/*
 
* Copyright (c) 1994, 2013, Oracle and/or its affiliates. All rights reserved.
 
* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
 
*
 
* This code is free software; you can redistribute it and/or modify it
 
* under the terms of the GNU General Public License version 2 only, as
 
* published by the Free Software Foundation.
  
Oracle designates this
 
* particular file as subject to the "Classpath" exception as provided
 
* by Oracle in the LICENSE file that accompanied this code.
 
*
 
* This code is distributed in the hope that it will be useful, but WITHOUT
 
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
 
* FITNESS FOR A PARTICULAR PURPOSE.
  
See the GNU General Public License
 
* version 2 for more details (a copy is included in the LICENSE file that
 
* accompanied this code).
 
*
 
* You should have received a copy of the GNU General Public License version
 
* 2 along with this work; if not, write to the Free Software Foundation,
 
* Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
 
*
 
* Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
 
* or visit www.oracle.com if you need additional information or have any
 
* questions.
 
*/

package java.lang;
import java.util.Random;

import sun.misc.FloatConsts;
import sun.misc.DoubleConsts;

/**
 
* The class {@code Math} contains methods for performing basic
 
* numeric operations such as the elementary exponential, logarithm,
 
* square root, and trigonometric functions.
 
*
 
* <p>Unlike some of the numeric methods of class
 
* {@code StrictMath}, all implementations of the equivalent
 
* functions of class {@code Math} are not defined to return the
 
* bit-for-bit same results.
  
This relaxation permits
 
* better-performing implementations where strict reproducibility is
 
* not required.
 
*
 
* <p>By default many of the {@code Math} methods simply call
 
* the equivalent method in {@code StrictMath} for their
 
* implementation.
  
Code generators are encouraged to use
 
* platform-specific native libraries or microprocessor instructions,
 
* where available, to provide higher-performance implementations of
 
* {@code Math} methods.
  
Such higher-performance
 
* implementations still must conform to the specification for
 
* {@code Math}.
 
*
 
* <p>The quality of implementation specifications concern two
 
* properties, accuracy of the returned result and monotonicity of the
 
* method.
  
Accuracy of the floating-point {@code Math} methods is
 
* measured in terms of <i>ulps</i>, units in the last place.
  
For a
 
* given floating-point format, an {@linkplain #ulp(double) ulp} of a
 
* specific real number value is the distance between the two
 
* floating-point values bracketing that numerical value.
  
When
 
* discussing the accuracy of a method as a whole rather than at a
 
* specific argument, the number of ulps cited is for the worst-case
 
* error at any argument.
  
If a method always has an error less than
 
* 0.5 ulps, the method always returns the floating-point number
 
* nearest the exact result; such a method is <i>correctly
 
* rounded</i>.
  
A correctly rounded method is generally the best a
 
* floating-point approximation can be; however, it is impractical for
 
* many floating-point methods to be correctly rounded.
  
Instead, for
 
* the {@code Math} class, a larger error bound of 1 or 2 ulps is
 
* allowed for certain methods.
  
Informally, with a 1 ulp error bound,
 
* when the exact result is a representable number, the exact result
 
* should be returned as the computed result; otherwise, either of the
 
* two floating-point values which bracket the exact result may be
 
* returned.
  
For exact results large in magnitude, one of the
 
* endpoints of the bracket may be infinite.
  
Besides accuracy at
 
* individual arguments, maintaining proper relations between the
 
* method at different arguments is also important.
  
Therefore, most
 
* methods with more than 0.5 ulp errors are required to be
 
* <i>semi-monotonic</i>: whenever the mathematical function is
 
* non-decreasing, so is the floating-point approximation, likewise,
 
* whenever the mathematical function is non-increasing, so is the
 
* floating-point approximation.
  
Not all approximations that have 1
 
* ulp accuracy will automatically meet the monotonicity requirements.
 
*
 
* <p>
 
* The platform uses signed two's complement integer arithmetic with
 
* int and long primitive types.
  
The developer should choose
 
* the primitive type to ensure that arithmetic operations consistently
 
* produce correct results, which in some cases means the operations
 
* will not overflow the range of values of the computation.
 
* The best practice is to choose the primitive type and algorithm to avoid
 
* overflow. In cases where the size is {@code int} or {@code long} and
 
* overflow errors need to be detected, the methods {@code addExact},
 
* {@code subtractExact}, {@code multiplyExact}, and {@code toIntExact}
 
* throw an {@code ArithmeticException} when the results overflow.
 
* For other arithmetic operations such as divide, absolute value,
 
* increment, decrement, and negation overflow occurs only with
 
* a specific minimum or maximum value and should be checked against
 
* the minimum or maximum as appropriate.
 
*
 
* @author
  
unascribed
 
* @author
  
Joseph D. Darcy
 
* @since
   
JDK1.0
 
*/


public final class Math {

    
/**
     
* Don't let anyone instantiate this class.
     
*/
    
private Math() {}

    
/**
     
* The {@code double} value that is closer than any other to
     
* <i>e</i>, the base of the natural logarithms.
     
*/

    
public static final double E = 2.7182818284590452354;

    
/**
     
* The {@code double} value that is closer than any other to
     
* <i>pi</i>, the ratio of the circumference of a circle to its
     
* diameter.
     
*/

    
public static final double PI = 3.14159265358979323846;

    
/**
     
* Returns the trigonometric sine of an angle.
  
Special cases:
     
* <ul><li>If the argument is NaN or an infinity, then the
     
* result is NaN.
     
* <li>If the argument is zero, then the result is a zero with the
     
* same sign as the argument.</ul>
     
*
     
* <p>The computed result must be within 1 ulp of the exact result.
     
* Results must be semi-monotonic.
     
*
     
* @param
   
aan angle, in radians.
     
* @return
  
the sine of the argument.
     
*/

    
public static double sin(double a) {
        
return StrictMath.sin(a); // default impl. delegates to StrictMath
    
}

    
/**
     
* Returns the trigonometric cosine of an angle. Special cases:
     
* <ul><li>If the argument is NaN or an infinity, then the
     
* result is NaN.</ul>
     
*
     
* <p>The computed result must be within 1 ulp of the exact result.
     
* Results must be semi-monotonic.
     
*
     
* @param
   
aan angle, in radians.
     
* @return
  
the cosine of the argument.
     
*/

    
public static double cos(double a) {
        
return StrictMath.cos(a); // default impl. delegates to StrictMath
    
}

    
/**
     
* Returns the trigonometric tangent of an angle.
  
Special cases:
     
* <ul><li>If the argument is NaN or an infinity, then the result
     
* is NaN.
     
* <li>If the argument is zero, then the result is a zero with the
     
* same sign as the argument.</ul>
     
*
     
* <p>The computed result must be within 1 ulp of the exact result.
     
* Results must be semi-monotonic.
     
*
     
* @param
   
aan angle, in radians.
     
* @return
  
the tangent of the argument.
     
*/

    
public static double tan(double a) {
        
return StrictMath.tan(a); // default impl. delegates to StrictMath
    
}

    
/**
     
* Returns the arc sine of a value; the returned angle is in the
     
* range -<i>pi</i>/2 through <i>pi</i>/2.
  
Special cases:
     
* <ul><li>If the argument is NaN or its absolute value is greater
     
* than 1, then the result is NaN.
     
* <li>If the argument is zero, then the result is a zero with the
     
* same sign as the argument.</ul>
     
*
     
* <p>The computed result must be within 1 ulp of the exact result.
     
* Results must be semi-monotonic.
     
*
     
* @param
   
athe value whose arc sine is to be returned.
     
* @return
  
the arc sine of the argument.
     
*/

    
public static double asin(double a) {
        
return StrictMath.asin(a); // default impl. delegates to StrictMath
    
}

    
/**
     
* Returns the arc cosine of a value; the returned angle is in the
     
* range 0.0 through <i>pi</i>.
  
Special case:
     
* <ul><li>If the argument is NaN or its absolute value is greater
     
* than 1, then the result is NaN.</ul>
     
*
     
* <p>The computed result must be within 1 ulp of the exact result.
     
* Results must be semi-monotonic.
     
*
     
* @param
   
athe value whose arc cosine is to be returned.
     
* @return
  
the arc cosine of the argument.
     
*/

    
public static double acos(double a) {
        
return StrictMath.acos(a); // default impl. delegates to StrictMath
    
}

    
/**
     
* Returns the arc tangent of a value; the returned angle is in the
     
* range -<i>pi</i>/2 through <i>pi</i>/2.
  
Special cases:
     
* <ul><li>If the argument is NaN, then the result is NaN.
     
* <li>If the argument is zero, then the result is a zero with the
     
* same sign as the argument.</ul>
     
*
     
* <p>The computed result must be within 1 ulp of the exact result.
     
* Results must be semi-monotonic.
     
*
     
* @param
   
athe value whose arc tangent is to be returned.
     
* @return
  
the arc tangent of the argument.
     
*/

    
public static double atan(double a) {
        
return StrictMath.atan(a); // default impl. delegates to StrictMath
    
}

    
/**
     
* Converts an angle measured in degrees to an approximately
     
* equivalent angle measured in radians.
  
The conversion from
     
* degrees to radians is generally inexact.
     
*
     
* @param
   
angdegan 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
   
angradan 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
   
athe exponent to raise <i>e</i> to.
     
* @return
  
the value <i>e</i><sup>{@code a}</sup>,
     
*
          
where <i>e</i> is the base of the natural logarithms.
     
*/

    
public static double exp(double a) {
        
return StrictMath.exp(a); // default impl. delegates to StrictMath
    
}

    
/**
     
* Returns the natural logarithm (base <i>e</i>) of a {@code double}
     
* value.
  
Special cases:
     
* <ul><li>If the argument is NaN or less than zero, then the result
     
* is NaN.
     
* <li>If the argument is positive infinity, then the result is
     
* positive infinity.
     
* <li>If the argument is positive zero or negative zero, then the
     
* result is negative infinity.</ul>
     
*
     
* <p>The computed result must be within 1 ulp of the exact result.
     
* Results must be semi-monotonic.
     
*
     
* @param
   
aa value
     
* @return
  
the value ln&nbsp;{@code a}, the natural logarithm of
     
*
          
{@code a}.
     
*/

    
public static double log(double a) {
        
return StrictMath.log(a); // default impl. delegates to StrictMath
    
}

    
/**
     
* Returns the base 10 logarithm of a {@code double} value.
     
* Special cases:
     
*
     
* <ul><li>If the argument is NaN or less than zero, then the result
     
* is NaN.
     
* <li>If the argument is positive infinity, then the result is
     
* positive infinity.
     
* <li>If the argument is positive zero or negative zero, then the
     
* result is negative infinity.
     
* <li> If the argument is equal to 10<sup><i>n</i></sup> for
     
* integer <i>n</i>, then the result is <i>n</i>.
     
* </ul>
     
*
     
* <p>The computed result must be within 1 ulp of the exact result.
     
* Results must be semi-monotonic.
     
*
     
* @param
   
aa value
     
* @return
  
the base 10 logarithm of{@code a}.
     
* @since 1.5
     
*/

    
public static double log10(double a) {
        
return StrictMath.log10(a); // default impl. delegates to StrictMath
    
}

    
/**
     
* Returns the correctly rounded positive square root of a
     
* {@code double} value.
     
* Special cases:
     
* <ul><li>If the argument is NaN or less than zero, then the result
     
* is NaN.
     
* <li>If the argument is positive infinity, then the result is positive
     
* infinity.
     
* <li>If the argument is positive zero or negative zero, then the
     
* result is the same as the argument.</ul>
     
* Otherwise, the result is the {@code double} value closest to
     
* the true mathematical square root of the argument value.
     
*
     
* @param
   
aa value.
     
* @return
  
the positive square root of {@code a}.
     
*
          
If the argument is NaN or less than zero, the result is NaN.
     
*/

    
public static double sqrt(double a) {
        
return StrictMath.sqrt(a); // default impl. delegates to StrictMath
                                   
// Note that hardware sqrt instructions
                                   
// frequently can be directly used by JITs
                                   
// and should be much faster than doing
                                   
// Math.sqrt in software.
    
}


    
/**
     
* Returns the cube root of a {@code double} value.
  
For
     
* positive finite {@code x}, {@code cbrt(-x) ==
     
* -cbrt(x)}; that is, the cube root of a negative value is
     
* the negative of the cube root of that value's magnitude.
     
*
     
* Special cases:
     
*
     
* <ul>
     
*
     
* <li>If the argument is NaN, then the result is NaN.
     
*
     
* <li>If the argument is infinite, then the result is an infinity
     
* with the same sign as the argument.
     
*
     
* <li>If the argument is zero, then the result is a zero with the
     
* same sign as the argument.
     
*
     
* </ul>
     
*
     
* <p>The computed result must be within 1 ulp of the exact result.
     
*
     
* @param
   
aa value.
     
* @return
  
the cube root of {@code a}.
     
* @since 1.5
     
*/

    
public static double cbrt(double a) {
        
return StrictMath.cbrt(a);
    
}

    
/**
     
* Computes the remainder operation on two arguments as prescribed
     
* by the IEEE 754 standard.
     
* The remainder value is mathematically equal to
     
* <code>f1&nbsp;-&nbsp;f2</code>&nbsp;&times;&nbsp;<i>n</i>,
     
* where <i>n</i> is the mathematical integer closest to the exact
     
* mathematical value of the quotient {@code f1/f2}, and if two
     
* mathematical integers are equally close to {@code f1/f2},
     
* then <i>n</i> is the integer that is even. If the remainder is
     
* zero, its sign is the same as the sign of the first argument.
     
* Special cases:
     
* <ul><li>If either argument is NaN, or the first argument is infinite,
     
* or the second argument is positive zero or negative zero, then the
     
* result is NaN.
     
* <li>If the first argument is finite and the second argument is
     
* infinite, then the result is the same as the first argument.</ul>
     
*
     
* @param
   
f1the dividend.
     
* @param
   
f2the divisor.
     
* @return
  
the remainder when {@code f1} is divided by
     
*
          
{@code f2}.
     
*/

    
public static double IEEEremainder(double f1, double f2) {
        
return StrictMath.IEEEremainder(f1, f2); // delegate to StrictMath
    
}

    
/**
     
* Returns the smallest (closest to negative infinity)
     
* {@code double} value that is greater than or equal to the
     
* argument and is equal to a mathematical integer. Special cases:
     
* <ul><li>If the argument value is already equal to a
     
* mathematical integer, then the result is the same as the
     
* argument.
  
<li>If the argument is NaN or an infinity or
     
* positive zero or negative zero, then the result is the same as
     
* the argument.
  
<li>If the argument value is less than zero but
     
* greater than -1.0, then the result is negative zero.</ul> Note
     
* that the value of {@code Math.ceil(x)} is exactly the
     
* value of {@code -Math.floor(-x)}.
     
*
     
*
     
* @param
   
aa value.
     
* @return
  
the smallest (closest to negative infinity)
     
*
          
floating-point value that is greater than or equal to
     
*
          
the argument and is equal to a mathematical integer.
     
*/

    
public static double ceil(double a) {
        
return StrictMath.ceil(a); // default impl. delegates to StrictMath
    
}

    
/**
     
* Returns the largest (closest to positive infinity)
     
* {@code double} value that is less than or equal to the
     
* argument and is equal to a mathematical integer. Special cases:
     
* <ul><li>If the argument value is already equal to a
     
* mathematical integer, then the result is the same as the
     
* argument.
  
<li>If the argument is NaN or an infinity or
     
* positive zero or negative zero, then the result is the same as
     
* the argument.</ul>
     
*
     
* @param
   
aa value.
     
* @return
  
the largest (closest to positive infinity)
     
*
          
floating-point value that less than or equal to the argument
     
*
          
and is equal to a mathematical integer.
     
*/

    
public static double floor(double a) {
        
return StrictMath.floor(a); // default impl. delegates to StrictMath
    
}

    
/**
     
* Returns the {@code double} value that is closest in value
     
* to the argument and is equal to a mathematical integer. If two
     
* {@code double} values that are mathematical integers are
     
* equally close, the result is the integer value that is
     
* even. Special cases:
     
* <ul><li>If the argument value is already equal to a mathematical
     
* integer, then the result is the same as the argument.
     
* <li>If the argument is NaN or an infinity or positive zero or negative
     
* zero, then the result is the same as the argument.</ul>
     
*
     
* @param
   
aa {@code double} value.
     
* @return
  
the closest floating-point value to {@code a} that is
     
*
          
equal to a mathematical integer.
     
*/

    
public static double rint(double a) {
        
return StrictMath.rint(a); // default impl. delegates to StrictMath
    
}

    
/**
     
* Returns the angle <i>theta</i> from the conversion of rectangular
     
* coordinates ({@code x},&nbsp;{@code y}) to polar
     
* coordinates (r,&nbsp;<i>theta</i>).
     
* This method computes the phase <i>theta</i> by computing an arc tangent
     
* of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special
     
* cases:
     
* <ul><li>If either argument is NaN, then the result is NaN.
     
* <li>If the first argument is positive zero and the second argument
     
* is positive, or the first argument is positive and finite and the
     
* second argument is positive infinity, then the result is positive
     
* zero.
     
* <li>If the first argument is negative zero and the second argument
     
* is positive, or the first argument is negative and finite and the
     
* second argument is positive infinity, then the result is negative zero.
     
* <li>If the first argument is positive zero and the second argument
     
* is negative, or the first argument is positive and finite and the
     
* second argument is negative infinity, then the result is the
     
* {@code double} value closest to <i>pi</i>.
     
* <li>If the first argument is negative zero and the second argument
     
* is negative, or the first argument is negative and finite and the
     
* second argument is negative infinity, then the result is the
     
* {@code double} value closest to -<i>pi</i>.
     
* <li>If the first argument is positive and the second argument is
     
* positive zero or negative zero, or the first argument is positive
     
* infinity and the second argument is finite, then the result is the
     
* {@code double} value closest to <i>pi</i>/2.
     
* <li>If the first argument is negative and the second argument is
     
* positive zero or negative zero, or the first argument is negative
     
* infinity and the second argument is finite, then the result is the
     
* {@code double} value closest to -<i>pi</i>/2.
     
* <li>If both arguments are positive infinity, then the result is the
     
* {@code double} value closest to <i>pi</i>/4.
     
* <li>If the first argument is positive infinity and the second argument
     
* is negative infinity, then the result is the {@code double}
     
* value closest to 3*<i>pi</i>/4.
     
* <li>If the first argument is negative infinity and the second argument
     
* is positive infinity, then the result is the {@code double} value
     
* closest to -<i>pi</i>/4.
     
* <li>If both arguments are negative infinity, then the result is the
     
* {@code double} value closest to -3*<i>pi</i>/4.</ul>
     
*
     
* <p>The computed result must be within 2 ulps of the exact result.
     
* Results must be semi-monotonic.
     
*
     
* @param
   
ythe ordinate coordinate
     
* @param
   
xthe abscissa coordinate
     
* @return
  
the <i>theta</i> component of the point
     
*
          
(<i>r</i>,&nbsp;<i>theta</i>)
     
*
          
in polar coordinates that corresponds to the point
     
*
          
(<i>x</i>,&nbsp;<i>y</i>) in Cartesian coordinates.
     
*/

    
public static double atan2(double y, double x) {
        
return StrictMath.atan2(y, x); // default impl. delegates to StrictMath
    
}

    
/**
     
* Returns the value of the first argument raised to the power of the
     
* second argument. Special cases:
     
*
     
* <ul><li>If the second argument is positive or negative zero, then the
     
* result is 1.0.
     
* <li>If the second argument is 1.0, then the result is the same as the
     
* first argument.
     
* <li>If the second argument is NaN, then the result is NaN.
     
* <li>If the first argument is NaN and the second argument is nonzero,
     
* then the result is NaN.
     
*
     
* <li>If
     
* <ul>
     
* <li>the absolute value of the first argument is greater than 1
     
* and the second argument is positive infinity, or
     
* <li>the absolute value of the first argument is less than 1 and
     
* the second argument is negative infinity,
     
* </ul>
     
* then the result is positive infinity.
     
*
     
* <li>If
     
* <ul>
     
* <li>the absolute value of the first argument is greater than 1 and
     
* the second argument is negative infinity, or
     
* <li>the absolute value of the
     
* first argument is less than 1 and the second argument is positive
     
* infinity,
     
* </ul>
     
* then the result is positive zero.
     
*
     
* <li>If the absolute value of the first argument equals 1 and the
     
* second argument is infinite, then the result is NaN.
     
*
     
* <li>If
     
* <ul>
     
* <li>the first argument is positive zero and the second argument
     
* is greater than zero, or
     
* <li>the first argument is positive infinity and the second
     
* argument is less than zero,
     
* </ul>
     
* then the result is positive zero.
     
*
     
* <li>If
     
* <ul>
     
* <li>the first argument is positive zero and the second argument
     
* is less than zero, or
     
* <li>the first argument is positive infinity and the second
     
* argument is greater than zero,
     
* </ul>
     
* then the result is positive infinity.
     
*
     
* <li>If
     
* <ul>
     
* <li>the first argument is negative zero and the second argument
     
* is greater than zero but not a finite odd integer, or
     
* <li>the first argument is negative infinity and the second
     
* argument is less than zero but not a finite odd integer,
     
* </ul>
     
* then the result is positive zero.
     
*
     
* <li>If
     
* <ul>
     
* <li>the first argument is negative zero and the second argument
     
* is a positive finite odd integer, or
     
* <li>the first argument is negative infinity and the second
     
* argument is a negative finite odd integer,
     
* </ul>
     
* then the result is negative zero.
     
*
     
* <li>If
     
* <ul>
     
* <li>the first argument is negative zero and the second argument
     
* is less than zero but not a finite odd integer, or
     
* <li>the first argument is negative infinity and the second
     
* argument is greater than zero but not a finite odd integer,
     
* </ul>
     
* then the result is positive infinity.
     
*
     
* <li>If
     
* <ul>
     
* <li>the first argument is negative zero and the second argument
     
* is a negative finite odd integer, or
     
* <li>the first argument is negative infinity and the second
     
* argument is a positive finite odd integer,
     
* </ul>
     
* then the result is negative infinity.
     
*
     
* <li>If the first argument is finite and less than zero
     
* <ul>
     
* <li> if the second argument is a finite even integer, the
     
* result is equal to the result of raising the absolute value of
     
* the first argument to the power of the second argument
     
*
     
* <li>if the second argument is a finite odd integer, the result
     
* is equal to the negative of the result of raising the absolute
     
* value of the first argument to the power of the second
     
* argument
     
*
     
* <li>if the second argument is finite and not an integer, then
     
* the result is NaN.
     
* </ul>
     
*
     
* <li>If both arguments are integers, then the result is exactly equal
     
* to the mathematical result of raising the first argument to the power
     
* of the second argument if that result can in fact be represented
     
* exactly as a {@code double} value.</ul>
     
*
     
* <p>(In the foregoing descriptions, a floating-point value is
     
* considered to be an integer if and only if it is finite and a
     
* fixed point of the method {@link #ceil ceil} or,
     
* equivalently, a fixed point of the method {@link #floor
     
* floor}. A value is a fixed point of a one-argument
     
* method if and only if the result of applying the method to the
     
* value is equal to the value.)
     
*
     
* <p>The computed result must be within 1 ulp of the exact result.
     
* Results must be semi-monotonic.
     
*
     
* @param
   
athe base.
     
* @param
   
bthe exponent.
     
* @return
  
the value {@code a}<sup>{@code b}</sup>.
     
*/

    
public static double pow(double a, double b) {
        
return StrictMath.pow(a, b); // default impl. delegates to StrictMath
    
}

    
/**
     
* Returns the closest {@code int} to the argument, with ties
     
* rounding to positive infinity.
     
*
     
* <p>
     
* Special cases:
     
* <ul><li>If the argument is NaN, the result is 0.
     
* <li>If the argument is negative infinity or any value less than or
     
* equal to the value of {@code Integer.MIN_VALUE}, the result is
     
* equal to the value of {@code Integer.MIN_VALUE}.
     
* <li>If the argument is positive infinity or any value greater than or
     
* equal to the value of {@code Integer.MAX_VALUE}, the result is
     
* equal to the value of {@code Integer.MAX_VALUE}.</ul>
     
*
     
* @param
   
aa floating-point value to be rounded to an integer.
     
* @return
  
the value of the argument rounded to the nearest
     
*
          
{@code int} value.
     
* @seejava.lang.Integer#MAX_VALUE
     
* @seejava.lang.Integer#MIN_VALUE
     
*/

    
public static int round(float a) {
        
int intBits = Float.floatToRawIntBits(a);
        
int biasedExp = (intBits & FloatConsts.EXP_BIT_MASK)
                
>> (FloatConsts.SIGNIFICAND_WIDTH - 1);
        
int shift = (FloatConsts.SIGNIFICAND_WIDTH - 2
                
+ FloatConsts.EXP_BIAS) - biasedExp;
        
if ((shift & -32) == 0) { // shift >= 0 && shift < 32
            
// a is a finite number such that pow(2,-32) <= ulp(a) < 1
            
int r = ((intBits & FloatConsts.SIGNIF_BIT_MASK)
                    
| (FloatConsts.SIGNIF_BIT_MASK + 1));
            
if (intBits < 0) {
                
r = -r;
            
}
            
// In the comments below each Java expression evaluates to the value
            
// the corresponding mathematical expression:
            
// (r) evaluates to a / ulp(a)
            
// (r >> shift) evaluates to floor(a * 2)
            
// ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
            
// (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
            
return ((r >> shift) + 1) >> 1;
        
} else {
            
// a is either
            
// - a finite number with abs(a) < exp(2,FloatConsts.SIGNIFICAND_WIDTH-32) < 1/2
            
// - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
            
// - an infinity or NaN
            
return (int) a;
        
}
    
}

    
/**
     
* Returns the closest {@code long} to the argument, with ties
     
* rounding to positive infinity.
     
*
     
* <p>Special cases:
     
* <ul><li>If the argument is NaN, the result is 0.
     
* <li>If the argument is negative infinity or any value less than or
     
* equal to the value of {@code Long.MIN_VALUE}, the result is
     
* equal to the value of {@code Long.MIN_VALUE}.
     
* <li>If the argument is positive infinity or any value greater than or
     
* equal to the value of {@code Long.MAX_VALUE}, the result is
     
* equal to the value of {@code Long.MAX_VALUE}.</ul>
     
*
     
* @param
   
aa floating-point value to be rounded to a
     
*
          
{@code long}.
     
* @return
  
the value of the argument rounded to the nearest
     
*
          
{@code long} value.
     
* @seejava.lang.Long#MAX_VALUE
     
* @seejava.lang.Long#MIN_VALUE
     
*/

    
public static long round(double a) {
        
long longBits = Double.doubleToRawLongBits(a);
        
long biasedExp = (longBits & DoubleConsts.EXP_BIT_MASK)
                
>> (DoubleConsts.SIGNIFICAND_WIDTH - 1);
        
long shift = (DoubleConsts.SIGNIFICAND_WIDTH - 2
                
+ DoubleConsts.EXP_BIAS) - biasedExp;
        
if ((shift & -64) == 0) { // shift >= 0 && shift < 64
            
// a is a finite number such that pow(2,-64) <= ulp(a) < 1
            
long r = ((longBits & DoubleConsts.SIGNIF_BIT_MASK)
                    
| (DoubleConsts.SIGNIF_BIT_MASK + 1));
            
if (longBits < 0) {
                
r = -r;
            
}
            
// In the comments below each Java expression evaluates to the value
            
// the corresponding mathematical expression:
            
// (r) evaluates to a / ulp(a)
            
// (r >> shift) evaluates to floor(a * 2)
            
// ((r >> shift) + 1) evaluates to floor((a + 1/2) * 2)
            
// (((r >> shift) + 1) >> 1) evaluates to floor(a + 1/2)
            
return ((r >> shift) + 1) >> 1;
        
} else {
            
// a is either
            
// - a finite number with abs(a) < exp(2,DoubleConsts.SIGNIFICAND_WIDTH-64) < 1/2
            
// - a finite number with ulp(a) >= 1 and hence a is a mathematical integer
            
// - an infinity or NaN
            
return (long) a;
        
}
    
}

    
private static final class RandomNumberGeneratorHolder {
        
static final Random randomNumberGenerator = new Random();
    
}

    
/**
     
* Returns a {@code double} value with a positive sign, greater
     
* than or equal to {@code 0.0} and less than {@code 1.0}.
     
* Returned values are chosen pseudorandomly with (approximately)
     
* uniform distribution from that range.
     
*
     
* <p>When this method is first called, it creates a single new
     
* pseudorandom-number generator, exactly as if by the expression
     
*
     
* <blockquote>{@code new java.util.Random()}</blockquote>
     
*
     
* This new pseudorandom-number generator is used thereafter for
     
* all calls to this method and is used nowhere else.
     
*
     
* <p>This method is properly synchronized to allow correct use by
     
* more than one thread. However, if many threads need to generate
     
* pseudorandom numbers at a great rate, it may reduce contention
     
* for each thread to have its own pseudorandom-number generator.
     
*
     
* @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();
    
}

    
/**
     
* 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
   
athe 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
   
athe 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
   
athe argument whose absolute value is to be determined
     
* @return
  
the absolute value of the argument.
     
*/

    
public static float abs(float a) {
        
return (a <= 0.0F) ? 0.0F - a : a;
    
}

    
/**
     
* Returns the absolute value of a {@code double} value.
     
* If the argument is not negative, the argument is returned.
     
* If the argument is negative, the negation of the argument is returned.
     
* Special cases:
     
* <ul><li>If the argument is positive zero or negative zero, the result
     
* is positive zero.
     
* <li>If the argument is infinite, the result is positive infinity.
     
* <li>If the argument is NaN, the result is NaN.</ul>
     
* In other words, the result is the same as the value of the expression:
     
* <p>{@code Double.longBitsToDouble((Double.doubleToLongBits(a)<<1)>>>1)}
     
*
     
* @param
   
athe argument whose absolute value is to be determined
     
* @return
  
the absolute value of the argument.
     
*/

    
public static double abs(double a) {
        
return (a <= 0.0D) ? 0.0D - a : a;
    
}

    
/**
     
* Returns the greater of two {@code int} values. That is, the
     
* result is the argument closer to the value of
     
* {@link Integer#MAX_VALUE}. If the arguments have the same value,
     
* the result is that same value.
     
*
     
* @param
   
aan argument.
     
* @param
   
banother 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
   
aan argument.
     
* @param
   
banother 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
   
aan argument.
     
* @param
   
banother 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
   
aan argument.
     
* @param
   
banother 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
   
aan argument.
     
* @param
   
banother 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
   
aan argument.
     
* @param
   
banother 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
   
aan argument.
     
* @param
   
banother 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
   
aan argument.
     
* @param
   
banother argument.
     
* @return
  
the smaller of {@code a} and {@code b}.
     
*/

    
public static double min(double a, double b) {
        
if (a != a)
            
return a;
   
// a is NaN
        
if ((a == 0.0d) &&
            
(b == 0.0d) &&
            
(Double.doubleToRawLongBits(b) == negativeZeroDoubleBits)) {
            
// Raw conversion ok since NaN can't map to -0.0.
            
return b;
        
}
        
return (a <= b) ? a : b;
    
}

    
/**
     
* Returns the size of an ulp of the argument.
  
An ulp, unit in
     
* the last place, of a {@code double} value is the positive
     
* distance between this floating-point value and the {@code
     
* double} value next larger in magnitude.
  
Note that for non-NaN
     
* <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
     
*
     
* <p>Special Cases:
     
* <ul>
     
* <li> If the argument is NaN, then the result is NaN.
     
* <li> If the argument is positive or negative infinity, then the
     
* result is positive infinity.
     
* <li> If the argument is positive or negative zero, then the result is
     
* {@code Double.MIN_VALUE}.
     
* <li> If the argument is &plusmn;{@code Double.MAX_VALUE}, then
     
* the result is equal to 2<sup>971</sup>.
     
* </ul>
     
*
     
* @param d the floating-point value whose ulp is to be returned
     
* @return the size of an ulp of the argument
     
* @author Joseph D. Darcy
     
* @since 1.5
     
*/

    
public static double ulp(double d) {
        
int exp = getExponent(d);

        
switch(exp) {
        
case DoubleConsts.MAX_EXPONENT+1:
       
// NaN or infinity
            
return Math.abs(d);

        
case DoubleConsts.MIN_EXPONENT-1:
       
// zero or subnormal
            
return Double.MIN_VALUE;

        
default:
            
assert exp <= DoubleConsts.MAX_EXPONENT && exp >= DoubleConsts.MIN_EXPONENT;

            
// ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
            
exp = exp - (DoubleConsts.SIGNIFICAND_WIDTH-1);
            
if (exp >= DoubleConsts.MIN_EXPONENT) {
                
return powerOfTwoD(exp);
            
}
            
else {
                
// return a subnormal result; left shift integer
                
// representation of Double.MIN_VALUE appropriate
                
// number of positions
                
return Double.longBitsToDouble(1L <<
                
(exp - (DoubleConsts.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1)) ));
            
}
        
}
    
}

    
/**
     
* Returns the size of an ulp of the argument.
  
An ulp, unit in
     
* the last place, of a {@code float} value is the positive
     
* distance between this floating-point value and the {@code
     
* float} value next larger in magnitude.
  
Note that for non-NaN
     
* <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
     
*
     
* <p>Special Cases:
     
* <ul>
     
* <li> If the argument is NaN, then the result is NaN.
     
* <li> If the argument is positive or negative infinity, then the
     
* result is positive infinity.
     
* <li> If the argument is positive or negative zero, then the result is
     
* {@code Float.MIN_VALUE}.
     
* <li> If the argument is &plusmn;{@code Float.MAX_VALUE}, then
     
* the result is equal to 2<sup>104</sup>.
     
* </ul>
     
*
     
* @param f the floating-point value whose ulp is to be returned
     
* @return the size of an ulp of the argument
     
* @author Joseph D. Darcy
     
* @since 1.5
     
*/

    
public static float ulp(float f) {
        
int exp = getExponent(f);

        
switch(exp) {
        
case FloatConsts.MAX_EXPONENT+1:
        
// NaN or infinity
            
return Math.abs(f);

        
case FloatConsts.MIN_EXPONENT-1:
        
// zero or subnormal
            
return FloatConsts.MIN_VALUE;

        
default:
            
assert exp <= FloatConsts.MAX_EXPONENT && exp >= FloatConsts.MIN_EXPONENT;

            
// ulp(x) is usually 2^(SIGNIFICAND_WIDTH-1)*(2^ilogb(x))
            
exp = exp - (FloatConsts.SIGNIFICAND_WIDTH-1);
            
if (exp >= FloatConsts.MIN_EXPONENT) {
                
return powerOfTwoF(exp);
            
}
            
else {
                
// return a subnormal result; left shift integer
                
// representation of FloatConsts.MIN_VALUE appropriate
                
// number of positions
                
return Float.intBitsToFloat(1 <<
                
(exp - (FloatConsts.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1)) ));
            
}
        
}
    
}

    
/**
     
* Returns the signum function of the argument; zero if the argument
     
* is zero, 1.0 if the argument is greater than zero, -1.0 if the
     
* argument is less than zero.
     
*
     
* <p>Special Cases:
     
* <ul>
     
* <li> If the argument is NaN, then the result is NaN.
     
* <li> If the argument is positive zero or negative zero, then the
     
*
      
result is the same as the argument.
     
* </ul>
     
*
     
* @param d the floating-point value whose signum is to be returned
     
* @return the signum function of the argument
     
* @author Joseph D. Darcy
     
* @since 1.5
     
*/

    
public static double signum(double d) {
        
return (d == 0.0 || Double.isNaN(d))?d:copySign(1.0, d);
    
}

    
/**
     
* Returns the signum function of the argument; zero if the argument
     
* is zero, 1.0f if the argument is greater than zero, -1.0f if the
     
* argument is less than zero.
     
*
     
* <p>Special Cases:
     
* <ul>
     
* <li> If the argument is NaN, then the result is NaN.
     
* <li> If the argument is positive zero or negative zero, then the
     
*
      
result is the same as the argument.
     
* </ul>
     
*
     
* @param f the floating-point value whose signum is to be returned
     
* @return the signum function of the argument
     
* @author Joseph D. Darcy
     
* @since 1.5
     
*/

    
public static float signum(float f) {
        
return (f == 0.0f || Float.isNaN(f))?f:copySign(1.0f, f);
    
}

    
/**
     
* Returns the hyperbolic sine of a {@code double} value.
     
* The hyperbolic sine of <i>x</i> is defined to be
     
* (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/2
     
* where <i>e</i> is {@linkplain Math#E Euler's number}.
     
*
     
* <p>Special cases:
     
* <ul>
     
*
     
* <li>If the argument is NaN, then the result is NaN.
     
*
     
* <li>If the argument is infinite, then the result is an infinity
     
* with the same sign as the argument.
     
*
     
* <li>If the argument is zero, then the result is a zero with the
     
* same sign as the argument.
     
*
     
* </ul>
     
*
     
* <p>The computed result must be within 2.5 ulps of the exact result.
     
*
     
* @param
   
x The number whose hyperbolic sine is to be returned.
     
* @return
  
The hyperbolic sine of {@code x}.
     
* @since 1.5
     
*/

    
public static double sinh(double x) {
        
return StrictMath.sinh(x);
    
}

    
/**
     
* Returns the hyperbolic cosine of a {@code double} value.
     
* The hyperbolic cosine of <i>x</i> is defined to be
     
* (<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>)/2
     
* where <i>e</i> is {@linkplain Math#E Euler's number}.
     
*
     
* <p>Special cases:
     
* <ul>
     
*
     
* <li>If the argument is NaN, then the result is NaN.
     
*
     
* <li>If the argument is infinite, then the result is positive
     
* infinity.
     
*
     
* <li>If the argument is zero, then the result is {@code 1.0}.
     
*
     
* </ul>
     
*
     
* <p>The computed result must be within 2.5 ulps of the exact result.
     
*
     
* @param
   
x The number whose hyperbolic cosine is to be returned.
     
* @return
  
The hyperbolic cosine of {@code x}.
     
* @since 1.5
     
*/

    
public static double cosh(double x) {
        
return StrictMath.cosh(x);
    
}

    
/**
     
* Returns the hyperbolic tangent of a {@code double} value.
     
* The hyperbolic tangent of <i>x</i> is defined to be
     
* (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/(<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>),
     
* in other words, {@linkplain Math#sinh
     
* sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}.
  
Note
     
* that the absolute value of the exact tanh is always less than
     
* 1.
     
*
     
* <p>Special cases:
     
* <ul>
     
*
     
* <li>If the argument is NaN, then the result is NaN.
     
*
     
* <li>If the argument is zero, then the result is a zero with the
     
* same sign as the argument.
     
*
     
* <li>If the argument is positive infinity, then the result is
     
* {@code +1.0}.
     
*
     
* <li>If the argument is negative infinity, then the result is
     
* {@code -1.0}.
     
*
     
* </ul>
     
*
     
* <p>The computed result must be within 2.5 ulps of the exact result.
     
* The result of {@code tanh} for any finite input must have
     
* an absolute value less than or equal to 1.
  
Note that once the
     
* exact result of tanh is within 1/2 of an ulp of the limit value
     
* of &plusmn;1, correctly signed &plusmn;{@code 1.0} should
     
* be returned.
     
*
     
* @param
   
x The number whose hyperbolic tangent is to be returned.
     
* @return
  
The hyperbolic tangent of {@code x}.
     
* @since 1.5
     
*/

    
public static double tanh(double x) {
        
return StrictMath.tanh(x);
    
}

    
/**
     
* Returns sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
     
* without intermediate overflow or underflow.
     
*
     
* <p>Special cases:
     
* <ul>
     
*
     
* <li> If either argument is infinite, then the result
     
* is positive infinity.
     
*
     
* <li> If either argument is NaN and neither argument is infinite,
     
* then the result is NaN.
     
*
     
* </ul>
     
*
     
* <p>The computed result must be within 1 ulp of the exact
     
* result.
  
If one parameter is held constant, the results must be
     
* semi-monotonic in the other parameter.
     
*
     
* @param x a value
     
* @param y a value
     
* @return sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
     
* without intermediate overflow or underflow
     
* @since 1.5
     
*/

    
public static double hypot(double x, double y) {
        
return StrictMath.hypot(x, y);
    
}

    
/**
     
* Returns <i>e</i><sup>x</sup>&nbsp;-1.
  
Note that for values of
     
* <i>x</i> near 0, the exact sum of
     
* {@code expm1(x)}&nbsp;+&nbsp;1 is much closer to the true
     
* result of <i>e</i><sup>x</sup> than {@code exp(x)}.
     
*
     
* <p>Special cases:
     
* <ul>
     
* <li>If the argument is NaN, the result is NaN.
     
*
     
* <li>If the argument is positive infinity, then the result is
     
* positive infinity.
     
*
     
* <li>If the argument is negative infinity, then the result is
     
* -1.0.
     
*
     
* <li>If the argument is zero, then the result is a zero with the
     
* same sign as the argument.
     
*
     
* </ul>
     
*
     
* <p>The computed result must be within 1 ulp of the exact result.
     
* Results must be semi-monotonic.
  
The result of
     
* {@code expm1} for any finite input must be greater than or
     
* equal to {@code -1.0}.
  
Note that once the exact result of
     
* <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1 is within 1/2
     
* ulp of the limit value -1, {@code -1.0} should be
     
* returned.
     
*
     
* @param
   
xthe exponent to raise <i>e</i> to in the computation of
     
*
              
<i>e</i><sup>{@code x}</sup>&nbsp;-1.
     
* @return
  
the value <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1.
     
* @since 1.5
     
*/

    
public static double expm1(double x) {
        
return StrictMath.expm1(x);
    
}

    
/**
     
* Returns the natural logarithm of the sum of the argument and 1.
     
* Note that for small values {@code x}, the result of
     
* {@code log1p(x)} is much closer to the true result of ln(1
     
* + {@code x}) than the floating-point evaluation of
     
* {@code log(1.0+x)}.
     
*
     
* <p>Special cases:
     
*
     
* <ul>
     
*
     
* <li>If the argument is NaN or less than -1, then the result is
     
* NaN.
     
*
     
* <li>If the argument is positive infinity, then the result is
     
* positive infinity.
     
*
     
* <li>If the argument is negative one, then the result is
     
* negative infinity.
     
*
     
* <li>If the argument is zero, then the result is a zero with the
     
* same sign as the argument.
     
*
     
* </ul>
     
*
     
* <p>The computed result must be within 1 ulp of the exact result.
     
* Results must be semi-monotonic.
     
*
     
* @param
   
xa value
     
* @return the value ln({@code x}&nbsp;+&nbsp;1), the natural
     
* log of {@code x}&nbsp;+&nbsp;1
     
* @since 1.5
     
*/

    
public static double log1p(double x) {
        
return StrictMath.log1p(x);
    
}

    
/**
     
* Returns the first floating-point argument with the sign of the
     
* second floating-point argument.
  
Note that unlike the {@link
     
* StrictMath#copySign(double, double) StrictMath.copySign}
     
* method, this method does not require NaN {@code sign}
     
* arguments to be treated as positive values; implementations are
     
* permitted to treat some NaN arguments as positive and other NaN
     
* arguments as negative to allow greater performance.
     
*
     
* @param magnitude
  
the parameter providing the magnitude of the result
     
* @param sign
   
the parameter providing the sign of the result
     
* @return a value with the magnitude of {@code magnitude}
     
* and the sign of {@code sign}.
     
* @since 1.6
     
*/

    
public static double copySign(double magnitude, double sign) {
        
return Double.longBitsToDouble((Double.doubleToRawLongBits(sign) &
                                        
(DoubleConsts.SIGN_BIT_MASK)) |
                                       
(Double.doubleToRawLongBits(magnitude) &
                                        
(DoubleConsts.EXP_BIT_MASK |
                                         
DoubleConsts.SIGNIF_BIT_MASK)));
    
}

    
/**
     
* Returns the first floating-point argument with the sign of the
     
* second floating-point argument.
  
Note that unlike the {@link
     
* StrictMath#copySign(float, float) StrictMath.copySign}
     
* method, this method does not require NaN {@code sign}
     
* arguments to be treated as positive values; implementations are
     
* permitted to treat some NaN arguments as positive and other NaN
     
* arguments as negative to allow greater performance.
     
*
     
* @param magnitude
  
the parameter providing the magnitude of the result
     
* @param sign
   
the parameter providing the sign of the result
     
* @return a value with the magnitude of {@code magnitude}
     
* and the sign of {@code sign}.
     
* @since 1.6
     
*/

    
public static float copySign(float magnitude, float sign) {
        
return Float.intBitsToFloat((Float.floatToRawIntBits(sign) &
                                     
(FloatConsts.SIGN_BIT_MASK)) |
                                    
(Float.floatToRawIntBits(magnitude) &
                                     
(FloatConsts.EXP_BIT_MASK |
                                      
FloatConsts.SIGNIF_BIT_MASK)));
    
}

    
/**
     
* Returns the unbiased exponent used in the representation of a
     
* {@code float}.
  
Special cases:
     
*
     
* <ul>
     
* <li>If the argument is NaN or infinite, then the result is
     
* {@link Float#MAX_EXPONENT} + 1.
     
* <li>If the argument is zero or subnormal, then the result is
     
* {@link Float#MIN_EXPONENT} -1.
     
* </ul>
     
* @param f a {@code float} value
     
* @return the unbiased exponent of the argument
     
* @since 1.6
     
*/

    
public static int getExponent(float f) {
        
/*
         
* Bitwise convert f to integer, mask out exponent bits, shift
         
* to the right and then subtract out float's bias adjust to
         
* get true exponent value
         
*/

        
return ((Float.floatToRawIntBits(f) & FloatConsts.EXP_BIT_MASK) >>
                
(FloatConsts.SIGNIFICAND_WIDTH - 1)) - FloatConsts.EXP_BIAS;
    
}

    
/**
     
* Returns the unbiased exponent used in the representation of a
     
* {@code double}.
  
Special cases:
     
*
     
* <ul>
     
* <li>If the argument is NaN or infinite, then the result is
     
* {@link Double#MAX_EXPONENT} + 1.
     
* <li>If the argument is zero or subnormal, then the result is
     
* {@link Double#MIN_EXPONENT} -1.
     
* </ul>
     
* @param d a {@code double} value
     
* @return the unbiased exponent of the argument
     
* @since 1.6
     
*/

    
public static int getExponent(double d) {
        
/*
         
* Bitwise convert d to long, mask out exponent bits, shift
         
* to the right and then subtract out double's bias adjust to
         
* get true exponent value.
         
*/

        
return (int)(((Double.doubleToRawLongBits(d) & DoubleConsts.EXP_BIT_MASK) >>
                      
(DoubleConsts.SIGNIFICAND_WIDTH - 1)) - DoubleConsts.EXP_BIAS);
    
}

    
/**
     
* Returns the floating-point number adjacent to the first
     
* argument in the direction of the second argument.
  
If both
     
* arguments compare as equal the second argument is returned.
     
*
     
* <p>
     
* Special cases:
     
* <ul>
     
* <li> If either argument is a NaN, then NaN is returned.
     
*
     
* <li> If both arguments are signed zeros, {@code direction}
     
* is returned unchanged (as implied by the requirement of
     
* returning the second argument if the arguments compare as
     
* equal).
     
*
     
* <li> If {@code start} is
     
* &plusmn;{@link Double#MIN_VALUE} and {@code direction}
     
* has a value such that the result should have a smaller
     
* magnitude, then a zero with the same sign as {@code start}
     
* is returned.
     
*
     
* <li> If {@code start} is infinite and
     
* {@code direction} has a value such that the result should
     
* have a smaller magnitude, {@link Double#MAX_VALUE} with the
     
* same sign as {@code start} is returned.
     
*
     
* <li> If {@code start} is equal to &plusmn;
     
* {@link Double#MAX_VALUE} and {@code direction} has a
     
* value such that the result should have a larger magnitude, an
     
* infinity with same sign as {@code start} is returned.
     
* </ul>
     
*
     
* @param start
  
starting floating-point value
     
* @param direction value indicating which of
     
* {@code start}'s neighbors or {@code start} should
     
* be returned
     
* @return The floating-point number adjacent to {@code start} in the
     
* direction of {@code direction}.
     
* @since 1.6
     
*/

    
public static double nextAfter(double start, double direction) {
        
/*
         
* The cases:
         
*
         
* nextAfter(+infinity, 0)
  
== MAX_VALUE
         
* nextAfter(+infinity, +infinity)
  
== +infinity
         
* nextAfter(-infinity, 0)
  
== -MAX_VALUE
         
* nextAfter(-infinity, -infinity)
  
== -infinity
         
*
         
* are naturally handled without any additional testing
         
*/


        
// 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);
    
}
}