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 * Copyright (c) 2003, 2010 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

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 */

package sun.misc;

import sun.misc.FloatConsts;

import sun.misc.DoubleConsts;

/**

 * The class {@code FpUtils} contains static utility methods for

 * manipulating and inspecting {@code float} and

 * {@code double} floating-point numbers.  These methods include

 * functionality recommended or required by the IEEE 754

 * floating-point standard.

 *

 * @author Joseph D. Darcy

 */

public class FpUtils {

    /*

     * The methods in this class are reasonably implemented using

     * direct or indirect bit-level manipulation of floating-point

     * values.  However, having access to the IEEE 754 recommended

     * functions would obviate the need for most programmers to engage

     * in floating-point bit-twiddling.

     *

     * An IEEE 754 number has three fields, from most significant bit

     * to to least significant, sign, exponent, and significand.

     *

     *  msb                                lsb

     * [sign|exponent|  fractional_significand]

     *

     * Using some encoding cleverness, explained below, the high order

     * bit of the logical significand does not need to be explicitly

     * stored, thus "fractional_significand" instead of simply

     * "significand" in the figure above.

     *

     * For finite normal numbers, the numerical value encoded is

     *

     * (-1)^sign * 2^(exponent)*(1.fractional_significand)

     *

     * Most finite floating-point numbers are normalized; the exponent

     * value is reduced until the leading significand bit is 1.

     * Therefore, the leading 1 is redundant and is not explicitly

     * stored.  If a numerical value is so small it cannot be

     * normalized, it has a subnormal representation. Subnormal

     * numbers don't have a leading 1 in their significand; subnormals

     * are encoding using a special exponent value.  In other words,

     * the high-order bit of the logical significand can be elided in

     * from the representation in either case since the bit's value is

     * implicit from the exponent value.

     *

     * The exponent field uses a biased representation; if the bits of

     * the exponent are interpreted as a unsigned integer E, the

     * exponent represented is E - E_bias where E_bias depends on the

     * floating-point format.  E can range between E_min and E_max,

     * constants which depend on the floating-point format.  E_min and

     * E_max are -126 and +127 for float, -1022 and +1023 for double.

     *

     * The 32-bit float format has 1 sign bit, 8 exponent bits, and 23

     * bits for the significand (which is logically 24 bits wide

     * because of the implicit bit).  The 64-bit double format has 1

     * sign bit, 11 exponent bits, and 52 bits for the significand

     * (logically 53 bits).

     *

     * Subnormal numbers and zero have the special exponent value

     * E_min -1; the numerical value represented by a subnormal is:

     *

     * (-1)^sign * 2^(E_min)*(0.fractional_significand)

     *

     * Zero is represented by all zero bits in the exponent and all

     * zero bits in the significand; zero can have either sign.

     *

     * Infinity and NaN are encoded using the exponent value E_max +

     * 1.  Signed infinities have all significand bits zero; NaNs have

     * at least one non-zero significand bit.

     *

     * The details of IEEE 754 floating-point encoding will be used in

     * the methods below without further comment.  For further

     * exposition on IEEE 754 numbers, see "IEEE Standard for Binary

     * Floating-Point Arithmetic" ANSI/IEEE Std 754-1985 or William

     * Kahan's "Lecture Notes on the Status of IEEE Standard 754 for

     * Binary Floating-Point Arithmetic",

     * http://www.cs.berkeley.edu/~wkahan/ieee754status/ieee754.ps.

     *

     * Many of this class's methods are members of the set of IEEE 754

     * recommended functions or similar functions recommended or

     * required by IEEE 754R.  Discussion of various implementation

     * techniques for these functions have occurred in:

     *

     * W.J. Cody and Jerome T. Coonen, "Algorithm 772 Functions to

     * Support the IEEE Standard for Binary Floating-Point

     * Arithmetic," ACM Transactions on Mathematical Software,

     * vol. 19, no. 4, December 1993, pp. 443-451.

     *

     * Joseph D. Darcy, "Writing robust IEEE recommended functions in

     * ``100% Pure Java''(TM)," University of California, Berkeley

     * technical report UCB//CSD-98-1009.

     */

    /**

     * Don't let anyone instantiate this class.

     */

    private FpUtils() {}

    // Constants used in scalb

    static double twoToTheDoubleScaleUp = powerOfTwoD(512);

    static double twoToTheDoubleScaleDown = powerOfTwoD(-512);

    // Helper Methods

    // The following helper methods are used in the implementation of

    // the public recommended functions; they generally omit certain

    // tests for exception cases.

    /**

     * Returns unbiased exponent of a {@code double}.

     */

    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 unbiased exponent of a {@code float}.

     */

    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 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);

    }

    /**

     * Returns the first floating-point argument with the sign of the

     * second floating-point argument.  Note that unlike the {@link

     * FpUtils#copySign(double, double) 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}.

     * @author Joseph D. Darcy

     */

    public static double rawCopySign(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

     * FpUtils#copySign(float, float) 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}.

     * @author Joseph D. Darcy

     */

    public static float rawCopySign(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 {@code true} if the argument is a finite

     * floating-point value; returns {@code false} otherwise (for

     * NaN and infinity arguments).

     *

     * @param d the {@code double} value to be tested

     * @return {@code true} if the argument is a finite

     * floating-point value, {@code false} otherwise.

     */

    public static boolean isFinite(double d) {

        return Math.abs(d) <= DoubleConsts.MAX_VALUE;

    }

    /**

     * Returns {@code true} if the argument is a finite

     * floating-point value; returns {@code false} otherwise (for

     * NaN and infinity arguments).

     *

     * @param f the {@code float} value to be tested

     * @return {@code true} if the argument is a finite

     * floating-point value, {@code false} otherwise.

     */

     public static boolean isFinite(float f) {

        return Math.abs(f) <= FloatConsts.MAX_VALUE;

    }

    /**

     * Returns {@code true} if the specified number is infinitely

     * large in magnitude, {@code false} otherwise.

     *

     * <p>Note that this method is equivalent to the {@link

     * Double#isInfinite(double) Double.isInfinite} method; the

     * functionality is included in this class for convenience.

     *

     * @param   d   the value to be tested.

     * @return  {@code true} if the value of the argument is positive

     *          infinity or negative infinity; {@code false} otherwise.

     */

    public static boolean isInfinite(double d) {

        return Double.isInfinite(d);

    }

    /**

     * Returns {@code true} if the specified number is infinitely

     * large in magnitude, {@code false} otherwise.

     *

     * <p>Note that this method is equivalent to the {@link

     * Float#isInfinite(float) Float.isInfinite} method; the

     * functionality is included in this class for convenience.

     *

     * @param   f   the value to be tested.

     * @return  {@code true} if the argument is positive infinity or

     *          negative infinity; {@code false} otherwise.

     */

     public static boolean isInfinite(float f) {

         return Float.isInfinite(f);

    }

    /**

     * Returns {@code true} if the specified number is a

     * Not-a-Number (NaN) value, {@code false} otherwise.

     *

     * <p>Note that this method is equivalent to the {@link

     * Double#isNaN(double) Double.isNaN} method; the functionality is

     * included in this class for convenience.

     *

     * @param   d   the value to be tested.

     * @return  {@code true} if the value of the argument is NaN;

     *          {@code false} otherwise.

     */

    public static boolean isNaN(double d) {

        return Double.isNaN(d);

    }

    /**

     * Returns {@code true} if the specified number is a

     * Not-a-Number (NaN) value, {@code false} otherwise.

     *

     * <p>Note that this method is equivalent to the {@link

     * Float#isNaN(float) Float.isNaN} method; the functionality is

     * included in this class for convenience.

     *

     * @param   f   the value to be tested.

     * @return  {@code true} if the argument is NaN;

     *          {@code false} otherwise.

     */

     public static boolean isNaN(float f) {

        return Float.isNaN(f);

    }

    /**

     * Returns {@code true} if the unordered relation holds

     * between the two arguments.  When two floating-point values are

     * unordered, one value is neither less than, equal to, nor

     * greater than the other.  For the unordered relation to be true,

     * at least one argument must be a {@code NaN}.

     *

     * @param arg1      the first argument

     * @param arg2      the second argument

     * @return {@code true} if at least one argument is a NaN,

     * {@code false} otherwise.

     */

    public static boolean isUnordered(double arg1, double arg2) {

        return isNaN(arg1) || isNaN(arg2);

    }

    /**

     * Returns {@code true} if the unordered relation holds

     * between the two arguments.  When two floating-point values are

     * unordered, one value is neither less than, equal to, nor

     * greater than the other.  For the unordered relation to be true,

     * at least one argument must be a {@code NaN}.

     *

     * @param arg1      the first argument

     * @param arg2      the second argument

     * @return {@code true} if at least one argument is a NaN,

     * {@code false} otherwise.

     */

     public static boolean isUnordered(float arg1, float arg2) {

        return isNaN(arg1) || isNaN(arg2);

    }

    /**

     * Returns unbiased exponent of a {@code double}; for

     * subnormal values, the number is treated as if it were

     * normalized.  That is for all finite, non-zero, positive numbers

     * <i>x</i>, <code>scalb(<i>x</i>, -ilogb(<i>x</i>))</code> is

     * always in the range [1, 2).

     * <p>

     * Special cases:

     * <ul>

     * <li> If the argument is NaN, then the result is 2<sup>30</sup>.

     * <li> If the argument is infinite, then the result is 2<sup>28</sup>.

     * <li> If the argument is zero, then the result is -(2<sup>28</sup>).

     * </ul>

     *

     * @param d floating-point number whose exponent is to be extracted

     * @return unbiased exponent of the argument.

     * @author Joseph D. Darcy

     */

    public static int ilogb(double d) {

        int exponent = getExponent(d);

        switch (exponent) {

        case DoubleConsts.MAX_EXPONENT+1:       // NaN or infinity

            if( isNaN(d) )

                return (1<<30);         // 2^30

            else // infinite value

                return (1<<28);         // 2^28

        case DoubleConsts.MIN_EXPONENT-1:       // zero or subnormal

            if(d == 0.0) {

                return -(1<<28);        // -(2^28)

            }

            else {

                long transducer = Double.doubleToRawLongBits(d);

                /*

                 * To avoid causing slow arithmetic on subnormals,

                 * the scaling to determine when d's significand

                 * is normalized is done in integer arithmetic.

                 * (there must be at least one "1" bit in the

                 * significand since zero has been screened out.

                 */

                // isolate significand bits

                transducer &= DoubleConsts.SIGNIF_BIT_MASK;

                assert(transducer != 0L);

                // This loop is simple and functional. We might be

                // able to do something more clever that was faster;

                // e.g. number of leading zero detection on

                // (transducer << (# exponent and sign bits).

                while (transducer <

                       (1L << (DoubleConsts.SIGNIFICAND_WIDTH - 1))) {

                    transducer *= 2;

                    exponent--;

                }

                exponent++;

                assert( exponent >=

                        DoubleConsts.MIN_EXPONENT - (DoubleConsts.SIGNIFICAND_WIDTH-1) &&

                        exponent < DoubleConsts.MIN_EXPONENT);

                return exponent;

            }

        default:

            assert( exponent >= DoubleConsts.MIN_EXPONENT &&

                    exponent <= DoubleConsts.MAX_EXPONENT);

            return exponent;

        }

    }

    /**

     * Returns unbiased exponent of a {@code float}; for

     * subnormal values, the number is treated as if it were

     * normalized.  That is for all finite, non-zero, positive numbers

     * <i>x</i>, <code>scalb(<i>x</i>, -ilogb(<i>x</i>))</code> is

     * always in the range [1, 2).

     * <p>

     * Special cases:

     * <ul>

     * <li> If the argument is NaN, then the result is 2<sup>30</sup>.

     * <li> If the argument is infinite, then the result is 2<sup>28</sup>.

     * <li> If the argument is zero, then the result is -(2<sup>28</sup>).

     * </ul>

     *

     * @param f floating-point number whose exponent is to be extracted

     * @return unbiased exponent of the argument.

     * @author Joseph D. Darcy

     */

     public static int ilogb(float f) {

        int exponent = getExponent(f);

        switch (exponent) {

        case FloatConsts.MAX_EXPONENT+1:        // NaN or infinity

            if( isNaN(f) )

                return (1<<30);         // 2^30

            else // infinite value

                return (1<<28);         // 2^28

        case FloatConsts.MIN_EXPONENT-1:        // zero or subnormal

            if(f == 0.0f) {

                return -(1<<28);        // -(2^28)

            }

            else {

                int transducer = Float.floatToRawIntBits(f);

                /*

                 * To avoid causing slow arithmetic on subnormals,

                 * the scaling to determine when f's significand

                 * is normalized is done in integer arithmetic.

                 * (there must be at least one "1" bit in the

                 * significand since zero has been screened out.

                 */

                // isolate significand bits

                transducer &= FloatConsts.SIGNIF_BIT_MASK;

                assert(transducer != 0);

                // This loop is simple and functional. We might be

                // able to do something more clever that was faster;

                // e.g. number of leading zero detection on

                // (transducer << (# exponent and sign bits).

                while (transducer <

                       (1 << (FloatConsts.SIGNIFICAND_WIDTH - 1))) {

                    transducer *= 2;

                    exponent--;

                }

                exponent++;

                assert( exponent >=

                        FloatConsts.MIN_EXPONENT - (FloatConsts.SIGNIFICAND_WIDTH-1) &&

                        exponent < FloatConsts.MIN_EXPONENT);

                return exponent;

            }

        default:

            assert( exponent >= FloatConsts.MIN_EXPONENT &&

                    exponent <= FloatConsts.MAX_EXPONENT);

            return exponent;

        }

    }

    /*

     * The scalb operation should be reasonably fast; however, there

     * are tradeoffs in writing a method to minimize the worst case

     * performance and writing a method to minimize the time for

     * expected common inputs.  Some processors operate very slowly on

     * subnormal operands, taking hundreds or thousands of cycles for

     * one floating-point add or multiply as opposed to, say, four

     * cycles for normal operands.  For processors with very slow

     * subnormal execution, scalb would be fastest if written entirely

     * with integer operations; in other words, scalb would need to

     * include the logic of performing correct rounding of subnormal

     * values.  This could be reasonably done in at most a few hundred

     * cycles.  However, this approach may penalize normal operations

     * since at least the exponent of the floating-point argument must

     * be examined.

     *

     * The approach taken in this implementation is a compromise.

     * Floating-point multiplication is used to do most of the work;

     * but knowingly multiplying by a subnormal scaling factor is

     * avoided.  However, the floating-point argument is not examined

     * to see whether or not it is subnormal since subnormal inputs

     * are assumed to be rare.  At most three multiplies are needed to

     * scale from the largest to smallest exponent ranges (scaling

     * down, at most two multiplies are needed if subnormal scaling

     * factors are allowed).  However, in this implementation an

     * expensive integer remainder operation is avoided at the cost of

     * requiring five floating-point multiplies in the worst case,