content: /*
   * Copyright (C) 2014 The Android Open Source Project
   * 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 dalvik.annotation.optimization.CriticalNative;
  import java.util.Random;
  
  import sun.misc.FloatConsts;
  import sun.misc.DoubleConsts;
  
  // Android-note: Document that the results from Math are based on libm's behavior.
  // For performance, Android implements many of the methods in this class in terms of the underlying
  // OS's libm functions. libm has well-defined behavior for special cases. Where known these are
  // marked with the tag above and the documentation has been modified as needed.
  
  /**
   * 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   JDK1.0
   */
  
  public final class Math {
  
      // Android-changed: Numerous methods in this class are re-implemented in native for performance.
      // Those methods are also annotated @CriticalNative.
  
      /**
       * 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.
       */
      @CriticalNative
      public static native double sin(double a);
  
      /**
       * 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.
       */
      @CriticalNative
      public static native double cos(double a);
  
      /**
       * 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.
       */
      @CriticalNative
      public static native double tan(double a);
  
      /**
       * 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.
       */
      @CriticalNative
      public static native double asin(double a);
  
      /**
       * 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.
       */
      @CriticalNative
      public static native double acos(double a);
  
      /**
       * 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.
       */
      @CriticalNative
      public static native double atan(double a);
  
      /**
       * 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.
       */
      @CriticalNative
      public static native double exp(double a);
  
      /**
       * 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}.
       */
      @CriticalNative
      public static native double log(double a);
  
      /**
       * 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
       */
      @CriticalNative
      public static native double log10(double a);
  
      /**
       * 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.
       */
      @CriticalNative
      public static native double sqrt(double a);
  
  
      /**
       * 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
       */
      @CriticalNative
      public static native double cbrt(double 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}.
       */
      @CriticalNative
      public static native double IEEEremainder(double f1, double f2);
  
      /**
       * 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.
       */
      @CriticalNative
      public static native double ceil(double a);
  
      /**
       * 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.
       */
      @CriticalNative
      public static native double floor(double a);
  
      /**
       * 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.
       */
      @CriticalNative
      public static native double rint(double a);
  
      /**
       * 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.
       */
      @CriticalNative
      public static native double atan2(double y, double x);
  
      // Android-changed: Document that the results from Math are based on libm's behavior.
      // The cases known to differ with libm's pow():
      //   If the first argument is 1.0 then result is always 1.0 (not NaN).
      //   If the first argument is -1.0 and the second argument is infinite, the result is 1.0 (not
      //   NaN).
      /**
       * 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 first argument is 1.0, then the result is 1.0.
       * <li>If the second argument is NaN, then the result is NaN except where the first argument is
       * 1.0.
       * <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 1.0.
       *
       * <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>.
       */
      @CriticalNative
      public static native double pow(double a, double b);
  
      /**
       * 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     Integer#MAX_VALUE
       * @see     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     Long#MAX_VALUE
       * @see     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.
       *
       * @return  a pseudorandom {@code double} greater than or equal
       * to {@code 0.0} and less than {@code 1.0}.
       * @see Random#nextDouble()
       */
      public static double random() {
          return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
      }
  
      // Android-added: setRandomSeedInternal(long), called after zygote forks.
      // This allows different processes to have different random seeds.
      /**
       * Set the seed for the pseudo random generator used by {@link #random()}
       * and {@link #randomIntInternal()}.
       *
       * @hide for internal use only.
       */
      public static void setRandomSeedInternal(long seed) {
          RandomNumberGeneratorHolder.randomNumberGenerator.setSeed(seed);
      }
  
      // Android-added: randomIntInternal() method: like random() but for int.
      /**
       * @hide for internal use only.
       */
      public static int randomIntInternal() {
          return RandomNumberGeneratorHolder.randomNumberGenerator.nextInt();
      }
  
      // Android-added: randomLongInternal() method: like random() but for long.
      /**
       * @hide for internal use only.
       */
      public static long randomLongInternal() {
          return RandomNumberGeneratorHolder.randomNumberGenerator.nextLong();
      }
  
      /**
       * 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>
       * <p>
       *
       * @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) {
          // Android-changed: Implementation modified to exactly match ART intrinsics behavior.
          // Note, as a "quality of implementation", rather than pure "spec compliance",
          // we require that Math.abs() clears the sign bit (but changes nothing else)
          // for all numbers, including NaN (signaling NaN may become quiet though).
          // http://b/30758343
          return Float.intBitsToFloat(0x7fffffff & Float.floatToRawIntBits(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) {
          // Android-changed: Implementation modified to exactly match ART intrinsics behavior.
          // Note, as a "quality of implementation", rather than pure "spec compliance",
          // we require that Math.abs() clears the sign bit (but changes nothing else)
          // for all numbers, including NaN (signaling NaN may become quiet though).
          // http://b/30758343
          return Double.longBitsToDouble(0x7fffffffffffffffL & Double.doubleToRawLongBits(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
       */
      @CriticalNative
      public static native double sinh(double 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
       */
      @CriticalNative
      public static native double cosh(double 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
       */
      @CriticalNative
      public static native double tanh(double 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
       */
      @CriticalNative
      public static native double hypot(double x, double 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
       */
      @CriticalNative
      public static native double expm1(double 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
       */
      @CriticalNative
      public static native double log1p(double 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
           */
  
          // First check for NaN values
          if (Double.isNaN(start) || Double.isNaN(direction)) {
              // return a NaN derived from the input NaN(s)
              return start + direction;
          } else if (start == direction) {
              return direction;
          } else {        // start > direction or start < direction
              // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
              // then bitwise convert start to integer.
              long transducer = Double.doubleToRawLongBits(start + 0.0d);
  
              /*
               * 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 values
               * is less than zero, the adjustment to the representation
               * is in the opposite direction than would be expected at
               * first .
               */
              if (direction > start) { // Calculate next greater value
                  transducer = transducer + (transducer >= 0L ? 1L:-1L);
              } else  { // Calculate next lesser value
                  assert direction < start;
                  if (transducer > 0L)
                      --transducer;
                  else
                      if (transducer < 0L )
                          ++transducer;
                      /*
                       * transducer==0, the result is -MIN_VALUE
                       *
                       * The transition from zero (implicitly
                       * positive) to the smallest negative
                       * signed magnitude value must be done
                       * explicitly.
                       */
                      else
                          transducer = DoubleConsts.SIGN_BIT_MASK | 1L;
              }
  
              return Double.longBitsToDouble(transducer);
          }
      }
  
      /**
       * 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
           */
  
          // First check for NaN values
          if (Float.isNaN(start) || Double.isNaN(direction)) {
              // return a NaN derived from the input NaN(s)
              return start + (float)direction;
          } else if (start == direction) {
              return (float)direction;
          } else {        // start > direction or start < direction
              // Add +0.0 to get rid of a -0.0 (+0.0 + -0.0 => +0.0)
              // then bitwise convert start to integer.
              int transducer = Float.floatToRawIntBits(start + 0.0f);
  
              /*
               * 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 values
               * is less than zero, the adjustment to the representation
               * is in the opposite direction than would be expected at
               * first.
               */
              if (direction > start) {// Calculate next greater value
                  transducer = transducer + (transducer >= 0 ? 1:-1);
              } else  { // Calculate next lesser value
                  assert direction < start;
                  if (transducer > 0)
                      --transducer;
                  else
                      if (transducer < 0 )
                          ++transducer;
                      /*
                       * transducer==0, the result is -MIN_VALUE
                       *
                       * The transition from zero (implicitly
                       * positive) to the smallest negative
                       * signed magnitude value must be done
                       * explicitly.
                       */
                      else
                          transducer = FloatConsts.SIGN_BIT_MASK | 1;
              }
  
              return Float.intBitsToFloat(transducer);
          }
      }
  
      /**
       * 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) {
          if( Double.isNaN(d) || d == Double.POSITIVE_INFINITY)
              return d;
          else {
              d += 0.0d;
              return Double.longBitsToDouble(Double.doubleToRawLongBits(d) +
                                             ((d >= 0.0d)?+1L:-1L));
          }
      }
  
      /**
       * 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) {
          if( Float.isNaN(f) || f == FloatConsts.POSITIVE_INFINITY)
              return f;
          else {
              f += 0.0f;
              return Float.intBitsToFloat(Float.floatToRawIntBits(f) +
                                          ((f >= 0.0f)?+1:-1));
          }
      }
  
      /**
       * 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);
      }
  }
  