jdk-sandbox: comparison jdk/src/java.base/share/classes/java/lang/Math.java

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+/*
+* Copyright (c) 1994, 2013, 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.util.Random;
+import sun.misc.FloatConsts;
+import sun.misc.DoubleConsts;
+/**
+* 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, decrement, 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;
+/**
+* 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.
+*/
+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.
+*/
+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.
+*/
+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 / 180.0 * PI;
+}
+/**
+* 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 * 180.0 / PI;
+}
+/**
+* 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.
+*/
+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&nbsp;{@code a}, the natural logarithm of
+*          {@code a}.
+*/
+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
+*/
+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.
+*/
+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&nbsp;-&nbsp;f2</code>&nbsp;&times;&nbsp;<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},&nbsp;{@code y}) to polar
+* coordinates (r,&nbsp;<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>,&nbsp;<i>theta</i>)
+*          in polar coordinates that corresponds to the point
+*          (<i>x</i>,&nbsp;<i>y</i>) in Cartesian coordinates.
+*/
+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>.
+*/
+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
+*/
+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
+*/
+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
+*/
+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 than 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
+*/
+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 than 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
+*/
+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 1.8
+*/
+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
+*/
+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
+*/
+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
+*/
+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
+*/
+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
+*/
+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
+*/
+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 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 the {@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 than 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 the {@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 than 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}; &nbsp; 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}; &nbsp; and {@code (+4 % -3) == +1} </li>
+*      <li>{@code floorMod(-4, +3) == +2}; &nbsp; and {@code (-4 % +3) == -1} </li>
+*      <li>{@code floorMod(-4, -3) == -1}; &nbsp; 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) {
+int r = x - floorDiv(x, y) * y;
+return r;
+}
+/**
+* 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>
+* In other words, the result is the same as the value of the expression:
+* <p>{@code Float.intBitsToFloat(0x7fffffff & Float.floatToIntBits(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>
+* In other words, the result is the same as the value of the expression:
+* <p>{@code Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)}
+*
+* @param   a   the argument whose absolute value is to be determined
+* @return  the absolute value of the argument.
+*/
+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}.
+*/
+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 long negativeZeroFloatBits  = Float.floatToRawIntBits(-0.0f);
+private static 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}.
+*/
+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 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 &plusmn;{@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 DoubleConsts.MAX_EXPONENT+1:       // NaN or infinity
+return Math.abs(d);
+case DoubleConsts.MIN_EXPONENT-1:       // zero or subnormal
+return Double.MIN_VALUE;
+default:
+assert exp <= DoubleConsts.MAX_EXPONENT && exp >= DoubleConsts.MIN_EXPONENT;
+// ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
+exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1);
+if (exp >= DoubleConsts.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 - (DoubleConsts.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 &plusmn;{@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 FloatConsts.MAX_EXPONENT+1:        // NaN or infinity
+return Math.abs(f);
+case FloatConsts.MIN_EXPONENT-1:        // zero or subnormal
+return FloatConsts.MIN_VALUE;
+default:
+assert exp <= FloatConsts.MAX_EXPONENT && exp >= FloatConsts.MIN_EXPONENT;
+// ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
+exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1);
+if (exp >= FloatConsts.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 - (FloatConsts.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>&nbsp;-&nbsp;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>&nbsp;+&nbsp;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>&nbsp;-&nbsp;e<sup>-x</sup></i>)/(<i>e<sup>x</sup>&nbsp;+&nbsp;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 &plusmn;1, correctly signed &plusmn;{@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>&nbsp;+<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>&nbsp;+<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>&nbsp;-1.  Note that for values of
+* <i>x</i> near 0, the exact sum of
+* {@code expm1(x)}&nbsp;+&nbsp;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>&nbsp;-&nbsp;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>&nbsp;-1.
+* @return  the value <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;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}&nbsp;+&nbsp;1), the natural
+* log of {@code x}&nbsp;+&nbsp;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
+* &plusmn;{@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 &plusmn;
+* {@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
+* &plusmn;{@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 &plusmn;
+* {@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} &times;
+* 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} &times; 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 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 takes an
+// additional power of two which is reflected here
+final int MAX_SCALE = DoubleConsts.MAX_EXPONENT + -DoubleConsts.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} &times;
+* 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} &times; 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 takes an
+// additional power of two which is reflected here
+final int MAX_SCALE = FloatConsts.MAX_EXPONENT + -FloatConsts.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 >= DoubleConsts.MIN_EXPONENT && n <= DoubleConsts.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 >= FloatConsts.MIN_EXPONENT && n <= FloatConsts.MAX_EXPONENT);
+return Float.intBitsToFloat(((n + FloatConsts.EXP_BIAS) <<
+(FloatConsts.SIGNIFICAND_WIDTH-1))
+& FloatConsts.EXP_BIT_MASK);
+}
+}

changeset 25859	3317bb8137f4
parent 24865	09b1d992ca72
child 26727	b4e26e7f964e