xref: /frameworks/base/core/java/android/util/Half.java

/*
 * Copyright (C) 2016 The Android Open Source Project
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

package android.util;

import android.annotation.HalfFloat;
import android.annotation.NonNull;
import android.annotation.Nullable;

import sun.misc.FloatingDecimal;

/**
 * <p>The {@code Half} class is a wrapper and a utility class to manipulate half-precision 16-bit
 * <a href="https://en.wikipedia.org/wiki/Half-precision_floating-point_format">IEEE 754</a>
 * floating point data types (also called fp16 or binary16). A half-precision float can be
 * created from or converted to single-precision floats, and is stored in a short data type.
 * To distinguish short values holding half-precision floats from regular short values,
 * it is recommended to use the <code>@HalfFloat</code> annotation.</p>
 *
 * <p>The IEEE 754 standard specifies an fp16 as having the following format:</p>
 * <ul>
 * <li>Sign bit: 1 bit</li>
 * <li>Exponent width: 5 bits</li>
 * <li>Significand: 10 bits</li>
 * </ul>
 *
 * <p>The format is laid out as follows:</p>
 * <pre>
 * 1   11111   1111111111
 * ^   --^--   -----^----
 * sign  |          |_______ significand
 *       |
 *       -- exponent
 * </pre>
 *
 * <p>Half-precision floating points can be useful to save memory and/or
 * bandwidth at the expense of range and precision when compared to single-precision
 * floating points (fp32).</p>
 * <p>To help you decide whether fp16 is the right storage type for you need, please
 * refer to the table below that shows the available precision throughout the range of
 * possible values. The <em>precision</em> column indicates the step size between two
 * consecutive numbers in a specific part of the range.</p>
 *
 * <table summary="Precision of fp16 across the range">
 *     <tr><th>Range start</th><th>Precision</th></tr>
 *     <tr><td>0</td><td>1 &frasl; 16,777,216</td></tr>
 *     <tr><td>1 &frasl; 16,384</td><td>1 &frasl; 16,777,216</td></tr>
 *     <tr><td>1 &frasl; 8,192</td><td>1 &frasl; 8,388,608</td></tr>
 *     <tr><td>1 &frasl; 4,096</td><td>1 &frasl; 4,194,304</td></tr>
 *     <tr><td>1 &frasl; 2,048</td><td>1 &frasl; 2,097,152</td></tr>
 *     <tr><td>1 &frasl; 1,024</td><td>1 &frasl; 1,048,576</td></tr>
 *     <tr><td>1 &frasl; 512</td><td>1 &frasl; 524,288</td></tr>
 *     <tr><td>1 &frasl; 256</td><td>1 &frasl; 262,144</td></tr>
 *     <tr><td>1 &frasl; 128</td><td>1 &frasl; 131,072</td></tr>
 *     <tr><td>1 &frasl; 64</td><td>1 &frasl; 65,536</td></tr>
 *     <tr><td>1 &frasl; 32</td><td>1 &frasl; 32,768</td></tr>
 *     <tr><td>1 &frasl; 16</td><td>1 &frasl; 16,384</td></tr>
 *     <tr><td>1 &frasl; 8</td><td>1 &frasl; 8,192</td></tr>
 *     <tr><td>1 &frasl; 4</td><td>1 &frasl; 4,096</td></tr>
 *     <tr><td>1 &frasl; 2</td><td>1 &frasl; 2,048</td></tr>
 *     <tr><td>1</td><td>1 &frasl; 1,024</td></tr>
 *     <tr><td>2</td><td>1 &frasl; 512</td></tr>
 *     <tr><td>4</td><td>1 &frasl; 256</td></tr>
 *     <tr><td>8</td><td>1 &frasl; 128</td></tr>
 *     <tr><td>16</td><td>1 &frasl; 64</td></tr>
 *     <tr><td>32</td><td>1 &frasl; 32</td></tr>
 *     <tr><td>64</td><td>1 &frasl; 16</td></tr>
 *     <tr><td>128</td><td>1 &frasl; 8</td></tr>
 *     <tr><td>256</td><td>1 &frasl; 4</td></tr>
 *     <tr><td>512</td><td>1 &frasl; 2</td></tr>
 *     <tr><td>1,024</td><td>1</td></tr>
 *     <tr><td>2,048</td><td>2</td></tr>
 *     <tr><td>4,096</td><td>4</td></tr>
 *     <tr><td>8,192</td><td>8</td></tr>
 *     <tr><td>16,384</td><td>16</td></tr>
 *     <tr><td>32,768</td><td>32</td></tr>
 * </table>
 *
 * <p>This table shows that numbers higher than 1024 lose all fractional precision.</p>
 */
@SuppressWarnings("SimplifiableIfStatement")
public final class Half extends Number implements Comparable<Half> {
    /**
     * The number of bits used to represent a half-precision float value.
     */
    public static final int SIZE = 16;

    /**
     * Epsilon is the difference between 1.0 and the next value representable
     * by a half-precision floating-point.
     */
    public static final @HalfFloat short EPSILON = (short) 0x1400;

    /**
     * Maximum exponent a finite half-precision float may have.
     */
    public static final int MAX_EXPONENT = 15;
    /**
     * Minimum exponent a normalized half-precision float may have.
     */
    public static final int MIN_EXPONENT = -14;

    /**
     * Smallest negative value a half-precision float may have.
     */
    public static final @HalfFloat short LOWEST_VALUE = (short) 0xfbff;
    /**
     * Maximum positive finite value a half-precision float may have.
     */
    public static final @HalfFloat short MAX_VALUE = (short) 0x7bff;
    /**
     * Smallest positive normal value a half-precision float may have.
     */
    public static final @HalfFloat short MIN_NORMAL = (short) 0x0400;
    /**
     * Smallest positive non-zero value a half-precision float may have.
     */
    public static final @HalfFloat short MIN_VALUE = (short) 0x0001;
    /**
     * A Not-a-Number representation of a half-precision float.
     */
    public static final @HalfFloat short NaN = (short) 0x7e00;
    /**
     * Negative infinity of type half-precision float.
     */
    public static final @HalfFloat short NEGATIVE_INFINITY = (short) 0xfc00;
    /**
     * Negative 0 of type half-precision float.
     */
    public static final @HalfFloat short NEGATIVE_ZERO = (short) 0x8000;
    /**
     * Positive infinity of type half-precision float.
     */
    public static final @HalfFloat short POSITIVE_INFINITY = (short) 0x7c00;
    /**
     * Positive 0 of type half-precision float.
     */
    public static final @HalfFloat short POSITIVE_ZERO = (short) 0x0000;

    private static final int FP16_SIGN_SHIFT        = 15;
    private static final int FP16_SIGN_MASK         = 0x8000;
    private static final int FP16_EXPONENT_SHIFT    = 10;
    private static final int FP16_EXPONENT_MASK     = 0x1f;
    private static final int FP16_SIGNIFICAND_MASK  = 0x3ff;
    private static final int FP16_EXPONENT_BIAS     = 15;
    private static final int FP16_COMBINED          = 0x7fff;
    private static final int FP16_EXPONENT_MAX      = 0x7c00;

    private static final int FP32_SIGN_SHIFT        = 31;
    private static final int FP32_EXPONENT_SHIFT    = 23;
    private static final int FP32_EXPONENT_MASK     = 0xff;
    private static final int FP32_SIGNIFICAND_MASK  = 0x7fffff;
    private static final int FP32_EXPONENT_BIAS     = 127;

    private static final int FP32_DENORMAL_MAGIC = 126 << 23;
    private static final float FP32_DENORMAL_FLOAT = Float.intBitsToFloat(FP32_DENORMAL_MAGIC);

    private final @HalfFloat short mValue;

    /**
     * Constructs a newly allocated {@code Half} object that represents the
     * half-precision float type argument.
     *
     * @param value The value to be represented by the {@code Half}
     */
    public Half(@HalfFloat short value) {
        mValue = value;
    }

    /**
     * Constructs a newly allocated {@code Half} object that represents the
     * argument converted to a half-precision float.
     *
     * @param value The value to be represented by the {@code Half}
     *
     * @see #toHalf(float)
     */
    public Half(float value) {
        mValue = toHalf(value);
    }

    /**
     * Constructs a newly allocated {@code Half} object that
     * represents the argument converted to a half-precision float.
     *
     * @param value The value to be represented by the {@code Half}
     *
     * @see #toHalf(float)
     */
    public Half(double value) {
        mValue = toHalf((float) value);
    }

    /**
     * <p>Constructs a newly allocated {@code Half} object that represents the
     * half-precision float value represented by the string.
     * The string is converted to a half-precision float value as if by the
     * {@link #valueOf(String)} method.</p>
     *
     * <p>Calling this constructor is equivalent to calling:</p>
     * <pre>
     *     new Half(Float.parseFloat(value))
     * </pre>
     *
     * @param value A string to be converted to a {@code Half}
     * @throws NumberFormatException if the string does not contain a parsable number
     *
     * @see Float#valueOf(java.lang.String)
     * @see #toHalf(float)
     */
    public Half(@NonNull String value) throws NumberFormatException {
        mValue = toHalf(Float.parseFloat(value));
    }

    /**
     * Returns the half-precision value of this {@code Half} as a {@code short}
     * containing the bit representation described in {@link Half}.
     *
     * @return The half-precision float value represented by this object
     */
    public @HalfFloat short halfValue() {
        return mValue;
    }

    /**
     * Returns the value of this {@code Half} as a {@code byte} after
     * a narrowing primitive conversion.
     *
     * @return The half-precision float value represented by this object
     *         converted to type {@code byte}
     */
    @Override
    public byte byteValue() {
        return (byte) toFloat(mValue);
    }

    /**
     * Returns the value of this {@code Half} as a {@code short} after
     * a narrowing primitive conversion.
     *
     * @return The half-precision float value represented by this object
     *         converted to type {@code short}
     */
    @Override
    public short shortValue() {
        return (short) toFloat(mValue);
    }

    /**
     * Returns the value of this {@code Half} as a {@code int} after
     * a narrowing primitive conversion.
     *
     * @return The half-precision float value represented by this object
     *         converted to type {@code int}
     */
    @Override
    public int intValue() {
        return (int) toFloat(mValue);
    }

    /**
     * Returns the value of this {@code Half} as a {@code long} after
     * a narrowing primitive conversion.
     *
     * @return The half-precision float value represented by this object
     *         converted to type {@code long}
     */
    @Override
    public long longValue() {
        return (long) toFloat(mValue);
    }

    /**
     * Returns the value of this {@code Half} as a {@code float} after
     * a widening primitive conversion.
     *
     * @return The half-precision float value represented by this object
     *         converted to type {@code float}
     */
    @Override
    public float floatValue() {
        return toFloat(mValue);
    }

    /**
     * Returns the value of this {@code Half} as a {@code double} after
     * a widening primitive conversion.
     *
     * @return The half-precision float value represented by this object
     *         converted to type {@code double}
     */
    @Override
    public double doubleValue() {
        return toFloat(mValue);
    }

    /**
     * Returns true if this {@code Half} value represents a Not-a-Number,
     * false otherwise.
     *
     * @return True if the value is a NaN, false otherwise
     */
    public boolean isNaN() {
        return isNaN(mValue);
    }

    /**
     * Compares this object against the specified object. The result is {@code true}
     * if and only if the argument is not {@code null} and is a {@code Half} object
     * that represents the same half-precision value as the this object. Two
     * half-precision values are considered to be the same if and only if the method
     * {@link #halfToIntBits(short)} returns an identical {@code int} value for both.
     *
     * @param o The object to compare
     * @return True if the objects are the same, false otherwise
     *
     * @see #halfToIntBits(short)
     */
    @Override
    public boolean equals(@Nullable Object o) {
        return (o instanceof Half) &&
                (halfToIntBits(((Half) o).mValue) == halfToIntBits(mValue));
    }

    /**
     * Returns a hash code for this {@code Half} object. The result is the
     * integer bit representation, exactly as produced by the method
     * {@link #halfToIntBits(short)}, of the primitive half-precision float
     * value represented by this {@code Half} object.
     *
     * @return A hash code value for this object
     */
    @Override
    public int hashCode() {
        return hashCode(mValue);
    }

    /**
     * Returns a string representation of the specified half-precision
     * float value. See {@link #toString(short)} for more information.
     *
     * @return A string representation of this {@code Half} object
     */
    @NonNull
    @Override
    public String toString() {
        return toString(mValue);
    }

    /**
     * <p>Compares the two specified half-precision float values. The following
     * conditions apply during the comparison:</p>
     *
     * <ul>
     * <li>{@link #NaN} is considered by this method to be equal to itself and greater
     * than all other half-precision float values (including {@code #POSITIVE_INFINITY})</li>
     * <li>{@link #POSITIVE_ZERO} is considered by this method to be greater than
     * {@link #NEGATIVE_ZERO}.</li>
     * </ul>
     *
     * @param h The half-precision float value to compare to the half-precision value
     *          represented by this {@code Half} object
     *
     * @return  The value {@code 0} if {@code x} is numerically equal to {@code y}; a
     *          value less than {@code 0} if {@code x} is numerically less than {@code y};
     *          and a value greater than {@code 0} if {@code x} is numerically greater
     *          than {@code y}
     */
    @Override
    public int compareTo(@NonNull Half h) {
        return compare(mValue, h.mValue);
    }

    /**
     * Returns a hash code for a half-precision float value.
     *
     * @param h The value to hash
     *
     * @return A hash code value for a half-precision float value
     */
    public static int hashCode(@HalfFloat short h) {
        return halfToIntBits(h);
    }

    /**
     * <p>Compares the two specified half-precision float values. The following
     * conditions apply during the comparison:</p>
     *
     * <ul>
     * <li>{@link #NaN} is considered by this method to be equal to itself and greater
     * than all other half-precision float values (including {@code #POSITIVE_INFINITY})</li>
     * <li>{@link #POSITIVE_ZERO} is considered by this method to be greater than
     * {@link #NEGATIVE_ZERO}.</li>
     * </ul>
     *
     * @param x The first half-precision float value to compare.
     * @param y The second half-precision float value to compare
     *
     * @return  The value {@code 0} if {@code x} is numerically equal to {@code y}, a
     *          value less than {@code 0} if {@code x} is numerically less than {@code y},
     *          and a value greater than {@code 0} if {@code x} is numerically greater
     *          than {@code y}
     */
    public static int compare(@HalfFloat short x, @HalfFloat short y) {
        if (less(x, y)) return -1;
        if (greater(x, y)) return 1;

        // Collapse NaNs, akin to halfToIntBits(), but we want to keep
        // (signed) short value types to preserve the ordering of -0.0
        // and +0.0
        short xBits = (x & FP16_COMBINED) > FP16_EXPONENT_MAX ? NaN : x;
        short yBits = (y & FP16_COMBINED) > FP16_EXPONENT_MAX ? NaN : y;

        return (xBits == yBits ? 0 : (xBits < yBits ? -1 : 1));
    }

    /**
     * <p>Returns a representation of the specified half-precision float value
     * according to the bit layout described in {@link Half}.</p>
     *
     * <p>Similar to {@link #halfToIntBits(short)}, this method collapses all
     * possible Not-a-Number values to a single canonical Not-a-Number value
     * defined by {@link #NaN}.</p>
     *
     * @param h A half-precision float value
     * @return The bits that represent the half-precision float value
     *
     * @see #halfToIntBits(short)
     */
    public static @HalfFloat short halfToShortBits(@HalfFloat short h) {
        return (h & FP16_COMBINED) > FP16_EXPONENT_MAX ? NaN : h;
    }

    /**
     * <p>Returns a representation of the specified half-precision float value
     * according to the bit layout described in {@link Half}.</p>
     *
     * <p>Unlike {@link #halfToRawIntBits(short)}, this method collapses all
     * possible Not-a-Number values to a single canonical Not-a-Number value
     * defined by {@link #NaN}.</p>
     *
     * @param h A half-precision float value
     * @return The bits that represent the half-precision float value
     *
     * @see #halfToRawIntBits(short)
     * @see #halfToShortBits(short)
     * @see #intBitsToHalf(int)
     */
    public static int halfToIntBits(@HalfFloat short h) {
        return (h & FP16_COMBINED) > FP16_EXPONENT_MAX ? NaN : h & 0xffff;
    }

    /**
     * <p>Returns a representation of the specified half-precision float value
     * according to the bit layout described in {@link Half}.</p>
     *
     * <p>The argument is considered to be a representation of a half-precision
     * float value according to the bit layout described in {@link Half}. The 16
     * most significant bits of the returned value are set to 0.</p>
     *
     * @param h A half-precision float value
     * @return The bits that represent the half-precision float value
     *
     * @see #halfToIntBits(short)
     * @see #intBitsToHalf(int)
     */
    public static int halfToRawIntBits(@HalfFloat short h) {
        return h & 0xffff;
    }

    /**
     * <p>Returns the half-precision float value corresponding to a given
     * bit representation.</p>
     *
     * <p>The argument is considered to be a representation of a half-precision
     * float value according to the bit layout described in {@link Half}. The 16
     * most significant bits of the argument are ignored.</p>
     *
     * @param bits An integer
     * @return The half-precision float value with the same bit pattern
     */
    public static @HalfFloat short intBitsToHalf(int bits) {
        return (short) (bits & 0xffff);
    }

    /**
     * Returns the first parameter with the sign of the second parameter.
     * This method treats NaNs as having a sign.
     *
     * @param magnitude A half-precision float value providing the magnitude of the result
     * @param sign  A half-precision float value providing the sign of the result
     * @return A value with the magnitude of the first parameter and the sign
     *         of the second parameter
     */
    public static @HalfFloat short copySign(@HalfFloat short magnitude, @HalfFloat short sign) {
        return (short) ((sign & FP16_SIGN_MASK) | (magnitude & FP16_COMBINED));
    }

    /**
     * Returns the absolute value of the specified half-precision float.
     * Special values are handled in the following ways:
     * <ul>
     * <li>If the specified half-precision float is NaN, the result is NaN</li>
     * <li>If the specified half-precision float is zero (negative or positive),
     * the result is positive zero (see {@link #POSITIVE_ZERO})</li>
     * <li>If the specified half-precision float is infinity (negative or positive),
     * the result is positive infinity (see {@link #POSITIVE_INFINITY})</li>
     * </ul>
     *
     * @param h A half-precision float value
     * @return The absolute value of the specified half-precision float
     */
    public static @HalfFloat short abs(@HalfFloat short h) {
        return (short) (h & FP16_COMBINED);
    }

    /**
     * Returns the closest integral half-precision float value to the specified
     * half-precision float value. Special values are handled in the
     * following ways:
     * <ul>
     * <li>If the specified half-precision float is NaN, the result is NaN</li>
     * <li>If the specified half-precision float is infinity (negative or positive),
     * the result is infinity (with the same sign)</li>
     * <li>If the specified half-precision float is zero (negative or positive),
     * the result is zero (with the same sign)</li>
     * </ul>
     *
     * @param h A half-precision float value
     * @return The value of the specified half-precision float rounded to the nearest
     *         half-precision float value
     */
    public static @HalfFloat short round(@HalfFloat short h) {
        int bits = h & 0xffff;
        int e = bits & 0x7fff;
        int result = bits;

        if (e < 0x3c00) {
            result &= FP16_SIGN_MASK;
            result |= (0x3c00 & (e >= 0x3800 ? 0xffff : 0x0));
        } else if (e < 0x6400) {
            e = 25 - (e >> 10);
            int mask = (1 << e) - 1;
            result += (1 << (e - 1));
            result &= ~mask;
        }

        return (short) result;
    }

    /**
     * Returns the smallest half-precision float value toward negative infinity
     * greater than or equal to the specified half-precision float value.
     * Special values are handled in the following ways:
     * <ul>
     * <li>If the specified half-precision float is NaN, the result is NaN</li>
     * <li>If the specified half-precision float is infinity (negative or positive),
     * the result is infinity (with the same sign)</li>
     * <li>If the specified half-precision float is zero (negative or positive),
     * the result is zero (with the same sign)</li>
     * </ul>
     *
     * @param h A half-precision float value
     * @return The smallest half-precision float value toward negative infinity
     *         greater than or equal to the specified half-precision float value
     */
    public static @HalfFloat short ceil(@HalfFloat short h) {
        int bits = h & 0xffff;
        int e = bits & 0x7fff;
        int result = bits;

        if (e < 0x3c00) {
            result &= FP16_SIGN_MASK;
            result |= 0x3c00 & -(~(bits >> 15) & (e != 0 ? 1 : 0));
        } else if (e < 0x6400) {
            e = 25 - (e >> 10);
            int mask = (1 << e) - 1;
            result += mask & ((bits >> 15) - 1);
            result &= ~mask;
        }

        return (short) result;
    }

    /**
     * Returns the largest half-precision float value toward positive infinity
     * less than or equal to the specified half-precision float value.
     * Special values are handled in the following ways:
     * <ul>
     * <li>If the specified half-precision float is NaN, the result is NaN</li>
     * <li>If the specified half-precision float is infinity (negative or positive),
     * the result is infinity (with the same sign)</li>
     * <li>If the specified half-precision float is zero (negative or positive),
     * the result is zero (with the same sign)</li>
     * </ul>
     *
     * @param h A half-precision float value
     * @return The largest half-precision float value toward positive infinity
     *         less than or equal to the specified half-precision float value
     */
    public static @HalfFloat short floor(@HalfFloat short h) {
        int bits = h & 0xffff;
        int e = bits & 0x7fff;
        int result = bits;

        if (e < 0x3c00) {
            result &= FP16_SIGN_MASK;
            result |= 0x3c00 & (bits > 0x8000 ? 0xffff : 0x0);
        } else if (e < 0x6400) {
            e = 25 - (e >> 10);
            int mask = (1 << e) - 1;
            result += mask & -(bits >> 15);
            result &= ~mask;
        }

        return (short) result;
    }

    /**
     * Returns the truncated half-precision float value of the specified
     * half-precision float value. Special values are handled in the following ways:
     * <ul>
     * <li>If the specified half-precision float is NaN, the result is NaN</li>
     * <li>If the specified half-precision float is infinity (negative or positive),
     * the result is infinity (with the same sign)</li>
     * <li>If the specified half-precision float is zero (negative or positive),
     * the result is zero (with the same sign)</li>
     * </ul>
     *
     * @param h A half-precision float value
     * @return The truncated half-precision float value of the specified
     *         half-precision float value
     */
    public static @HalfFloat short trunc(@HalfFloat short h) {
        int bits = h & 0xffff;
        int e = bits & 0x7fff;
        int result = bits;

        if (e < 0x3c00) {
            result &= FP16_SIGN_MASK;
        } else if (e < 0x6400) {
            e = 25 - (e >> 10);
            int mask = (1 << e) - 1;
            result &= ~mask;
        }

        return (short) result;
    }

    /**
     * Returns the smaller of two half-precision float values (the value closest
     * to negative infinity). Special values are handled in the following ways:
     * <ul>
     * <li>If either value is NaN, the result is NaN</li>
     * <li>{@link #NEGATIVE_ZERO} is smaller than {@link #POSITIVE_ZERO}</li>
     * </ul>
     *
     * @param x The first half-precision value
     * @param y The second half-precision value
     * @return The smaller of the two specified half-precision values
     */
    public static @HalfFloat short min(@HalfFloat short x, @HalfFloat short y) {
        if ((x & FP16_COMBINED) > FP16_EXPONENT_MAX) return NaN;
        if ((y & FP16_COMBINED) > FP16_EXPONENT_MAX) return NaN;

        if ((x & FP16_COMBINED) == 0 && (y & FP16_COMBINED) == 0) {
            return (x & FP16_SIGN_MASK) != 0 ? x : y;
        }

        return ((x & FP16_SIGN_MASK) != 0 ? 0x8000 - (x & 0xffff) : x & 0xffff) <
               ((y & FP16_SIGN_MASK) != 0 ? 0x8000 - (y & 0xffff) : y & 0xffff) ? x : y;
    }

    /**
     * Returns the larger of two half-precision float values (the value closest
     * to positive infinity). Special values are handled in the following ways:
     * <ul>
     * <li>If either value is NaN, the result is NaN</li>
     * <li>{@link #POSITIVE_ZERO} is greater than {@link #NEGATIVE_ZERO}</li>
     * </ul>
     *
     * @param x The first half-precision value
     * @param y The second half-precision value
     *
     * @return The larger of the two specified half-precision values
     */
    public static @HalfFloat short max(@HalfFloat short x, @HalfFloat short y) {
        if ((x & FP16_COMBINED) > FP16_EXPONENT_MAX) return NaN;
        if ((y & FP16_COMBINED) > FP16_EXPONENT_MAX) return NaN;

        if ((x & FP16_COMBINED) == 0 && (y & FP16_COMBINED) == 0) {
            return (x & FP16_SIGN_MASK) != 0 ? y : x;
        }

        return ((x & FP16_SIGN_MASK) != 0 ? 0x8000 - (x & 0xffff) : x & 0xffff) >
               ((y & FP16_SIGN_MASK) != 0 ? 0x8000 - (y & 0xffff) : y & 0xffff) ? x : y;
    }

    /**
     * Returns true if the first half-precision float value is less (smaller
     * toward negative infinity) than the second half-precision float value.
     * If either of the values is NaN, the result is false.
     *
     * @param x The first half-precision value
     * @param y The second half-precision value
     *
     * @return True if x is less than y, false otherwise
     */
    public static boolean less(@HalfFloat short x, @HalfFloat short y) {
        if ((x & FP16_COMBINED) > FP16_EXPONENT_MAX) return false;
        if ((y & FP16_COMBINED) > FP16_EXPONENT_MAX) return false;

        return ((x & FP16_SIGN_MASK) != 0 ? 0x8000 - (x & 0xffff) : x & 0xffff) <
               ((y & FP16_SIGN_MASK) != 0 ? 0x8000 - (y & 0xffff) : y & 0xffff);
    }

    /**
     * Returns true if the first half-precision float value is less (smaller
     * toward negative infinity) than or equal to the second half-precision
     * float value. If either of the values is NaN, the result is false.
     *
     * @param x The first half-precision value
     * @param y The second half-precision value
     *
     * @return True if x is less than or equal to y, false otherwise
     */
    public static boolean lessEquals(@HalfFloat short x, @HalfFloat short y) {
        if ((x & FP16_COMBINED) > FP16_EXPONENT_MAX) return false;
        if ((y & FP16_COMBINED) > FP16_EXPONENT_MAX) return false;

        return ((x & FP16_SIGN_MASK) != 0 ? 0x8000 - (x & 0xffff) : x & 0xffff) <=
               ((y & FP16_SIGN_MASK) != 0 ? 0x8000 - (y & 0xffff) : y & 0xffff);
    }

    /**
     * Returns true if the first half-precision float value is greater (larger
     * toward positive infinity) than the second half-precision float value.
     * If either of the values is NaN, the result is false.
     *
     * @param x The first half-precision value
     * @param y The second half-precision value
     *
     * @return True if x is greater than y, false otherwise
     */
    public static boolean greater(@HalfFloat short x, @HalfFloat short y) {
        if ((x & FP16_COMBINED) > FP16_EXPONENT_MAX) return false;
        if ((y & FP16_COMBINED) > FP16_EXPONENT_MAX) return false;

        return ((x & FP16_SIGN_MASK) != 0 ? 0x8000 - (x & 0xffff) : x & 0xffff) >
               ((y & FP16_SIGN_MASK) != 0 ? 0x8000 - (y & 0xffff) : y & 0xffff);
    }

    /**
     * Returns true if the first half-precision float value is greater (larger
     * toward positive infinity) than or equal to the second half-precision float
     * value. If either of the values is NaN, the result is false.
     *
     * @param x The first half-precision value
     * @param y The second half-precision value
     *
     * @return True if x is greater than y, false otherwise
     */
    public static boolean greaterEquals(@HalfFloat short x, @HalfFloat short y) {
        if ((x & FP16_COMBINED) > FP16_EXPONENT_MAX) return false;
        if ((y & FP16_COMBINED) > FP16_EXPONENT_MAX) return false;

        return ((x & FP16_SIGN_MASK) != 0 ? 0x8000 - (x & 0xffff) : x & 0xffff) >=
               ((y & FP16_SIGN_MASK) != 0 ? 0x8000 - (y & 0xffff) : y & 0xffff);
    }

    /**
     * Returns true if the two half-precision float values are equal.
     * If either of the values is NaN, the result is false. {@link #POSITIVE_ZERO}
     * and {@link #NEGATIVE_ZERO} are considered equal.
     *
     * @param x The first half-precision value
     * @param y The second half-precision value
     *
     * @return True if x is equal to y, false otherwise
     */
    public static boolean equals(@HalfFloat short x, @HalfFloat short y) {
        if ((x & FP16_COMBINED) > FP16_EXPONENT_MAX) return false;
        if ((y & FP16_COMBINED) > FP16_EXPONENT_MAX) return false;

        return x == y || ((x | y) & FP16_COMBINED) == 0;
    }

    /**
     * Returns the sign of the specified half-precision float.
     *
     * @param h A half-precision float value
     * @return 1 if the value is positive, -1 if the value is negative
     */
    public static int getSign(@HalfFloat short h) {
        return (h & FP16_SIGN_MASK) == 0 ? 1 : -1;
    }

    /**
     * Returns the unbiased exponent used in the representation of
     * the specified  half-precision float value. if the value is NaN
     * or infinite, this* method returns {@link #MAX_EXPONENT} + 1.
     * If the argument is 0 or a subnormal representation, this method
     * returns {@link #MIN_EXPONENT} - 1.
     *
     * @param h A half-precision float value
     * @return The unbiased exponent of the specified value
     */
    public static int getExponent(@HalfFloat short h) {
        return ((h >>> FP16_EXPONENT_SHIFT) & FP16_EXPONENT_MASK) - FP16_EXPONENT_BIAS;
    }

    /**
     * Returns the significand, or mantissa, used in the representation
     * of the specified half-precision float value.
     *
     * @param h A half-precision float value
     * @return The significand, or significand, of the specified vlaue
     */
    public static int getSignificand(@HalfFloat short h) {
        return h & FP16_SIGNIFICAND_MASK;
    }

    /**
     * Returns true if the specified half-precision float value represents
     * infinity, false otherwise.
     *
     * @param h A half-precision float value
     * @return True if the value is positive infinity or negative infinity,
     *         false otherwise
     */
    public static boolean isInfinite(@HalfFloat short h) {
        return (h & FP16_COMBINED) == FP16_EXPONENT_MAX;
    }

    /**
     * Returns true if the specified half-precision float value represents
     * a Not-a-Number, false otherwise.
     *
     * @param h A half-precision float value
     * @return True if the value is a NaN, false otherwise
     */
    public static boolean isNaN(@HalfFloat short h) {
        return (h & FP16_COMBINED) > FP16_EXPONENT_MAX;
    }

    /**
     * Returns true if the specified half-precision float value is normalized
     * (does not have a subnormal representation). If the specified value is
     * {@link #POSITIVE_INFINITY}, {@link #NEGATIVE_INFINITY},
     * {@link #POSITIVE_ZERO}, {@link #NEGATIVE_ZERO}, NaN or any subnormal
     * number, this method returns false.
     *
     * @param h A half-precision float value
     * @return True if the value is normalized, false otherwise
     */
    public static boolean isNormalized(@HalfFloat short h) {
        return (h & FP16_EXPONENT_MAX) != 0 && (h & FP16_EXPONENT_MAX) != FP16_EXPONENT_MAX;
    }

    /**
     * <p>Converts the specified half-precision float value into a
     * single-precision float value. The following special cases are handled:</p>
     * <ul>
     * <li>If the input is {@link #NaN}, the returned value is {@link Float#NaN}</li>
     * <li>If the input is {@link #POSITIVE_INFINITY} or
     * {@link #NEGATIVE_INFINITY}, the returned value is respectively
     * {@link Float#POSITIVE_INFINITY} or {@link Float#NEGATIVE_INFINITY}</li>
     * <li>If the input is 0 (positive or negative), the returned value is +/-0.0f</li>
     * <li>Otherwise, the returned value is a normalized single-precision float value</li>
     * </ul>
     *
     * @param h The half-precision float value to convert to single-precision
     * @return A normalized single-precision float value
     */
    public static float toFloat(@HalfFloat short h) {
        int bits = h & 0xffff;
        int s = bits & FP16_SIGN_MASK;
        int e = (bits >>> FP16_EXPONENT_SHIFT) & FP16_EXPONENT_MASK;
        int m = (bits                        ) & FP16_SIGNIFICAND_MASK;

        int outE = 0;
        int outM = 0;

        if (e == 0) { // Denormal or 0
            if (m != 0) {
                // Convert denorm fp16 into normalized fp32
                float o = Float.intBitsToFloat(FP32_DENORMAL_MAGIC + m);
                o -= FP32_DENORMAL_FLOAT;
                return s == 0 ? o : -o;
            }
        } else {
            outM = m << 13;
            if (e == 0x1f) { // Infinite or NaN
                outE = 0xff;
            } else {
                outE = e - FP16_EXPONENT_BIAS + FP32_EXPONENT_BIAS;
            }
        }

        int out = (s << 16) | (outE << FP32_EXPONENT_SHIFT) | outM;
        return Float.intBitsToFloat(out);
    }

    /**
     * <p>Converts the specified single-precision float value into a
     * half-precision float value. The following special cases are handled:</p>
     * <ul>
     * <li>If the input is NaN (see {@link Float#isNaN(float)}), the returned
     * value is {@link #NaN}</li>
     * <li>If the input is {@link Float#POSITIVE_INFINITY} or
     * {@link Float#NEGATIVE_INFINITY}, the returned value is respectively
     * {@link #POSITIVE_INFINITY} or {@link #NEGATIVE_INFINITY}</li>
     * <li>If the input is 0 (positive or negative), the returned value is
     * {@link #POSITIVE_ZERO} or {@link #NEGATIVE_ZERO}</li>
     * <li>If the input is a less than {@link #MIN_VALUE}, the returned value
     * is flushed to {@link #POSITIVE_ZERO} or {@link #NEGATIVE_ZERO}</li>
     * <li>If the input is a less than {@link #MIN_NORMAL}, the returned value
     * is a denorm half-precision float</li>
     * <li>Otherwise, the returned value is rounded to the nearest
     * representable half-precision float value</li>
     * </ul>
     *
     * @param f The single-precision float value to convert to half-precision
     * @return A half-precision float value
     */
    @SuppressWarnings("StatementWithEmptyBody")
    public static @HalfFloat short toHalf(float f) {
        int bits = Float.floatToRawIntBits(f);
        int s = (bits >>> FP32_SIGN_SHIFT    );
        int e = (bits >>> FP32_EXPONENT_SHIFT) & FP32_EXPONENT_MASK;
        int m = (bits                        ) & FP32_SIGNIFICAND_MASK;

        int outE = 0;
        int outM = 0;

        if (e == 0xff) { // Infinite or NaN
            outE = 0x1f;
            outM = m != 0 ? 0x200 : 0;
        } else {
            e = e - FP32_EXPONENT_BIAS + FP16_EXPONENT_BIAS;
            if (e >= 0x1f) { // Overflow
                outE = 0x31;
            } else if (e <= 0) { // Underflow
                if (e < -10) {
                    // The absolute fp32 value is less than MIN_VALUE, flush to +/-0
                } else {
                    // The fp32 value is a normalized float less than MIN_NORMAL,
                    // we convert to a denorm fp16
                    m = (m | 0x800000) >> (1 - e);
                    if ((m & 0x1000) != 0) m += 0x2000;
                    outM = m >> 13;
                }
            } else {
                outE = e;
                outM = m >> 13;
                if ((m & 0x1000) != 0) {
                    // Round to nearest "0.5" up
                    int out = (outE << FP16_EXPONENT_SHIFT) | outM;
                    out++;
                    return (short) (out | (s << FP16_SIGN_SHIFT));
                }
            }
        }

        return (short) ((s << FP16_SIGN_SHIFT) | (outE << FP16_EXPONENT_SHIFT) | outM);
    }

    /**
     * Returns a {@code Half} instance representing the specified
     * half-precision float value.
     *
     * @param h A half-precision float value
     * @return a {@code Half} instance representing {@code h}
     */
    public static @NonNull Half valueOf(@HalfFloat short h) {
        return new Half(h);
    }

    /**
     * Returns a {@code Half} instance representing the specified float value.
     *
     * @param f A float value
     * @return a {@code Half} instance representing {@code f}
     */
    public static @NonNull Half valueOf(float f) {
        return new Half(f);
    }

    /**
     * Returns a {@code Half} instance representing the specified string value.
     * Calling this method is equivalent to calling
     * <code>toHalf(Float.parseString(h))</code>. See {@link Float#valueOf(String)}
     * for more information on the format of the string representation.
     *
     * @param s The string to be parsed
     * @return a {@code Half} instance representing {@code h}
     * @throws NumberFormatException if the string does not contain a parsable
     *         half-precision float value
     */
    public static @NonNull Half valueOf(@NonNull String s) {
        return new Half(s);
    }

    /**
     * Returns the half-precision float value represented by the specified string.
     * Calling this method is equivalent to calling
     * <code>toHalf(Float.parseString(h))</code>. See {@link Float#valueOf(String)}
     * for more information on the format of the string representation.
     *
     * @param s The string to be parsed
     * @return A half-precision float value represented by the string
     * @throws NumberFormatException if the string does not contain a parsable
     *         half-precision float value
     */
    public static @HalfFloat short parseHalf(@NonNull String s) throws NumberFormatException {
        return toHalf(FloatingDecimal.parseFloat(s));
    }

    /**
     * Returns a string representation of the specified half-precision
     * float value. Calling this method is equivalent to calling
     * <code>Float.toString(toFloat(h))</code>. See {@link Float#toString(float)}
     * for more information on the format of the string representation.
     *
     * @param h A half-precision float value
     * @return A string representation of the specified value
     */
    @NonNull
    public static String toString(@HalfFloat short h) {
        return Float.toString(toFloat(h));
    }

    /**
     * <p>Returns a hexadecimal string representation of the specified half-precision
     * float value. If the value is a NaN, the result is <code>"NaN"</code>,
     * otherwise the result follows this format:</p>
     * <ul>
     * <li>If the sign is positive, no sign character appears in the result</li>
     * <li>If the sign is negative, the first character is <code>'-'</code></li>
     * <li>If the value is inifinity, the string is <code>"Infinity"</code></li>
     * <li>If the value is 0, the string is <code>"0x0.0p0"</code></li>
     * <li>If the value has a normalized representation, the exponent and
     * significand are represented in the string in two fields. The significand
     * starts with <code>"0x1."</code> followed by its lowercase hexadecimal
     * representation. Trailing zeroes are removed unless all digits are 0, then
     * a single zero is used. The significand representation is followed by the
     * exponent, represented by <code>"p"</code>, itself followed by a decimal
     * string of the unbiased exponent</li>
     * <li>If the value has a subnormal representation, the significand starts
     * with <code>"0x0."</code> followed by its lowercase hexadecimal
     * representation. Trailing zeroes are removed unless all digits are 0, then
     * a single zero is used. The significand representation is followed by the
     * exponent, represented by <code>"p-14"</code></li>
     * </ul>
     *
     * @param h A half-precision float value
     * @return A hexadecimal string representation of the specified value
     */
    @NonNull
    public static String toHexString(@HalfFloat short h) {
        StringBuilder o = new StringBuilder();

        int bits = h & 0xffff;
        int s = (bits >>> FP16_SIGN_SHIFT    );
        int e = (bits >>> FP16_EXPONENT_SHIFT) & FP16_EXPONENT_MASK;
        int m = (bits                        ) & FP16_SIGNIFICAND_MASK;

        if (e == 0x1f) { // Infinite or NaN
            if (m == 0) {
                if (s != 0) o.append('-');
                o.append("Infinity");
            } else {
                o.append("NaN");
            }
        } else {
            if (s == 1) o.append('-');
            if (e == 0) {
                if (m == 0) {
                    o.append("0x0.0p0");
                } else {
                    o.append("0x0.");
                    String significand = Integer.toHexString(m);
                    o.append(significand.replaceFirst("0{2,}$", ""));
                    o.append("p-14");
                }
            } else {
                o.append("0x1.");
                String significand = Integer.toHexString(m);
                o.append(significand.replaceFirst("0{2,}$", ""));
                o.append('p');
                o.append(Integer.toString(e - FP16_EXPONENT_BIAS));
            }
        }

        return o.toString();
    }
}