @param a a value * @return the value ln {@code a}, the natural logarithm of * {@code a}. */ @HotSpotIntrinsicCandidate public static double log(double a) { return StrictMath.log(a); // default impl. delegates to StrictMath } /** * Returns the base 10 logarithm of a {@code double} value. * Special cases: * * <ul><li>If the argument is NaN or less than zero, then the result * is NaN. * <li>If the argument is positive infinity, then the result is * positive infinity. * <li>If the argument is positive zero or negative zero, then the * result is negative infinity. * <li> If the argument is equal to 10<sup><i>n</i></sup> for * integer <i>n</i>, then the result is <i>n</i>. * </ul> * * <p>The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. * * @param a a value * @return the base 10 logarithm of {@code a}. * @since 1.5 */ @HotSpotIntrinsicCandidate public static double log10(double a) { return StrictMath.log10(a); // default impl. delegates to StrictMath } /** * Returns the correctly rounded positive square root of a * {@code double} value. * Special cases: * <ul><li>If the argument is NaN or less than zero, then the result * is NaN. * <li>If the argument is positive infinity, then the result is positive * infinity. * <li>If the argument is positive zero or negative zero, then the * result is the same as the argument.</ul> * Otherwise, the result is the {@code double} value closest to * the true mathematical square root of the argument value. * * @param a a value. * @return the positive square root of {@code a}. * If the argument is NaN or less than zero, the result is NaN. */ @HotSpotIntrinsicCandidate public static double sqrt(double a) { return StrictMath.sqrt(a); // default impl. delegates to StrictMath // Note that hardware sqrt instructions // frequently can be directly used by JITs // and should be much faster than doing // Math.sqrt in software. } /** * Returns the cube root of a {@code double} value. For * positive finite {@code x}, {@code cbrt(-x) == * -cbrt(x)}; that is, the cube root of a negative value is * the negative of the cube root of that value's magnitude. * * Special cases: * * <ul> * * <li>If the argument is NaN, then the result is NaN. * * <li>If the argument is infinite, then the result is an infinity * with the same sign as the argument. * * <li>If the argument is zero, then the result is a zero with the * same sign as the argument. * * </ul> * * <p>The computed result must be within 1 ulp of the exact result. * * @param a a value. * @return the cube root of {@code a}. * @since 1.5 */ public static double cbrt(double a) { return StrictMath.cbrt(a); } /** * Computes the remainder operation on two arguments as prescribed * by the IEEE 754 standard. * The remainder value is mathematically equal to * <code>f1 - f2</code> × <i>n</i>, * where <i>n</i> is the mathematical integer closest to the exact * mathematical value of the quotient {@code f1/f2}, and if two * mathematical integers are equally close to {@code f1/f2}, * then <i>n</i> is the integer that is even. If the remainder is * zero, its sign is the same as the sign of the first argument. * Special cases: * <ul><li>If either argument is NaN, or the first argument is infinite, * or the second argument is positive zero or negative zero, then the * result is NaN. * <li>If the first argument is finite and the second argument is * infinite, then the result is the same as the first argument.</ul> * * @param f1 the dividend. * @param f2 the divisor. * @return the remainder when {@code f1} is divided by * {@code f2}. */ public static double IEEEremainder(double f1, double f2) { return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath } /** * Returns the smallest (closest to negative infinity) * {@code double} value that is greater than or equal to the * argument and is equal to a mathematical integer. Special cases: * <ul><li>If the argument value is already equal to a * mathematical integer, then the result is the same as the * argument. <li>If the argument is NaN or an infinity or * positive zero or negative zero, then the result is the same as * the argument. <li>If the argument value is less than zero but * greater than -1.0, then the result is negative zero.</ul> Note * that the value of {@code Math.ceil(x)} is exactly the * value of {@code -Math.floor(-x)}. * * * @param a a value. * @return the smallest (closest to negative infinity) * floating-point value that is greater than or equal to * the argument and is equal to a mathematical integer. */ public static double ceil(double a) { return StrictMath.ceil(a); // default impl. delegates to StrictMath } /** * Returns the largest (closest to positive infinity) * {@code double} value that is less than or equal to the * argument and is equal to a mathematical integer. Special cases: * <ul><li>If the argument value is already equal to a * mathematical integer, then the result is the same as the * argument. <li>If the argument is NaN or an infinity or * positive zero or negative zero, then the result is the same as * the argument.</ul> * * @param a a value. * @return the largest (closest to positive infinity) * floating-point value that less than or equal to the argument * and is equal to a mathematical integer. */ public static double floor(double a) { return StrictMath.floor(a); // default impl. delegates to StrictMath } /** * Returns the {@code double} value that is closest in value * to the argument and is equal to a mathematical integer. If two * {@code double} values that are mathematical integers are * equally close, the result is the integer value that is * even. Special cases: * <ul><li>If the argument value is already equal to a mathematical * integer, then the result is the same as the argument. * <li>If the argument is NaN or an infinity or positive zero or negative * zero, then the result is the same as the argument.</ul> * * @param a a {@code double} value. * @return the closest floating-point value to {@code a} that is * equal to a mathematical integer. */ public static double rint(double a) { return StrictMath.rint(a); // default impl. delegates to StrictMath } /** * Returns the angle <i>theta</i> from the conversion of rectangular * coordinates ({@code x}, {@code y}) to polar * coordinates (r, <i>theta</i>). * This method computes the phase <i>theta</i> by computing an arc tangent * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special * cases: * <ul><li>If either argument is NaN, then the result is NaN. * <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>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>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 * {@code double} value closest to <i>pi</i>. * <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 * {@code double} value closest to -<i>pi</i>. * <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 * {@code double} value closest to <i>pi</i>/2. * <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 * {@code double} value closest to -<i>pi</i>/2. * <li>If both arguments are positive infinity, then the result is the * {@code double} value closest to <i>pi</i>/4. * <li>If the first argument is positive infinity and the second argument * is negative infinity, then the result is the {@code double} * value closest to 3*<i>pi</i>/4. * <li>If the first argument is negative infinity and the second argument * is positive infinity, then the result is the {@code double} value * closest to -<i>pi</i>/4. * <li>If both arguments are negative infinity, then the result is the * {@code double} value closest to -3*<i>pi</i>/4.</ul> * * <p>The computed result must be within 2 ulps of the exact result. * Results must be semi-monotonic. * * @param y the ordinate coordinate * @param x the abscissa coordinate * @return the <i>theta</i> component of the point * (<i>r</i>, <i>theta</i>) * in polar coordinates that corresponds to the point * (<i>x</i>, <i>y</i>) in Cartesian coordinates. */ @HotSpotIntrinsicCandidate public static double atan2(double y, double x) { return StrictMath.atan2(y, x); // default impl. delegates to StrictMath } /** * Returns the value of the first argument raised to the power of the * second argument. Special cases: * * <ul><li>If the second argument is positive or negative zero, then the * result is 1.0. * <li>If the second argument is 1.0, then the result is the same as the * first argument. * <li>If the second argument is NaN, then the result is NaN. * <li>If the first argument is NaN and the second argument is nonzero, * then the result is NaN. * * <li>If * <ul> * <li>the absolute value of the first argument is greater than 1 * and the second argument is positive infinity, or * <li>the absolute value of the first argument is less than 1 and * the second argument is negative infinity, * </ul> * then the result is positive infinity. * * <li>If * <ul> * <li>the absolute value of the first argument is greater than 1 and * the second argument is negative infinity, or * <li>the absolute value of the * first argument is less than 1 and the second argument is positive * infinity, * </ul> * then the result is positive zero. * * <li>If the absolute value of the first argument equals 1 and the * second argument is infinite, then the result is NaN. * * <li>If * <ul> * <li>the first argument is positive zero and the second argument * is greater than zero, or * <li>the first argument is positive infinity and the second * argument is less than zero, * </ul> * then the result is positive zero. * * <li>If * <ul> * <li>the first argument is positive zero and the second argument * is less than zero, or * <li>the first argument is positive infinity and the second * argument is greater than zero, * </ul> * then the result is positive infinity. * * <li>If * <ul> * <li>the first argument is negative zero and the second argument * is greater than zero but not a finite odd integer, or * <li>the first argument is negative infinity and the second * argument is less than zero but not a finite odd integer, * </ul> * then the result is positive zero. * * <li>If * <ul> * <li>the first argument is negative zero and the second argument * is a positive finite odd integer, or * <li>the first argument is negative infinity and the second * argument is a negative finite odd integer, * </ul> * then the result is negative zero. * * <li>If * <ul> * <li>the first argument is negative zero and the second argument * is less than zero but not a finite odd integer, or * <li>the first argument is negative infinity and the second * argument is greater than zero but not a finite odd integer, * </ul> * then the result is positive infinity. * * <li>If * <ul> * <li>the first argument is negative zero and the second argument * is a negative finite odd integer, or * <li>the first argument is negative infinity and the second * argument is a positive finite odd integer, * </ul> * then the result is negative infinity. * * <li>If the first argument is finite and less than zero * <ul> * <li> if the second argument is a finite even integer, the * result is equal to the result of raising the absolute value of * the first argument to the power of the second argument * * <li>if the second argument is a finite odd integer, 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>if the second argument is finite and not an integer, then * the result is NaN. * </ul> * * <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 {@code double} value.</ul> * * <p>(In the foregoing descriptions, a floating-point value is * considered to be an integer if and only if it is finite and a * fixed point of the method {@link #ceil ceil} or, * equivalently, a fixed point of the method {@link #floor * floor}. 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.) * * <p>The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. * * @param a the base. * @param b the exponent. * @return the value {@code a}<sup>{@code b}</sup>. */ @HotSpotIntrinsicCandidate public static double pow(double a, double b) { return StrictMath.pow(a, b); // default impl. delegates to StrictMath } /** * Returns the closest {@code int} to the argument, with ties * rounding to positive infinity. * * <p> * Special cases: * <ul><li>If the argument is NaN, the result is 0. * <li>If the argument is negative infinity or any value less than or * equal to the value of {@code Integer.MIN_VALUE}, the result is * equal to the value of {@code Integer.MIN_VALUE}. * <li>If the argument is positive infinity or any value greater than or * equal to the value of {@code Integer.MAX_VALUE}, the result is * equal to the value of {@code Integer.MAX_VALUE}.</ul> * * @param a a floating-point value to be rounded to an integer. * @return the value of the argument rounded to the nearest * {@code int} value. * @see java.lang.Integer#MAX_VALUE * @see java.lang.Integer#MIN_VALUE */ public static int round(float a) { int intBits = Float.floatToRawIntBits(a); int biasedExp = (intBits & FloatConsts.EXP_BIT_MASK) >> (FloatConsts.SIGNIFICAND_WIDTH - 1); int shift = (FloatConsts.SIGNIFICAND_WIDTH - 2 + FloatConsts.EXP_BIAS) - biasedExp; if ((shift & -32) == 0) { // shift >= 0 && shift < 32 // a is a finite number such that pow(2,-32) <= ulp(a) < 1 int r = ((intBits & FloatConsts.SIGNIF_BIT_MASK) | (FloatConsts.SIGNIF_BIT_MASK + 1)); if (intBits < 0) { r = -r; } // In the comments below each Java expression evaluates to the value // the corresponding mathematical expression: // (r) evaluates to a / ulp(a) // (r >> shift) evaluates to floor(a * 2) // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2) // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2) return ((r >> shift) + 1) >> 1; } else { // a is either // - a finite number with abs(a) < exp(2,FloatConsts.SIGNIFICAND_WIDTH-32) < 1/2 // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer // - an infinity or NaN return (int) a; } } /** * Returns the closest {@code long} to the argument, with ties * rounding to positive infinity. * * <p>Special cases: * <ul><li>If the argument is NaN, the result is 0. * <li>If the argument is negative infinity or any value less than or * equal to the value of {@code Long.MIN_VALUE}, the result is * equal to the value of {@code Long.MIN_VALUE}. * <li>If the argument is positive infinity or any value greater than or * equal to the value of {@code Long.MAX_VALUE}, the result is * equal to the value of {@code Long.MAX_VALUE}.</ul> * * @param a a floating-point value to be rounded to a * {@code long}. * @return the value of the argument rounded to the nearest * {@code long} value. * @see java.lang.Long#MAX_VALUE * @see java.lang.Long#MIN_VALUE */ public static long round(double a) { long longBits = Double.doubleToRawLongBits(a); long biasedExp = (longBits & DoubleConsts.EXP_BIT_MASK) >> (DoubleConsts.SIGNIFICAND_WIDTH - 1); long shift = (DoubleConsts.SIGNIFICAND_WIDTH - 2 + DoubleConsts.EXP_BIAS) - biasedExp; if ((shift & -64) == 0) { // shift >= 0 && shift < 64 // a is a finite number such that pow(2,-64) <= ulp(a) < 1 long r = ((longBits & DoubleConsts.SIGNIF_BIT_MASK) | (DoubleConsts.SIGNIF_BIT_MASK + 1)); if (longBits < 0) { r = -r; } // In the comments below each Java expression evaluates to the value // the corresponding mathematical expression: // (r) evaluates to a / ulp(a) // (r >> shift) evaluates to floor(a * 2) // ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2) // (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2) return ((r >> shift) + 1) >> 1; } else { // a is either // - a finite number with abs(a) < exp(2,DoubleConsts.SIGNIFICAND_WIDTH-64) < 1/2 // - a finite number with ulp(a) >= 1 and hence a is a mathematical integer // - an infinity or NaN return (long) a; } } private static final class RandomNumberGeneratorHolder { static final Random randomNumberGenerator = new Random(); } /** * Returns a {@code double} value with a positive sign, greater * than or equal to {@code 0.0} and less than {@code 1.0}. * Returned values are chosen pseudorandomly with (approximately) * uniform distribution from that range. * * <p>When this method is first called, it creates a single new * pseudorandom-number generator, exactly as if by the expression * * <blockquote>{@code new java.util.Random()}</blockquote> * * This new pseudorandom-number generator is used thereafter for * all calls to this method and is used nowhere else. * * <p>This method is properly synchronized to allow correct use by * more than one thread. However, if many threads need to generate * pseudorandom numbers at a great rate, it may reduce contention * for each thread to have its own pseudorandom-number generator. * * @apiNote * As the largest {@code double} value less than {@code 1.0} * is {@code Math.nextDown(1.0)}, a value {@code x} in the closed range * {@code [x1,x2]} where {@code x1<=x2} may be defined by the statements * * <blockquote><pre>{@code * double f = Math.random()/Math.nextDown(1.0); * double x = x1*(1.0 - f) + x2*f; * }</pre></blockquote> * * @return a pseudorandom {@code double} greater than or equal * to {@code 0.0} and less than {@code 1.0}. * @see #nextDown(double) * @see Random#nextDouble() */ public static double random() { return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble(); } /** * Returns the sum of its arguments, * throwing an exception if the result overflows an {@code int}. * * @param x the first value * @param y the second value * @return the result * @throws ArithmeticException if the result overflows an int * @since 1.8 */ @HotSpotIntrinsicCandidate public static int addExact(int x, int y) { int r = x + y; // HD 2-12 Overflow iff both arguments have the opposite sign of the result if (((x ^ r) & (y ^ r)) < 0) { throw new ArithmeticException("integer overflow"); } return r; } /** * Returns the sum of its arguments, * throwing an exception if the result overflows a {@code long}. * * @param x the first value * @param y the second value * @return the result * @throws ArithmeticException if the result overflows a long * @since 1.8 */ @HotSpotIntrinsicCandidate public static long addExact(long x, long y) { long r = x + y; // HD 2-12 Overflow iff both arguments have the opposite sign of the result if (((x ^ r) & (y ^ r)) < 0) { throw new ArithmeticException("long overflow"); } return r; } /** * Returns the difference of the arguments, * throwing an exception if the result overflows an {@code int}. * * @param x the first value * @param y the second value to subtract from the first * @return the result * @throws ArithmeticException if the result overflows an int * @since 1.8 */ @HotSpotIntrinsicCandidate public static int subtractExact(int x, int y) { int r = x - y; // HD 2-12 Overflow iff the arguments have different signs and // the sign of the result is different from the sign of x if (((x ^ y) & (x ^ r)) < 0) { throw new ArithmeticException("integer overflow"); } return r; } /** * Returns the difference of the arguments, * throwing an exception if the result overflows a {@code long}. * * @param x the first value * @param y the second value to subtract from the first * @return the result * @throws ArithmeticException if the result overflows a long * @since 1.8 */ @HotSpotIntrinsicCandidate public static long subtractExact(long x, long y) { long r = x - y; // HD 2-12 Overflow iff the arguments have different signs and // the sign of the result is different from the sign of x if (((x ^ y) & (x ^ r)) < 0) { throw new ArithmeticException("long overflow"); } return r; } /** * Returns the product of the arguments, * throwing an exception if the result overflows an {@code int}. * * @param x the first value * @param y the second value * @return the result * @throws ArithmeticException if the result overflows an int * @since 1.8 */ @HotSpotIntrinsicCandidate public static int multiplyExact(int x, int y) { long r = (long)x * (long)y; if ((int)r != r) { throw new ArithmeticException("integer overflow"); } return (int)r; } /** * Returns the product of the arguments, throwing an exception if the result * overflows a {@code long}. * * @param x the first value * @param y the second value * @return the result * @throws ArithmeticException if the result overflows a long * @since 9 */ public static long multiplyExact(long x, int y) { return multiplyExact(x, (long)y); } /** * Returns the product of the arguments, * throwing an exception if the result overflows a {@code long}. * * @param x the first value * @param y the second value * @return the result * @throws ArithmeticException if the result overflows a long * @since 1.8 */ @HotSpotIntrinsicCandidate public static long multiplyExact(long x, long y) { long r = x * y; long ax = Math.abs(x); long ay = Math.abs(y); if (((ax | ay) >>> 31 != 0)) { // Some bits greater than 2^31 that might cause overflow // Check the result using the divide operator // and check for the special case of Long.MIN_VALUE * -1 if (((y != 0) && (r / y != x)) || (x == Long.MIN_VALUE && y == -1)) { throw new ArithmeticException("long overflow"); } } return r; } /** * Returns the argument incremented by one, throwing an exception if the * result overflows an {@code int}. * * @param a the value to increment * @return the result * @throws ArithmeticException if the result overflows an int * @since 1.8 */ @HotSpotIntrinsicCandidate public static int incrementExact(int a) { if (a == Integer.MAX_VALUE) { throw new ArithmeticException("integer overflow"); } return a + 1; } /** * Returns the argument incremented by one, throwing an exception if the * result overflows a {@code long}. * * @param a the value to increment * @return the result * @throws ArithmeticException if the result overflows a long * @since 1.8 */ @HotSpotIntrinsicCandidate public static long incrementExact(long a) { if (a == Long.MAX_VALUE) { throw new ArithmeticException("long overflow"); } return a + 1L; } /** * Returns the argument decremented by one, throwing an exception if the * result overflows an {@code int}. * * @param a the value to decrement * @return the result * @throws ArithmeticException if the result overflows an int * @since 1.8 */ @HotSpotIntrinsicCandidate public static int decrementExact(int a) { if (a == Integer.MIN_VALUE) { throw new ArithmeticException("integer overflow"); } return a - 1; } /** * Returns the argument decremented by one, throwing an exception if the * result overflows a {@code long}. * * @param a the value to decrement * @return the result * @throws ArithmeticException if the result overflows a long * @since 1.8 */ @HotSpotIntrinsicCandidate public static long decrementExact(long a) { if (a == Long.MIN_VALUE) { throw new ArithmeticException("long overflow"); } return a - 1L; } /** * Returns the negation of the argument, throwing an exception if the * result overflows an {@code int}. * * @param a the value to negate * @return the result * @throws ArithmeticException if the result overflows an int * @since 1.8 */ @HotSpotIntrinsicCandidate public static int negateExact(int a) { if (a == Integer.MIN_VALUE) { throw new ArithmeticException("integer overflow"); } return -a; } /** * Returns the negation of the argument, throwing an exception if the * result overflows a {@code long}. * * @param a the value to negate * @return the result * @throws ArithmeticException if the result overflows a long * @since 1.8 */ @HotSpotIntrinsicCandidate public static long negateExact(long a) { if (a == Long.MIN_VALUE) { throw new ArithmeticException("long overflow"); } return -a; } /** * Returns the value of the {@code long} argument; * throwing an exception if the value overflows an {@code int}. * * @param value the long value * @return the argument as an int * @throws ArithmeticException if the {@code argument} overflows an int * @since 1.8 */ public static int toIntExact(long value) { if ((int)value != value) { throw new ArithmeticException("integer overflow"); } return (int)value; } /** * Returns the exact mathematical product of the arguments. * * @param x the first value * @param y the second value * @return the result * @since 9 */ public static long multiplyFull(int x, int y) { return (long)x * (long)y; } /** * Returns as a {@code long} the most significant 64 bits of the 128-bit * product of two 64-bit factors. * * @param x the first value * @param y the second value * @return the result * @since 9 */ @HotSpotIntrinsicCandidate public static long multiplyHigh(long x, long y) { if (x < 0 || y < 0) { // Use technique from section 8-2 of Henry S. Warren, Jr., // Hacker's Delight (2nd ed.) (Addison Wesley, 2013), 173-174. long x1 = x >> 32; long x2 = x & 0xFFFFFFFFL; long y1 = y >> 32; long y2 = y & 0xFFFFFFFFL; long z2 = x2 * y2; long t = x1 * y2 + (z2 >>> 32); long z1 = t & 0xFFFFFFFFL; long z0 = t >> 32; z1 += x2 * y1; return x1 * y1 + z0 + (z1 >> 32); } else { // Use Karatsuba technique with two base 2^32 digits. long x1 = x >>> 32; long y1 = y >>> 32; long x2 = x & 0xFFFFFFFFL; long y2 = y & 0xFFFFFFFFL; long A = x1 * y1; long B = x2 * y2; long C = (x1 + x2) * (y1 + y2); long K = C - A - B; return (((B >>> 32) + K) >>> 32) + A; } } /** * Returns the largest (closest to positive infinity) * {@code int} value that is less than or equal to the algebraic quotient. * There is one special case, if the dividend is the * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1}, * then integer overflow occurs and * the result is equal to {@code Integer.MIN_VALUE}. * <p> * Normal integer division operates under the round to zero rounding mode * (truncation). This operation instead acts under the round toward * negative infinity (floor) rounding mode. * The floor rounding mode gives different results from truncation * when the exact result is negative. * <ul> * <li>If the signs of the arguments are the same, the results of * {@code floorDiv} and the {@code /} operator are the same. <br> * For example, {@code floorDiv(4, 3) == 1} and {@code (4 / 3) == 1}.</li> * <li>If the signs of the arguments are different, the quotient is negative and * {@code floorDiv} returns the integer less than or equal to the quotient * and the {@code /} operator returns the integer closest to zero.<br> * For example, {@code floorDiv(-4, 3) == -2}, * whereas {@code (-4 / 3) == -1}. * </li> * </ul> * * @param x the dividend * @param y the divisor * @return the largest (closest to positive infinity) * {@code int} value that is less than or equal to the algebraic quotient. * @throws ArithmeticException if the divisor {@code y} is zero * @see #floorMod(int, int) * @see #floor(double) * @since 1.8 */ public static int floorDiv(int x, int y) { int r = x / y; // if the signs are different and modulo not zero, round down if ((x ^ y) < 0 && (r * y != x)) { r--; } return r; } /** * Returns the largest (closest to positive infinity) * {@code long} value that is less than or equal to the algebraic quotient. * There is one special case, if the dividend is the * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1}, * then integer overflow occurs and * the result is equal to {@code Long.MIN_VALUE}. * <p> * Normal integer division operates under the round to zero rounding mode * (truncation). This operation instead acts under the round toward * negative infinity (floor) rounding mode. * The floor rounding mode gives different results from truncation * when the exact result is negative. * <p> * For examples, see {@link #floorDiv(int, int)}. * * @param x the dividend * @param y the divisor * @return the largest (closest to positive infinity) * {@code int} value that is less than or equal to the algebraic quotient. * @throws ArithmeticException if the divisor {@code y} is zero * @see #floorMod(long, int) * @see #floor(double) * @since 9 */ public static long floorDiv(long x, int y) { return floorDiv(x, (long)y); } /** * Returns the largest (closest to positive infinity) * {@code long} value that is less than or equal to the algebraic quotient. * There is one special case, if the dividend is the * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1}, * then integer overflow occurs and * the result is equal to {@code Long.MIN_VALUE}. * <p> * Normal integer division operates under the round to zero rounding mode * (truncation). This operation instead acts under the round toward * negative infinity (floor) rounding mode. * The floor rounding mode gives different results from truncation * when the exact result is negative. * <p> * For examples, see {@link #floorDiv(int, int)}. * * @param x the dividend * @param y the divisor * @return the largest (closest to positive infinity) * {@code long} value that is less than or equal to the algebraic quotient. * @throws ArithmeticException if the divisor {@code y} is zero * @see #floorMod(long, long) * @see #floor(double) * @since 1.8 */ public static long floorDiv(long x, long y) { long r = x / y; // if the signs are different and modulo not zero, round down if ((x ^ y) < 0 && (r * y != x)) { r--; } return r; } /** * Returns the floor modulus of the {@code int} arguments. * <p> * The floor modulus is {@code x - (floorDiv(x, y) * y)}, * has the same sign as the divisor {@code y}, and * is in the range of {@code -abs(y) < r < +abs(y)}. * * <p> * The relationship between {@code floorDiv} and {@code floorMod} is such that: * <ul> * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x} * </ul> * <p> * The difference in values between {@code floorMod} and * the {@code %} operator is due to the difference between * {@code floorDiv} that returns the integer less than or equal to the quotient * and the {@code /} operator that returns the integer closest to zero. * <p> * Examples: * <ul> * <li>If the signs of the arguments are the same, the results * of {@code floorMod} and the {@code %} operator are the same. <br> * <ul> * <li>{@code floorMod(4, 3) == 1}; and {@code (4 % 3) == 1}</li> * </ul> * <li>If the signs of the arguments are different, the results differ from the {@code %} operator.<br> * <ul> * <li>{@code floorMod(+4, -3) == -2}; and {@code (+4 % -3) == +1} </li> * <li>{@code floorMod(-4, +3) == +2}; and {@code (-4 % +3) == -1} </li> * <li>{@code floorMod(-4, -3) == -1}; and {@code (-4 % -3) == -1 } </li> * </ul> * </li> * </ul> * <p> * If the signs of arguments are unknown and a positive modulus * is needed it can be computed as {@code (floorMod(x, y) + abs(y)) % abs(y)}. * * @param x the dividend * @param y the divisor * @return the floor modulus {@code x - (floorDiv(x, y) * y)} * @throws ArithmeticException if the divisor {@code y} is zero * @see #floorDiv(int, int) * @since 1.8 */ public static int floorMod(int x, int y) { return x - floorDiv(x, y) * y; } /** * Returns the floor modulus of the {@code long} and {@code int} arguments. * <p> * The floor modulus is {@code x - (floorDiv(x, y) * y)}, * has the same sign as the divisor {@code y}, and * is in the range of {@code -abs(y) < r < +abs(y)}. * * <p> * The relationship between {@code floorDiv} and {@code floorMod} is such that: * <ul> * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x} * </ul> * <p> * For examples, see {@link #floorMod(int, int)}. * * @param x the dividend * @param y the divisor * @return the floor modulus {@code x - (floorDiv(x, y) * y)} * @throws ArithmeticException if the divisor {@code y} is zero * @see #floorDiv(long, int) * @since 9 */ public static int floorMod(long x, int y) { // Result cannot overflow the range of int. return (int)(x - floorDiv(x, y) * y); } /** * Returns the floor modulus of the {@code long} arguments. * <p> * The floor modulus is {@code x - (floorDiv(x, y) * y)}, * has the same sign as the divisor {@code y}, and * is in the range of {@code -abs(y) < r < +abs(y)}. * * <p> * The relationship between {@code floorDiv} and {@code floorMod} is such that: * <ul> * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x} * </ul> * <p> * For examples, see {@link #floorMod(int, int)}. * * @param x the dividend * @param y the divisor * @return the floor modulus {@code x - (floorDiv(x, y) * y)} * @throws ArithmeticException if the divisor {@code y} is zero * @see #floorDiv(long, long) * @since 1.8 */ public static long floorMod(long x, long y) { return x - floorDiv(x, y) * y; } /** * Returns the absolute value of an {@code int} value. * If the argument is not negative, the argument is returned. * If the argument is negative, the negation of the argument is returned. * * <p>Note that if the argument is equal to the value of * {@link Integer#MIN_VALUE}, the most negative representable * {@code int} value, the result is that same value, which is * negative. * * @param a the argument whose absolute value is to be determined * @return the absolute value of the argument. */ public static int abs(int a) { return (a < 0) ? -a : a; } /** * Returns the absolute value of a {@code long} value. * If the argument is not negative, the argument is returned. * If the argument is negative, the negation of the argument is returned. * * <p>Note that if the argument is equal to the value of * {@link Long#MIN_VALUE}, the most negative representable * {@code long} value, the result is that same value, which * is negative. * * @param a the argument whose absolute value is to be determined * @return the absolute value of the argument. */ public static long abs(long a) { return (a < 0) ? -a : a; } /** * Returns the absolute value of a {@code float} value. * If the argument is not negative, the argument is returned. * If the argument is negative, the negation of the argument is returned. * Special cases: * <ul><li>If the argument is positive zero or negative zero, the * result is positive zero. * <li>If the argument is infinite, the result is positive infinity. * <li>If the argument is NaN, the result is NaN.</ul> * * @apiNote As implied by the above, one valid implementation of * this method is given by the expression below which computes a * {@code float} with the same exponent and significand as the * argument but with a guaranteed zero sign bit indicating a * positive value:<br> * {@code Float.intBitsToFloat(0x7fffffff & Float.floatToRawIntBits(a))} * * @param a the argument whose absolute value is to be determined * @return the absolute value of the argument. */ public static float abs(float a) { return (a <= 0.0F) ? 0.0F - a : a; } /** * Returns the absolute value of a {@code double} value. * If the argument is not negative, the argument is returned. * If the argument is negative, the negation of the argument is returned. * Special cases: * <ul><li>If the argument is positive zero or negative zero, the result * is positive zero. * <li>If the argument is infinite, the result is positive infinity. * <li>If the argument is NaN, the result is NaN.</ul> * * @apiNote As implied by the above, one valid implementation of * this method is given by the expression below which computes a * {@code double} with the same exponent and significand as the * argument but with a guaranteed zero sign bit indicating a * positive value:<br> * {@code Double.longBitsToDouble((Double.doubleToRawLongBits(a)<<1)>>>1)} * * @param a the argument whose absolute value is to be determined * @return the absolute value of the argument. */ @HotSpotIntrinsicCandidate public static double abs(double a) { return (a <= 0.0D) ? 0.0D - a : a; } /** * Returns the greater of two {@code int} values. That is, the * result is the argument closer to the value of * {@link Integer#MAX_VALUE}. If the arguments have the same value, * the result is that same value. * * @param a an argument. * @param b another argument. * @return the larger of {@code a} and {@code b}. */ @HotSpotIntrinsicCandidate public static int max(int a, int b) { return (a >= b) ? a : b; } /** * Returns the greater of two {@code long} values. That is, the * result is the argument closer to the value of * {@link Long#MAX_VALUE}. If the arguments have the same value, * the result is that same value. * * @param a an argument. * @param b another argument. * @return the larger of {@code a} and {@code b}. */ public static long max(long a, long b) { return (a >= b) ? a : b; } // Use raw bit-wise conversions on guaranteed non-NaN arguments. private static final long negativeZeroFloatBits = Float.floatToRawIntBits(-0.0f); private static final long negativeZeroDoubleBits = Double.doubleToRawLongBits(-0.0d); /** * Returns the greater of two {@code float} values. That is, * the result is the argument closer to positive infinity. If the * arguments have the same value, the result is that same * value. If either value is NaN, then the result is NaN. Unlike * the numerical comparison operators, this method considers * negative zero to be strictly smaller than positive zero. If one * argument is positive zero and the other negative zero, the * result is positive zero. * * @param a an argument. * @param b another argument. * @return the larger of {@code a} and {@code b}. */ public static float max(float a, float b) { if (a != a) return a; // a is NaN if ((a == 0.0f) && (b == 0.0f) && (Float.floatToRawIntBits(a) == negativeZeroFloatBits)) { // Raw conversion ok since NaN can't map to -0.0. return b; } return (a >= b) ? a : b; } /** * Returns the greater of two {@code double} values. That * is, the result is the argument closer to positive infinity. If * the arguments have the same value, the result is that same * value. If either value is NaN, then the result is NaN. Unlike * the numerical comparison operators, this method considers * negative zero to be strictly smaller than positive zero. If one * argument is positive zero and the other negative zero, the * result is positive zero. * * @param a an argument. * @param b another argument. * @return the larger of {@code a} and {@code b}. */ public static double max(double a, double b) { if (a != a) return a; // a is NaN if ((a == 0.0d) && (b == 0.0d) && (Double.doubleToRawLongBits(a) == negativeZeroDoubleBits)) { // Raw conversion ok since NaN can't map to -0.0. return b; } return (a >= b) ? a : b; } /** * Returns the smaller of two {@code int} values. That is, * the result the argument closer to the value of * {@link Integer#MIN_VALUE}. If the arguments have the same * value, the result is that same value. * * @param a an argument. * @param b another argument. * @return the smaller of {@code a} and {@code b}. */ @HotSpotIntrinsicCandidate public static int min(int a, int b) { return (a <= b) ? a : b; } /** * Returns the smaller of two {@code long} values. That is, * the result is the argument closer to the value of * {@link Long#MIN_VALUE}. If the arguments have the same * value, the result is that same value. * * @param a an argument. * @param b another argument. * @return the smaller of {@code a} and {@code b}. */ public static long min(long a, long b) { return (a <= b) ? a : b; } /** * Returns the smaller of two {@code float} values. That is, * the result is the value closer to negative infinity. If the * arguments have the same value, the result is that same * value. If either value is NaN, then the result is NaN. Unlike * the numerical comparison operators, this method considers * negative zero to be strictly smaller than positive zero. If * one argument is positive zero and the other is negative zero, * the result is negative zero. * * @param a an argument. * @param b another argument. * @return the smaller of {@code a} and {@code b}. */ public static float min(float a, float b) { if (a != a) return a; // a is NaN if ((a == 0.0f) && (b == 0.0f) && (Float.floatToRawIntBits(b) == negativeZeroFloatBits)) { // Raw conversion ok since NaN can't map to -0.0. return b; } return (a <= b) ? a : b; } /** * Returns the smaller of two {@code double} values. That * is, the result is the value closer to negative infinity. If the * arguments have the same value, the result is that same * value. If either value is NaN, then the result is NaN. Unlike * the numerical comparison operators, this method considers * negative zero to be strictly smaller than positive zero. If one * argument is positive zero and the other is negative zero, the * result is negative zero. * * @param a an argument. * @param b another argument. * @return the smaller of {@code a} and {@code b}. */ public static double min(double a, double b) { if (a != a) return a; // a is NaN if ((a == 0.0d) && (b == 0.0d) && (Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) { // Raw conversion ok since NaN can't map to -0.0. return b; } return (a <= b) ? a : b; } /** * Returns the fused multiply add of the three arguments; that is, * returns the exact product of the first two arguments summed * with the third argument and then rounded once to the nearest * {@code double}. * * The rounding is done using the {@linkplain * java.math.RoundingMode#HALF_EVEN round to nearest even * rounding mode}. * * In contrast, if {@code a * b + c} is evaluated as a regular * floating-point expression, two rounding errors are involved, * the first for the multiply operation, the second for the * addition operation. * * <p>Special cases: * <ul> * <li> If any argument is NaN, the result is NaN. * * <li> If one of the first two arguments is infinite and the * other is zero, the result is NaN. * * <li> If the exact product of the first two arguments is infinite * (in other words, at least one of the arguments is infinite and * the other is neither zero nor NaN) and the third argument is an * infinity of the opposite sign, the result is NaN. * * </ul> * * <p>Note that {@code fma(a, 1.0, c)} returns the same * result as ({@code a + c}). However, * {@code fma(a, b, +0.0)} does <em>not</em> always return the * same result as ({@code a * b}) since * {@code fma(-0.0, +0.0, +0.0)} is {@code +0.0} while * ({@code -0.0 * +0.0}) is {@code -0.0}; {@code fma(a, b, -0.0)} is * equivalent to ({@code a * b}) however. * * @apiNote This method corresponds to the fusedMultiplyAdd * operation defined in IEEE 754-2008. * * @param a a value * @param b a value * @param c a value * * @return (<i>a</i> × <i>b</i> + <i>c</i>) * computed, as if with unlimited range and precision, and rounded * once to the nearest {@code double} value * * @since 9 */ @HotSpotIntrinsicCandidate public static double fma(double a, double b, double c) { /* * Infinity and NaN arithmetic is not quite the same with two * roundings as opposed to just one so the simple expression * "a * b + c" cannot always be used to compute the correct * result. With two roundings, the product can overflow and * if the addend is infinite, a spurious NaN can be produced * if the infinity from the overflow and the infinite addend * have opposite signs. */ // First, screen for and handle non-finite input values whose // arithmetic is not supported by BigDecimal. if (Double.isNaN(a) || Double.isNaN(b) || Double.isNaN(c)) { return Double.NaN; } else { // All inputs non-NaN boolean infiniteA = Double.isInfinite(a); boolean infiniteB = Double.isInfinite(b); boolean infiniteC = Double.isInfinite(c); double result; if (infiniteA || infiniteB || infiniteC) { if (infiniteA && b == 0.0 || infiniteB && a == 0.0 ) { return Double.NaN; } // Store product in a double field to cause an // overflow even if non-strictfp evaluation is being // used. double product = a * b; if (Double.isInfinite(product) && !infiniteA && !infiniteB) { // Intermediate overflow; might cause a // spurious NaN if added to infinite c. assert Double.isInfinite(c); return c; } else { result = product + c; assert !Double.isFinite(result); return result; } } else { // All inputs finite BigDecimal product = (new BigDecimal(a)).multiply(new BigDecimal(b)); if (c == 0.0) { // Positive or negative zero // If the product is an exact zero, use a // floating-point expression to compute the sign // of the zero final result. The product is an // exact zero if and only if at least one of a and // b is zero. if (a == 0.0 || b == 0.0) { return a * b + c; } else { // The sign of a zero addend doesn't matter if // the product is nonzero. The sign of a zero // addend is not factored in the result if the // exact product is nonzero but underflows to // zero; see IEEE-754 2008 section 6.3 "The // sign bit". return product.doubleValue(); } } else { return product.add(new BigDecimal(c)).doubleValue(); } } } } /** * Returns the fused multiply add of the three arguments; that is, * returns the exact product of the first two arguments summed * with the third argument and then rounded once to the nearest * {@code float}. * * The rounding is done using the {@linkplain * java.math.RoundingMode#HALF_EVEN round to nearest even * rounding mode}. * * In contrast, if {@code a * b + c} is evaluated as a regular * floating-point expression, two rounding errors are involved, * the first for the multiply operation, the second for the * addition operation. * * <p>Special cases: * <ul> * <li> If any argument is NaN, the result is NaN. * * <li> If one of the first two arguments is infinite and the * other is zero, the result is NaN. * * <li> If the exact product of the first two arguments is infinite * (in other words, at least one of the arguments is infinite and * the other is neither zero nor NaN) and the third argument is an * infinity of the opposite sign, the result is NaN. * * </ul> * * <p>Note that {@code fma(a, 1.0f, c)} returns the same * result as ({@code a + c}). However, * {@code fma(a, b, +0.0f)} does <em>not</em> always return the * same result as ({@code a * b}) since * {@code fma(-0.0f, +0.0f, +0.0f)} is {@code +0.0f} while * ({@code -0.0f * +0.0f}) is {@code -0.0f}; {@code fma(a, b, -0.0f)} is * equivalent to ({@code a * b}) however. * * @apiNote This method corresponds to the fusedMultiplyAdd * operation defined in IEEE 754-2008. * * @param a a value * @param b a value * @param c a value * * @return (<i>a</i> × <i>b</i> + <i>c</i>) * computed, as if with unlimited range and precision, and rounded * once to the nearest {@code float} value * * @since 9 */ @HotSpotIntrinsicCandidate public static float fma(float a, float b, float c) { /* * Since the double format has more than twice the precision * of the float format, the multiply of a * b is exact in * double. The add of c to the product then incurs one * rounding error. Since the double format moreover has more * than (2p + 2) precision bits compared to the p bits of the * float format, the two roundings of (a * b + c), first to * the double format and then secondarily to the float format, * are equivalent to rounding the intermediate result directly * to the float format. * * In terms of strictfp vs default-fp concerns related to * overflow and underflow, since * * (Float.MAX_VALUE * Float.MAX_VALUE) << Double.MAX_VALUE * (Float.MIN_VALUE * Float.MIN_VALUE) >> Double.MIN_VALUE * * neither the multiply nor add will overflow or underflow in * double. Therefore, it is not necessary for this method to * be declared strictfp to have reproducible * behavior. However, it is necessary to explicitly store down * to a float variable to avoid returning a value in the float * extended value set. */ float result = (float)(((double) a * (double) b ) + (double) c); return result; } /** * Returns the size of an ulp of the argument. An ulp, unit in * the last place, of a {@code double} value is the positive * distance between this floating-point value and the {@code * double} value next larger in magnitude. Note that for non-NaN * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>. * * <p>Special Cases: * <ul> * <li> If the argument is NaN, then the result is NaN. * <li> If the argument is positive or negative infinity, then the * result is positive infinity. * <li> If the argument is positive or negative zero, then the result is * {@code Double.MIN_VALUE}. * <li> If the argument is ±{@code Double.MAX_VALUE}, then * the result is equal to 2<sup>971</sup>. * </ul> * * @param d the floating-point value whose ulp is to be returned * @return the size of an ulp of the argument * @author Joseph D. Darcy * @since 1.5 */ public static double ulp(double d) { int exp = getExponent(d); switch(exp) { case Double.MAX_EXPONENT + 1: // NaN or infinity return Math.abs(d); case Double.MIN_EXPONENT - 1: // zero or subnormal return Double.MIN_VALUE; default: assert exp <= Double.MAX_EXPONENT && exp >= Double.MIN_EXPONENT; // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x)) exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1); if (exp >= Double.MIN_EXPONENT) { return powerOfTwoD(exp); } else { // return a subnormal result; left shift integer // representation of Double.MIN_VALUE appropriate // number of positions return Double.longBitsToDouble(1L << (exp - (Double.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) )); } } } /** * Returns the size of an ulp of the argument. An ulp, unit in * the last place, of a {@code float} value is the positive * distance between this floating-point value and the {@code * float} value next larger in magnitude. Note that for non-NaN * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>. * * <p>Special Cases: * <ul> * <li> If the argument is NaN, then the result is NaN. * <li> If the argument is positive or negative infinity, then the * result is positive infinity. * <li> If the argument is positive or negative zero, then the result is * {@code Float.MIN_VALUE}. * <li> If the argument is ±{@code Float.MAX_VALUE}, then * the result is equal to 2<sup>104</sup>. * </ul> * * @param f the floating-point value whose ulp is to be returned * @return the size of an ulp of the argument * @author Joseph D. Darcy * @since 1.5 */ public static float ulp(float f) { int exp = getExponent(f); switch(exp) { case Float.MAX_EXPONENT+1: // NaN or infinity return Math.abs(f); case Float.MIN_EXPONENT-1: // zero or subnormal return Float.MIN_VALUE; default: assert exp <= Float.MAX_EXPONENT && exp >= Float.MIN_EXPONENT; // ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x)) exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1); if (exp >= Float.MIN_EXPONENT) { return powerOfTwoF(exp); } else { // return a subnormal result; left shift integer // representation of FloatConsts.MIN_VALUE appropriate // number of positions return Float.intBitsToFloat(1 << (exp - (Float.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) )); } } } /** * Returns the signum function of the argument; zero if the argument * is zero, 1.0 if the argument is greater than zero, -1.0 if the * argument is less than zero. * * <p>Special Cases: * <ul> * <li> If the argument is NaN, then the result is NaN. * <li> If the argument is positive zero or negative zero, then the * result is the same as the argument. * </ul> * * @param d the floating-point value whose signum is to be returned * @return the signum function of the argument * @author Joseph D. Darcy * @since 1.5 */ public static double signum(double d) { return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d); } /** * Returns the signum function of the argument; zero if the argument * is zero, 1.0f if the argument is greater than zero, -1.0f if the * argument is less than zero. * * <p>Special Cases: * <ul> * <li> If the argument is NaN, then the result is NaN. * <li> If the argument is positive zero or negative zero, then the * result is the same as the argument. * </ul> * * @param f the floating-point value whose signum is to be returned * @return the signum function of the argument * @author Joseph D. Darcy * @since 1.5 */ public static float signum(float f) { return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f); } /** * Returns the hyperbolic sine of a {@code double} value. * The hyperbolic sine of <i>x</i> is defined to be * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/2 * where <i>e</i> is {@linkplain Math#E Euler's number}. * * <p>Special cases: * <ul> * * <li>If the argument is NaN, then the result is NaN. * * <li>If the argument is infinite, then the result is an infinity * with the same sign as the argument. * * <li>If the argument is zero, then the result is a zero with the * same sign as the argument. * * </ul> * * <p>The computed result must be within 2.5 ulps of the exact result. * * @param x The number whose hyperbolic sine is to be returned. * @return The hyperbolic sine of {@code x}. * @since 1.5 */ public static double sinh(double x) { return StrictMath.sinh(x); } /** * Returns the hyperbolic cosine of a {@code double} value. * The hyperbolic cosine of <i>x</i> is defined to be * (<i>e<sup>x</sup> + e<sup>-x</sup></i>)/2 * where <i>e</i> is {@linkplain Math#E Euler's number}. * * <p>Special cases: * <ul> * * <li>If the argument is NaN, then the result is NaN. * * <li>If the argument is infinite, then the result is positive * infinity. * * <li>If the argument is zero, then the result is {@code 1.0}. * * </ul> * * <p>The computed result must be within 2.5 ulps of the exact result. * * @param x The number whose hyperbolic cosine is to be returned. * @return The hyperbolic cosine of {@code x}. * @since 1.5 */ public static double cosh(double x) { return StrictMath.cosh(x); } /** * Returns the hyperbolic tangent of a {@code double} value. * The hyperbolic tangent of <i>x</i> is defined to be * (<i>e<sup>x</sup> - e<sup>-x</sup></i>)/(<i>e<sup>x</sup> + e<sup>-x</sup></i>), * in other words, {@linkplain Math#sinh * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}. Note * that the absolute value of the exact tanh is always less than * 1. * * <p>Special cases: * <ul> * * <li>If the argument is NaN, then the result is NaN. * * <li>If the argument is zero, then the result is a zero with the * same sign as the argument. * * <li>If the argument is positive infinity, then the result is * {@code +1.0}. * * <li>If the argument is negative infinity, then the result is * {@code -1.0}. * * </ul> * * <p>The computed result must be within 2.5 ulps of the exact result. * The result of {@code tanh} for any finite input must have * an absolute value less than or equal to 1. Note that once the * exact result of tanh is within 1/2 of an ulp of the limit value * of ±1, correctly signed ±{@code 1.0} should * be returned. * * @param x The number whose hyperbolic tangent is to be returned. * @return The hyperbolic tangent of {@code x}. * @since 1.5 */ public static double tanh(double x) { return StrictMath.tanh(x); } /** * Returns sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>) * without intermediate overflow or underflow. * * <p>Special cases: * <ul> * * <li> If either argument is infinite, then the result * is positive infinity. * * <li> If either argument is NaN and neither argument is infinite, * then the result is NaN. * * </ul> * * <p>The computed result must be within 1 ulp of the exact * result. If one parameter is held constant, the results must be * semi-monotonic in the other parameter. * * @param x a value * @param y a value * @return sqrt(<i>x</i><sup>2</sup> +<i>y</i><sup>2</sup>) * without intermediate overflow or underflow * @since 1.5 */ public static double hypot(double x, double y) { return StrictMath.hypot(x, y); } /** * Returns <i>e</i><sup>x</sup> -1. Note that for values of * <i>x</i> near 0, the exact sum of * {@code expm1(x)} + 1 is much closer to the true * result of <i>e</i><sup>x</sup> than {@code exp(x)}. * * <p>Special cases: * <ul> * <li>If the argument is NaN, the result is NaN. * * <li>If the argument is positive infinity, then the result is * positive infinity. * * <li>If the argument is negative infinity, then the result is * -1.0. * * <li>If the argument is zero, then the result is a zero with the * same sign as the argument. * * </ul> * * <p>The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. The result of * {@code expm1} for any finite input must be greater than or * equal to {@code -1.0}. Note that once the exact result of * <i>e</i><sup>{@code x}</sup> - 1 is within 1/2 * ulp of the limit value -1, {@code -1.0} should be * returned. * * @param x the exponent to raise <i>e</i> to in the computation of * <i>e</i><sup>{@code x}</sup> -1. * @return the value <i>e</i><sup>{@code x}</sup> - 1. * @since 1.5 */ public static double expm1(double x) { return StrictMath.expm1(x); } /** * Returns the natural logarithm of the sum of the argument and 1. * Note that for small values {@code x}, the result of * {@code log1p(x)} is much closer to the true result of ln(1 * + {@code x}) than the floating-point evaluation of * {@code log(1.0+x)}. * * <p>Special cases: * * <ul> * * <li>If the argument is NaN or less than -1, then the result is * NaN. * * <li>If the argument is positive infinity, then the result is * positive infinity. * * <li>If the argument is negative one, then the result is * negative infinity. * * <li>If the argument is zero, then the result is a zero with the * same sign as the argument. * * </ul> * * <p>The computed result must be within 1 ulp of the exact result. * Results must be semi-monotonic. * * @param x a value * @return the value ln({@code x} + 1), the natural * log of {@code x} + 1 * @since 1.5 */ public static double log1p(double x) { return StrictMath.log1p(x); } /** * Returns the first floating-point argument with the sign of the * second floating-point argument. Note that unlike the {@link * StrictMath#copySign(double, double) StrictMath.copySign} * method, this method does not require NaN {@code sign} * arguments to be treated as positive values; implementations are * permitted to treat some NaN arguments as positive and other NaN * arguments as negative to allow greater performance. * * @param magnitude the parameter providing the magnitude of the result * @param sign the parameter providing the sign of the result * @return a value with the magnitude of {@code magnitude} * and the sign of {@code sign}. * @since 1.6 */ public static double copySign(double magnitude, double sign) { return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) & (DoubleConsts.SIGN_BIT_MASK)) | (Double.doubleToRawLongBits(magnitude) & (DoubleConsts.EXP_BIT_MASK | DoubleConsts.SIGNIF_BIT_MASK))); } /** * Returns the first floating-point argument with the sign of the * second floating-point argument. Note that unlike the {@link * StrictMath#copySign(float, float) StrictMath.copySign} * method, this method does not require NaN {@code sign} * arguments to be treated as positive values; implementations are * permitted to treat some NaN arguments as positive and other NaN * arguments as negative to allow greater performance. * * @param magnitude the parameter providing the magnitude of the result * @param sign the parameter providing the sign of the result * @return a value with the magnitude of {@code magnitude} * and the sign of {@code sign}. * @since 1.6 */ public static float copySign(float magnitude, float sign) { return Float.intBitsToFloat((Float.floatToRawIntBits(sign) & (FloatConsts.SIGN_BIT_MASK)) | (Float.floatToRawIntBits(magnitude) & (FloatConsts.EXP_BIT_MASK | FloatConsts.SIGNIF_BIT_MASK))); } /** * Returns the unbiased exponent used in the representation of a * {@code float}. Special cases: * * <ul> * <li>If the argument is NaN or infinite, then the result is * {@link Float#MAX_EXPONENT} + 1. * <li>If the argument is zero or subnormal, then the result is * {@link Float#MIN_EXPONENT} -1. * </ul> * @param f a {@code float} value * @return the unbiased exponent of the argument * @since 1.6 */ public static int getExponent(float f) { /* * Bitwise convert f to integer, mask out exponent bits, shift * to the right and then subtract out float's bias adjust to * get true exponent value */ return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >> (FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS; } /** * Returns the unbiased exponent used in the representation of a * {@code double}. Special cases: * * <ul> * <li>If the argument is NaN or infinite, then the result is * {@link Double#MAX_EXPONENT} + 1. * <li>If the argument is zero or subnormal, then the result is * {@link Double#MIN_EXPONENT} -1. * </ul> * @param d a {@code double} value * @return the unbiased exponent of the argument * @since 1.6 */ public static int getExponent(double d) { /* * Bitwise convert d to long, mask out exponent bits, shift * to the right and then subtract out double's bias adjust to * get true exponent value. */ return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >> (DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS); } /** * Returns the floating-point number adjacent to the first * argument in the direction of the second argument. If both * arguments compare as equal the second argument is returned. * * <p> * Special cases: * <ul> * <li> If either argument is a NaN, then NaN is returned. * * <li> If both arguments are signed zeros, {@code direction} * is returned unchanged (as implied by the requirement of * returning the second argument if the arguments compare as * equal). * * <li> If {@code start} is * ±{@link Double#MIN_VALUE} and {@code direction} * has a value such that the result should have a smaller * magnitude, then a zero with the same sign as {@code start} * is returned. * * <li> If {@code start} is infinite and * {@code direction} has a value such that the result should * have a smaller magnitude, {@link Double#MAX_VALUE} with the * same sign as {@code start} is returned. * * <li> If {@code start} is equal to ± * {@link Double#MAX_VALUE} and {@code direction} has a * value such that the result should have a larger magnitude, an * infinity with same sign as {@code start} is returned. * </ul> * * @param start starting floating-point value * @param direction value indicating which of * {@code start}'s neighbors or {@code start} should * be returned * @return The floating-point number adjacent to {@code start} in the * direction of {@code direction}. * @since 1.6 */ public static double nextAfter(double start, double direction) { /* * The cases: * * nextAfter(+infinity, 0) == MAX_VALUE * nextAfter(+infinity, +infinity) == +infinity * nextAfter(-infinity, 0) == -MAX_VALUE * nextAfter(-infinity, -infinity) == -infinity * * are naturally handled without any additional testing */ /* * IEEE 754 floating-point numbers are lexicographically * ordered if treated as signed-magnitude integers. * Since Java's integers are two's complement, * incrementing the two's complement representation of a * logically negative floating-point value *decrements* * the signed-magnitude representation. Therefore, when * the integer representation of a floating-point value * is negative, the adjustment to the representation is in * the opposite direction from what would initially be expected. */ // Branch to descending case first as it is more costly than ascending // case due to start != 0.0d conditional. if (start > direction) { // descending if (start != 0.0d) { final long transducer = Double.doubleToRawLongBits(start); return Double.longBitsToDouble(transducer + ((transducer > 0L) ? -1L : 1L)); } else { // start == 0.0d && direction < 0.0d return -Double.MIN_VALUE; } } else if (start < direction) { // ascending // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0) // then bitwise convert start to integer. final long transducer = Double.doubleToRawLongBits(start + 0.0d); return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L)); } else if (start == direction) { return direction; } else { // isNaN(start) || isNaN(direction) return start + direction; } } /** * Returns the floating-point number adjacent to the first * argument in the direction of the second argument. If both * arguments compare as equal a value equivalent to the second argument * is returned. * * <p> * Special cases: * <ul> * <li> If either argument is a NaN, then NaN is returned. * * <li> If both arguments are signed zeros, a value equivalent * to {@code direction} is returned. * * <li> If {@code start} is * ±{@link Float#MIN_VALUE} and {@code direction} * has a value such that the result should have a smaller * magnitude, then a zero with the same sign as {@code start} * is returned. * * <li> If {@code start} is infinite and * {@code direction} has a value such that the result should * have a smaller magnitude, {@link Float#MAX_VALUE} with the * same sign as {@code start} is returned. * * <li> If {@code start} is equal to ± * {@link Float#MAX_VALUE} and {@code direction} has a * value such that the result should have a larger magnitude, an * infinity with same sign as {@code start} is returned. * </ul> * * @param start starting floating-point value * @param direction value indicating which of * {@code start}'s neighbors or {@code start} should * be returned * @return The floating-point number adjacent to {@code start} in the * direction of {@code direction}. * @since 1.6 */ public static float nextAfter(float start, double direction) { /* * The cases: * * nextAfter(+infinity, 0) == MAX_VALUE * nextAfter(+infinity, +infinity) == +infinity * nextAfter(-infinity, 0) == -MAX_VALUE * nextAfter(-infinity, -infinity) == -infinity * * are naturally handled without any additional testing */ /* * IEEE 754 floating-point numbers are lexicographically * ordered if treated as signed-magnitude integers. * Since Java's integers are two's complement, * incrementing the two's complement representation of a * logically negative floating-point value *decrements* * the signed-magnitude representation. Therefore, when * the integer representation of a floating-point value * is negative, the adjustment to the representation is in * the opposite direction from what would initially be expected. */ // Branch to descending case first as it is more costly than ascending // case due to start != 0.0f conditional. if (start > direction) { // descending if (start != 0.0f) { final int transducer = Float.floatToRawIntBits(start); return Float.intBitsToFloat(transducer + ((transducer > 0) ? -1 : 1)); } else { // start == 0.0f && direction < 0.0f return -Float.MIN_VALUE; } } else if (start < direction) { // ascending // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0) // then bitwise convert start to integer. final int transducer = Float.floatToRawIntBits(start + 0.0f); return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1)); } else if (start == direction) { return (float)direction; } else { // isNaN(start) || isNaN(direction) return start + (float)direction; } } /** * Returns the floating-point value adjacent to {@code d} in * the direction of positive infinity. This method is * semantically equivalent to {@code nextAfter(d, * Double.POSITIVE_INFINITY)}; however, a {@code nextUp} * implementation may run faster than its equivalent * {@code nextAfter} call. * * <p>Special Cases: * <ul> * <li> If the argument is NaN, the result is NaN. * * <li> If the argument is positive infinity, the result is * positive infinity. * * <li> If the argument is zero, the result is * {@link Double#MIN_VALUE} * * </ul> * * @param d starting floating-point value * @return The adjacent floating-point value closer to positive * infinity. * @since 1.6 */ public static double nextUp(double d) { // Use a single conditional and handle the likely cases first. if (d < Double.POSITIVE_INFINITY) { // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0). final long transducer = Double.doubleToRawLongBits(d + 0.0D); return Double.longBitsToDouble(transducer + ((transducer >= 0L) ? 1L : -1L)); } else { // d is NaN or +Infinity return d; } } /** * Returns the floating-point value adjacent to {@code f} in * the direction of positive infinity. This method is * semantically equivalent to {@code nextAfter(f, * Float.POSITIVE_INFINITY)}; however, a {@code nextUp} * implementation may run faster than its equivalent * {@code nextAfter} call. * * <p>Special Cases: * <ul> * <li> If the argument is NaN, the result is NaN. * * <li> If the argument is positive infinity, the result is * positive infinity. * * <li> If the argument is zero, the result is * {@link Float#MIN_VALUE} * * </ul> * * @param f starting floating-point value * @return The adjacent floating-point value closer to positive * infinity. * @since 1.6 */ public static float nextUp(float f) { // Use a single conditional and handle the likely cases first. if (f < Float.POSITIVE_INFINITY) { // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0). final int transducer = Float.floatToRawIntBits(f + 0.0F); return Float.intBitsToFloat(transducer + ((transducer >= 0) ? 1 : -1)); } else { // f is NaN or +Infinity return f; } } /** * Returns the floating-point value adjacent to {@code d} in * the direction of negative infinity. This method is * semantically equivalent to {@code nextAfter(d, * Double.NEGATIVE_INFINITY)}; however, a * {@code nextDown} implementation may run faster than its * equivalent {@code nextAfter} call. * * <p>Special Cases: * <ul> * <li> If the argument is NaN, the result is NaN. * * <li> If the argument is negative infinity, the result is * negative infinity. * * <li> If the argument is zero, the result is * {@code -Double.MIN_VALUE} * * </ul> * * @param d starting floating-point value * @return The adjacent floating-point value closer to negative * infinity. * @since 1.8 */ public static double nextDown(double d) { if (Double.isNaN(d) || d == Double.NEGATIVE_INFINITY) return d; else { if (d == 0.0) return -Double.MIN_VALUE; else return Double.longBitsToDouble(Double.doubleToRawLongBits(d) + ((d > 0.0d)?-1L:+1L)); } } /** * Returns the floating-point value adjacent to {@code f} in * the direction of negative infinity. This method is * semantically equivalent to {@code nextAfter(f, * Float.NEGATIVE_INFINITY)}; however, a * {@code nextDown} implementation may run faster than its * equivalent {@code nextAfter} call. * * <p>Special Cases: * <ul> * <li> If the argument is NaN, the result is NaN. * * <li> If the argument is negative infinity, the result is * negative infinity. * * <li> If the argument is zero, the result is * {@code -Float.MIN_VALUE} * * </ul> * * @param f starting floating-point value * @return The adjacent floating-point value closer to negative * infinity. * @since 1.8 */ public static float nextDown(float f) { if (Float.isNaN(f) || f == Float.NEGATIVE_INFINITY) return f; else { if (f == 0.0f) return -Float.MIN_VALUE; else return Float.intBitsToFloat(Float.floatToRawIntBits(f) + ((f > 0.0f)?-1:+1)); } } /** * Returns {@code d} × * 2<sup>{@code scaleFactor}</sup> rounded as if performed * by a single correctly rounded floating-point multiply to a * member of the double value set. See the Java * Language Specification for a discussion of floating-point * value sets. If the exponent of the result is between {@link * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the * answer is calculated exactly. If the exponent of the result * would be larger than {@code Double.MAX_EXPONENT}, an * infinity is returned. Note that if the result is subnormal, * precision may be lost; that is, when {@code scalb(x, n)} * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal * <i>x</i>. When the result is non-NaN, the result has the same * sign as {@code d}. * * <p>Special cases: * <ul> * <li> If the first argument is NaN, NaN is returned. * <li> If the first argument is infinite, then an infinity of the * same sign is returned. * <li> If the first argument is zero, then a zero of the same * sign is returned. * </ul> * * @param d number to be scaled by a power of two. * @param scaleFactor power of 2 used to scale {@code d} * @return {@code d} × 2<sup>{@code scaleFactor}</sup> * @since 1.6 */ public static double scalb(double d, int scaleFactor) { /* * This method does not need to be declared strictfp to * compute the same correct result on all platforms. When * scaling up, it does not matter what order the * multiply-store operations are done; the result will be * finite or overflow regardless of the operation ordering. * However, to get the correct result when scaling down, a * particular ordering must be used. * * When scaling down, the multiply-store operations are * sequenced so that it is not possible for two consecutive * multiply-stores to return subnormal results. If one * multiply-store result is subnormal, the next multiply will * round it away to zero. This is done by first multiplying * by 2 ^ (scaleFactor % n) and then multiplying several * times by 2^n as needed where n is the exponent of number * that is a covenient power of two. In this way, at most one * real rounding error occurs. If the double value set is * being used exclusively, the rounding will occur on a * multiply. If the double-extended-exponent value set is * being used, the products will (perhaps) be exact but the * stores to d are guaranteed to round to the double value * set. * * It is _not_ a valid implementation to first multiply d by * 2^MIN_EXPONENT and then by 2 ^ (scaleFactor % * MIN_EXPONENT) since even in a strictfp program double * rounding on underflow could occur; e.g. if the scaleFactor * argument was (MIN_EXPONENT - n) and the exponent of d was a * little less than -(MIN_EXPONENT - n), meaning the final * result would be subnormal. * * Since exact reproducibility of this method can be achieved * without any undue performance burden, there is no * compelling reason to allow double rounding on underflow in * scalb. */ // magnitude of a power of two so large that scaling a finite // nonzero value by it would be guaranteed to over or // underflow; due to rounding, scaling down takes an // additional power of two which is reflected here final int MAX_SCALE = Double.MAX_EXPONENT + -Double.MIN_EXPONENT + DoubleConsts.SIGNIFICAND_WIDTH + 1; int exp_adjust = 0; int scale_increment = 0; double exp_delta = Double.NaN; // Make sure scaling factor is in a reasonable range if(scaleFactor < 0) { scaleFactor = Math.max(scaleFactor, -MAX_SCALE); scale_increment = -512; exp_delta = twoToTheDoubleScaleDown; } else { scaleFactor = Math.min(scaleFactor, MAX_SCALE); scale_increment = 512; exp_delta = twoToTheDoubleScaleUp; } // Calculate (scaleFactor % +/-512), 512 = 2^9, using // technique from "Hacker's Delight" section 10-2. int t = (scaleFactor >> 9-1) >>> 32 - 9; exp_adjust = ((scaleFactor + t) & (512 -1)) - t; d *= powerOfTwoD(exp_adjust); scaleFactor -= exp_adjust; while(scaleFactor != 0) { d *= exp_delta; scaleFactor -= scale_increment; } return d; } /** * Returns {@code f} × * 2<sup>{@code scaleFactor}</sup> rounded as if performed * by a single correctly rounded floating-point multiply to a * member of the float value set. See the Java * Language Specification for a discussion of floating-point * value sets. If the exponent of the result is between {@link * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the * answer is calculated exactly. If the exponent of the result * would be larger than {@code Float.MAX_EXPONENT}, an * infinity is returned. Note that if the result is subnormal, * precision may be lost; that is, when {@code scalb(x, n)} * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal * <i>x</i>. When the result is non-NaN, the result has the same * sign as {@code f}. * * <p>Special cases: * <ul> * <li> If the first argument is NaN, NaN is returned. * <li> If the first argument is infinite, then an infinity of the * same sign is returned. * <li> If the first argument is zero, then a zero of the same * sign is returned. * </ul> * * @param f number to be scaled by a power of two. * @param scaleFactor power of 2 used to scale {@code f} * @return {@code f} × 2<sup>{@code scaleFactor}</sup> * @since 1.6 */ public static float scalb(float f, int scaleFactor) { // magnitude of a power of two so large that scaling a finite // nonzero value by it would be guaranteed to over or // underflow; due to rounding, scaling down takes an // additional power of two which is reflected here final int MAX_SCALE = Float.MAX_EXPONENT + -Float.MIN_EXPONENT + FloatConsts.SIGNIFICAND_WIDTH + 1; // Make sure scaling factor is in a reasonable range scaleFactor = Math.max(Math.min(scaleFactor, MAX_SCALE), -MAX_SCALE); /* * Since + MAX_SCALE for float fits well within the double * exponent range and + float -> double conversion is exact * the multiplication below will be exact. Therefore, the * rounding that occurs when the double product is cast to * float will be the correctly rounded float result. Since * all operations other than the final multiply will be exact, * it is not necessary to declare this method strictfp. */ return (float)((double)f*powerOfTwoD(scaleFactor)); } // Constants used in scalb static double twoToTheDoubleScaleUp = powerOfTwoD(512); static double twoToTheDoubleScaleDown = powerOfTwoD(-512); /** * Returns a floating-point power of two in the normal range. */ static double powerOfTwoD(int n) { assert(n >= Double.MIN_EXPONENT && n <= Double.MAX_EXPONENT); return Double.longBitsToDouble((((long)n + (long)DoubleConsts.EXP_BIAS) << (DoubleConsts.SIGNIFICAND_WIDTH-1)) & DoubleConsts.EXP_BIT_MASK); } /** * Returns a floating-point power of two in the normal range. */ static float powerOfTwoF(int n) { assert(n >= Float.MIN_EXPONENT && n <= Float.MAX_EXPONENT); return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) << (FloatConsts.SIGNIFICAND_WIDTH-1)) & FloatConsts.EXP_BIT_MASK); } }

**⏎ java/lang/Math.java**

2020-05-29, 44541👍, 0💬

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