diff -r 4ebc2e2fb97c -r 71c04702a3d5 src/java.base/share/classes/java/lang/Math.java --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/java.base/share/classes/java/lang/Math.java Tue Sep 12 19:03:39 2017 +0200 @@ -0,0 +1,2714 @@ +/* + * Copyright (c) 1994, 2017, Oracle and/or its affiliates. All rights reserved. + * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. + * + * This code is free software; you can redistribute it and/or modify it + * under the terms of the GNU General Public License version 2 only, as + * published by the Free Software Foundation. Oracle designates this + * particular file as subject to the "Classpath" exception as provided + * by Oracle in the LICENSE file that accompanied this code. + * + * This code 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 + * version 2 for more details (a copy is included in the LICENSE file that + * accompanied this code). + * + * You should have received a copy of the GNU General Public License version + * 2 along with this work; if not, write to the Free Software Foundation, + * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. + * + * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA + * or visit www.oracle.com if you need additional information or have any + * questions. + */ + +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. + * + *

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. + * + *

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}. + * + *

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 ulps, 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 correctly + * rounded. 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 + * semi-monotonic: 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. + * + *

+ * 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 + * e, the base of the natural logarithms. + */ + public static final double E = 2.7182818284590452354; + + /** + * The {@code double} value that is closer than any other to + * pi, 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: + *

+ * + *

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: + *

+ * + *

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: + *

+ * + *

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 -pi/2 through pi/2. Special cases: + *

+ * + *

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 pi. Special case: + *

+ * + *

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 -pi/2 through pi/2. Special cases: + *

+ * + *

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 + * not 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 e raised to the power of a + * {@code double} value. Special cases: + *

+ * + *

The computed result must be within 1 ulp of the exact result. + * Results must be semi-monotonic. + * + * @param a the exponent to raise e to. + * @return the value e{@code a}, + * where e 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 e) of a {@code double} + * value. Special cases: + *

+ * + *

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: + * + *

+ * + *

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: + *

+ * 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: + * + * + * + *

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 + * f1 - f2 × n, + * where n 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 n 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: + *

+ * + * @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: + * 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: + * + * + * @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: + * + * + * @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 theta from the conversion of rectangular + * coordinates ({@code x}, {@code y}) to polar + * coordinates (r, theta). + * This method computes the phase theta by computing an arc tangent + * of {@code y/x} in the range of -pi to pi. Special + * cases: + * + * + *

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 theta component of the point + * (r, theta) + * in polar coordinates that corresponds to the point + * (x, y) 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: + * + *

+ * + *

(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.) + * + *

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}{@code b}. + */ + @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. + * + *

+ * Special cases: + *

+ * + * @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. + * + *

Special cases: + *

+ * + * @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. + * + *

When this method is first called, it creates a single new + * pseudorandom-number generator, exactly as if by the expression + * + *

{@code new java.util.Random()}
+ * + * This new pseudorandom-number generator is used thereafter for + * all calls to this method and is used nowhere else. + * + *

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 + * + * 

{@code
+     * double f = Math.random()/Math.nextDown(1.0);
+     * double x = x1*(1.0 - f) + x2*f;
+     * }
+ * + * @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 + */ + 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}. + *

+ * 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. + *

+ * + * @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}. + *

+ * 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. + *

+ * 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}. + *

+ * 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. + *

+ * 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. + *

+ * 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)}. + * + *

+ * The relationship between {@code floorDiv} and {@code floorMod} is such that: + *

+ *

+ * 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. + *

+ * Examples: + *

+ *

+ * 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. + *

+ * 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)}. + * + *

+ * The relationship between {@code floorDiv} and {@code floorMod} is such that: + *

+ *

+ * 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. + *

+ * 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)}. + * + *

+ * The relationship between {@code floorDiv} and {@code floorMod} is such that: + *

+ *

+ * 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. + * + *

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. + * + *

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: + *

+ * + * @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:
+ * {@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: + * + * + * @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:
+ * {@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. + * + *

Special cases: + *

+ * + *

Note that {@code fma(a, 1.0, c)} returns the same + * result as ({@code a + c}). However, + * {@code fma(a, b, +0.0)} does not 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 (a × b + c) + * 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. + * + *

Special cases: + *

+ * + *

Note that {@code fma(a, 1.0f, c)} returns the same + * result as ({@code a + c}). However, + * {@code fma(a, b, +0.0f)} does not 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 (a × b + c) + * 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 + * x, ulp(-x) == ulp(x). + * + *

Special Cases: + *

+ * + * @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 + * x, ulp(-x) == ulp(x). + * + *

Special Cases: + *

+ * + * @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. + * + *

Special Cases: + *

+ * + * @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. + * + *

Special Cases: + *

+ * + * @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 x is defined to be + * (ex - e-x)/2 + * where e is {@linkplain Math#E Euler's number}. + * + *

Special cases: + *

+ * + *

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 x is defined to be + * (ex + e-x)/2 + * where e is {@linkplain Math#E Euler's number}. + * + *

Special cases: + *

+ * + *

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 x is defined to be + * (ex - e-x)/(ex + e-x), + * in other words, {@linkplain Math#sinh + * sinh(x)}/{@linkplain Math#cosh cosh(x)}. Note + * that the absolute value of the exact tanh is always less than + * 1. + * + *

Special cases: + *

+ * + *

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(x2 +y2) + * without intermediate overflow or underflow. + * + *

Special cases: + *

+ * + *

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(x2 +y2) + * without intermediate overflow or underflow + * @since 1.5 + */ + public static double hypot(double x, double y) { + return StrictMath.hypot(x, y); + } + + /** + * Returns ex -1. Note that for values of + * x near 0, the exact sum of + * {@code expm1(x)} + 1 is much closer to the true + * result of ex than {@code exp(x)}. + * + *

Special cases: + *

+ * + *

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 + * e{@code x} - 1 is within 1/2 + * ulp of the limit value -1, {@code -1.0} should be + * returned. + * + * @param x the exponent to raise e to in the computation of + * e{@code x} -1. + * @return the value e{@code x} - 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)}. + * + *

Special cases: + * + *

+ * + *

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: + * + *

+ * @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: + * + * + * @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. + * + *

+ * Special cases: + *

+ * + * @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. + * + *

+ * Special cases: + *

+ * + * @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. + * + *

Special Cases: + *

+ * + * @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. + * + *

Special Cases: + *

+ * + * @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. + * + *

Special Cases: + *

+ * + * @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. + * + *

Special Cases: + *

+ * + * @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{@code scaleFactor} 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 + * x. When the result is non-NaN, the result has the same + * sign as {@code d}. + * + *

Special cases: + *

+ * + * @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{@code scaleFactor} + * @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{@code scaleFactor} 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 + * x. When the result is non-NaN, the result has the same + * sign as {@code f}. + * + *

Special cases: + *

+ * + * @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{@code scaleFactor} + * @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); + } +}