1 /*
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   3  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
   4  *
   5  * This code is free software; you can redistribute it and/or modify it
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   7  * published by the Free Software Foundation.  Oracle designates this
   8  * particular file as subject to the "Classpath" exception as provided
   9  * by Oracle in the LICENSE file that accompanied this code.
  10  *
  11  * This code is distributed in the hope that it will be useful, but WITHOUT
  12  * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
  13  * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
  14  * version 2 for more details (a copy is included in the LICENSE file that
  15  * accompanied this code).
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  25 
  26 package java.lang;
  27 
  28 import java.lang.invoke.MethodHandles;
  29 import java.lang.constant.Constable;
  30 import java.lang.constant.ConstantDesc;
  31 import java.util.Optional;
  32 
  33 import jdk.internal.math.FloatConsts;
  34 import jdk.internal.math.FloatingDecimal;
  35 import jdk.internal.math.FloatToDecimal;
  36 import jdk.internal.vm.annotation.IntrinsicCandidate;
  37 
  38 /**
  39  * The {@code Float} class is the {@linkplain
  40  * java.lang##wrapperClass wrapper class} for values of the primitive
  41  * type {@code float}. An object of type {@code Float} contains a
  42  * single field whose type is {@code float}.
  43  *
  44  * <p>In addition, this class provides several methods for converting a
  45  * {@code float} to a {@code String} and a
  46  * {@code String} to a {@code float}, as well as other
  47  * constants and methods useful when dealing with a
  48  * {@code float}.
  49  *
  50  * <p>This is a <a href="{@docRoot}/java.base/java/lang/doc-files/ValueBased.html">value-based</a>
  51  * class; programmers should treat instances that are
  52  * {@linkplain #equals(Object) equal} as interchangeable and should not
  53  * use instances for synchronization, or unpredictable behavior may
  54  * occur. For example, in a future release, synchronization may fail.
  55  *
  56  * <h2><a id=equivalenceRelation>Floating-point Equality, Equivalence,
  57  * and Comparison</a></h2>
  58  *
  59  * The class {@code java.lang.Double} has a {@linkplain
  60  * Double##equivalenceRelation discussion of equality,
  61  * equivalence, and comparison of floating-point values} that is
  62  * equally applicable to {@code float} values.
  63  *
  64  * <h2><a id=decimalToBinaryConversion>Decimal &harr; Binary Conversion Issues</a></h2>
  65  *
  66  * The {@linkplain Double##decimalToBinaryConversion discussion of binary to
  67  * decimal conversion issues} in {@code java.lang.Double} is also
  68  * applicable to {@code float} values.
  69  *
  70  * @see <a href="https://standards.ieee.org/ieee/754/6210/">
  71  *      <cite>IEEE Standard for Floating-Point Arithmetic</cite></a>
  72  *
  73  * @author  Lee Boynton
  74  * @author  Arthur van Hoff
  75  * @author  Joseph D. Darcy
  76  * @since 1.0
  77  */
  78 @jdk.internal.ValueBased
  79 public final class Float extends Number
  80         implements Comparable<Float>, Constable, ConstantDesc {
  81     /**
  82      * A constant holding the positive infinity of type
  83      * {@code float}. It is equal to the value returned by
  84      * {@code Float.intBitsToFloat(0x7f800000)}.
  85      */
  86     public static final float POSITIVE_INFINITY = 1.0f / 0.0f;
  87 
  88     /**
  89      * A constant holding the negative infinity of type
  90      * {@code float}. It is equal to the value returned by
  91      * {@code Float.intBitsToFloat(0xff800000)}.
  92      */
  93     public static final float NEGATIVE_INFINITY = -1.0f / 0.0f;
  94 
  95     /**
  96      * A constant holding a Not-a-Number (NaN) value of type
  97      * {@code float}.  It is equivalent to the value returned by
  98      * {@code Float.intBitsToFloat(0x7fc00000)}.
  99      */
 100     public static final float NaN = 0.0f / 0.0f;
 101 
 102     /**
 103      * A constant holding the largest positive finite value of type
 104      * {@code float}, (2-2<sup>-23</sup>)&middot;2<sup>127</sup>.
 105      * It is equal to the hexadecimal floating-point literal
 106      * {@code 0x1.fffffeP+127f} and also equal to
 107      * {@code Float.intBitsToFloat(0x7f7fffff)}.
 108      */
 109     public static final float MAX_VALUE = 0x1.fffffeP+127f; // 3.4028235e+38f
 110 
 111     /**
 112      * A constant holding the smallest positive normal value of type
 113      * {@code float}, 2<sup>-126</sup>.  It is equal to the
 114      * hexadecimal floating-point literal {@code 0x1.0p-126f} and also
 115      * equal to {@code Float.intBitsToFloat(0x00800000)}.
 116      *
 117      * @since 1.6
 118      */
 119     public static final float MIN_NORMAL = 0x1.0p-126f; // 1.17549435E-38f
 120 
 121     /**
 122      * A constant holding the smallest positive nonzero value of type
 123      * {@code float}, 2<sup>-149</sup>. It is equal to the
 124      * hexadecimal floating-point literal {@code 0x0.000002P-126f}
 125      * and also equal to {@code Float.intBitsToFloat(0x1)}.
 126      */
 127     public static final float MIN_VALUE = 0x0.000002P-126f; // 1.4e-45f
 128 
 129     /**
 130      * The number of bits used to represent a {@code float} value,
 131      * {@value}.
 132      *
 133      * @since 1.5
 134      */
 135     public static final int SIZE = 32;
 136 
 137     /**
 138      * The number of bits in the significand of a {@code float} value,
 139      * {@value}.  This is the parameter N in section {@jls 4.2.3} of
 140      * <cite>The Java Language Specification</cite>.
 141      *
 142      * @since 19
 143      */
 144     public static final int PRECISION = 24;
 145 
 146     /**
 147      * Maximum exponent a finite {@code float} variable may have,
 148      * {@value}.  It is equal to the value returned by {@code
 149      * Math.getExponent(Float.MAX_VALUE)}.
 150      *
 151      * @since 1.6
 152      */
 153     public static final int MAX_EXPONENT = (1 << (SIZE - PRECISION - 1)) - 1; // 127
 154 
 155     /**
 156      * Minimum exponent a normalized {@code float} variable may have,
 157      * {@value}.  It is equal to the value returned by {@code
 158      * Math.getExponent(Float.MIN_NORMAL)}.
 159      *
 160      * @since 1.6
 161      */
 162     public static final int MIN_EXPONENT = 1 - MAX_EXPONENT; // -126
 163 
 164     /**
 165      * The number of bytes used to represent a {@code float} value,
 166      * {@value}.
 167      *
 168      * @since 1.8
 169      */
 170     public static final int BYTES = SIZE / Byte.SIZE;
 171 
 172     /**
 173      * The {@code Class} instance representing the primitive type
 174      * {@code float}.
 175      *
 176      * @since 1.1
 177      */
 178     public static final Class<Float> TYPE = Class.getPrimitiveClass("float");
 179 
 180     /**
 181      * Returns a string representation of the {@code float}
 182      * argument. All characters mentioned below are ASCII characters.
 183      * <ul>
 184      * <li>If the argument is NaN, the result is the string
 185      * "{@code NaN}".
 186      * <li>Otherwise, the result is a string that represents the sign and
 187      *     magnitude (absolute value) of the argument. If the sign is
 188      *     negative, the first character of the result is
 189      *     '{@code -}' ({@code '\u005Cu002D'}); if the sign is
 190      *     positive, no sign character appears in the result. As for
 191      *     the magnitude <i>m</i>:
 192      * <ul>
 193      * <li>If <i>m</i> is infinity, it is represented by the characters
 194      *     {@code "Infinity"}; thus, positive infinity produces
 195      *     the result {@code "Infinity"} and negative infinity
 196      *     produces the result {@code "-Infinity"}.
 197      * <li>If <i>m</i> is zero, it is represented by the characters
 198      *     {@code "0.0"}; thus, negative zero produces the result
 199      *     {@code "-0.0"} and positive zero produces the result
 200      *     {@code "0.0"}.
 201      *
 202      * <li> Otherwise <i>m</i> is positive and finite.
 203      * It is converted to a string in two stages:
 204      * <ul>
 205      * <li> <em>Selection of a decimal</em>:
 206      * A well-defined decimal <i>d</i><sub><i>m</i></sub>
 207      * is selected to represent <i>m</i>.
 208      * This decimal is (almost always) the <em>shortest</em> one that
 209      * rounds to <i>m</i> according to the round to nearest
 210      * rounding policy of IEEE 754 floating-point arithmetic.
 211      * <li> <em>Formatting as a string</em>:
 212      * The decimal <i>d</i><sub><i>m</i></sub> is formatted as a string,
 213      * either in plain or in computerized scientific notation,
 214      * depending on its value.
 215      * </ul>
 216      * </ul>
 217      * </ul>
 218      *
 219      * <p>A <em>decimal</em> is a number of the form
 220      * <i>s</i>&times;10<sup><i>i</i></sup>
 221      * for some (unique) integers <i>s</i> &gt; 0 and <i>i</i> such that
 222      * <i>s</i> is not a multiple of 10.
 223      * These integers are the <em>significand</em> and
 224      * the <em>exponent</em>, respectively, of the decimal.
 225      * The <em>length</em> of the decimal is the (unique)
 226      * positive integer <i>n</i> meeting
 227      * 10<sup><i>n</i>-1</sup> &le; <i>s</i> &lt; 10<sup><i>n</i></sup>.
 228      *
 229      * <p>The decimal <i>d</i><sub><i>m</i></sub> for a finite positive <i>m</i>
 230      * is defined as follows:
 231      * <ul>
 232      * <li>Let <i>R</i> be the set of all decimals that round to <i>m</i>
 233      * according to the usual <em>round to nearest</em> rounding policy of
 234      * IEEE 754 floating-point arithmetic.
 235      * <li>Let <i>p</i> be the minimal length over all decimals in <i>R</i>.
 236      * <li>When <i>p</i> &ge; 2, let <i>T</i> be the set of all decimals
 237      * in <i>R</i> with length <i>p</i>.
 238      * Otherwise, let <i>T</i> be the set of all decimals
 239      * in <i>R</i> with length 1 or 2.
 240      * <li>Define <i>d</i><sub><i>m</i></sub> as the decimal in <i>T</i>
 241      * that is closest to <i>m</i>.
 242      * Or if there are two such decimals in <i>T</i>,
 243      * select the one with the even significand.
 244      * </ul>
 245      *
 246      * <p>The (uniquely) selected decimal <i>d</i><sub><i>m</i></sub>
 247      * is then formatted.
 248      * Let <i>s</i>, <i>i</i> and <i>n</i> be the significand, exponent and
 249      * length of <i>d</i><sub><i>m</i></sub>, respectively.
 250      * Further, let <i>e</i> = <i>n</i> + <i>i</i> - 1 and let
 251      * <i>s</i><sub>1</sub>&hellip;<i>s</i><sub><i>n</i></sub>
 252      * be the usual decimal expansion of <i>s</i>.
 253      * Note that <i>s</i><sub>1</sub> &ne; 0
 254      * and <i>s</i><sub><i>n</i></sub> &ne; 0.
 255      * Below, the decimal point {@code '.'} is {@code '\u005Cu002E'}
 256      * and the exponent indicator {@code 'E'} is {@code '\u005Cu0045'}.
 257      * <ul>
 258      * <li>Case -3 &le; <i>e</i> &lt; 0:
 259      * <i>d</i><sub><i>m</i></sub> is formatted as
 260      * <code>0.0</code>&hellip;<code>0</code><!--
 261      * --><i>s</i><sub>1</sub>&hellip;<i>s</i><sub><i>n</i></sub>,
 262      * where there are exactly -(<i>n</i> + <i>i</i>) zeroes between
 263      * the decimal point and <i>s</i><sub>1</sub>.
 264      * For example, 123 &times; 10<sup>-4</sup> is formatted as
 265      * {@code 0.0123}.
 266      * <li>Case 0 &le; <i>e</i> &lt; 7:
 267      * <ul>
 268      * <li>Subcase <i>i</i> &ge; 0:
 269      * <i>d</i><sub><i>m</i></sub> is formatted as
 270      * <i>s</i><sub>1</sub>&hellip;<i>s</i><sub><i>n</i></sub><!--
 271      * --><code>0</code>&hellip;<code>0.0</code>,
 272      * where there are exactly <i>i</i> zeroes
 273      * between <i>s</i><sub><i>n</i></sub> and the decimal point.
 274      * For example, 123 &times; 10<sup>2</sup> is formatted as
 275      * {@code 12300.0}.
 276      * <li>Subcase <i>i</i> &lt; 0:
 277      * <i>d</i><sub><i>m</i></sub> is formatted as
 278      * <i>s</i><sub>1</sub>&hellip;<!--
 279      * --><i>s</i><sub><i>n</i>+<i>i</i></sub><code>.</code><!--
 280      * --><i>s</i><sub><i>n</i>+<i>i</i>+1</sub>&hellip;<!--
 281      * --><i>s</i><sub><i>n</i></sub>,
 282      * where there are exactly -<i>i</i> digits to the right of
 283      * the decimal point.
 284      * For example, 123 &times; 10<sup>-1</sup> is formatted as
 285      * {@code 12.3}.
 286      * </ul>
 287      * <li>Case <i>e</i> &lt; -3 or <i>e</i> &ge; 7:
 288      * computerized scientific notation is used to format
 289      * <i>d</i><sub><i>m</i></sub>.
 290      * Here <i>e</i> is formatted as by {@link Integer#toString(int)}.
 291      * <ul>
 292      * <li>Subcase <i>n</i> = 1:
 293      * <i>d</i><sub><i>m</i></sub> is formatted as
 294      * <i>s</i><sub>1</sub><code>.0E</code><i>e</i>.
 295      * For example, 1 &times; 10<sup>23</sup> is formatted as
 296      * {@code 1.0E23}.
 297      * <li>Subcase <i>n</i> &gt; 1:
 298      * <i>d</i><sub><i>m</i></sub> is formatted as
 299      * <i>s</i><sub>1</sub><code>.</code><i>s</i><sub>2</sub><!--
 300      * -->&hellip;<i>s</i><sub><i>n</i></sub><code>E</code><i>e</i>.
 301      * For example, 123 &times; 10<sup>-21</sup> is formatted as
 302      * {@code 1.23E-19}.
 303      * </ul>
 304      * </ul>
 305      *
 306      * <p>To create localized string representations of a floating-point
 307      * value, use subclasses of {@link java.text.NumberFormat}.
 308      *
 309      * @apiNote
 310      * This method corresponds to the general functionality of the
 311      * convertToDecimalCharacter operation defined in IEEE 754;
 312      * however, that operation is defined in terms of specifying the
 313      * number of significand digits used in the conversion.
 314      * Code to do such a conversion in the Java platform includes
 315      * converting the {@code float} to a {@link java.math.BigDecimal
 316      * BigDecimal} exactly and then rounding the {@code BigDecimal} to
 317      * the desired number of digits; sample code:
 318      * {@snippet lang=java :
 319      * floatf = 0.1f;
 320      * int digits = 15;
 321      * BigDecimal bd = new BigDecimal(f);
 322      * String result = bd.round(new MathContext(digits,  RoundingMode.HALF_UP));
 323      * // 0.100000001490116
 324      * }
 325      *
 326      * @param   f   the {@code float} to be converted.
 327      * @return a string representation of the argument.
 328      */
 329     public static String toString(float f) {
 330         return FloatToDecimal.toString(f);
 331     }
 332 
 333     /**
 334      * Returns a hexadecimal string representation of the
 335      * {@code float} argument. All characters mentioned below are
 336      * ASCII characters.
 337      *
 338      * <ul>
 339      * <li>If the argument is NaN, the result is the string
 340      *     "{@code NaN}".
 341      * <li>Otherwise, the result is a string that represents the sign and
 342      * magnitude (absolute value) of the argument. If the sign is negative,
 343      * the first character of the result is '{@code -}'
 344      * ({@code '\u005Cu002D'}); if the sign is positive, no sign character
 345      * appears in the result. As for the magnitude <i>m</i>:
 346      *
 347      * <ul>
 348      * <li>If <i>m</i> is infinity, it is represented by the string
 349      * {@code "Infinity"}; thus, positive infinity produces the
 350      * result {@code "Infinity"} and negative infinity produces
 351      * the result {@code "-Infinity"}.
 352      *
 353      * <li>If <i>m</i> is zero, it is represented by the string
 354      * {@code "0x0.0p0"}; thus, negative zero produces the result
 355      * {@code "-0x0.0p0"} and positive zero produces the result
 356      * {@code "0x0.0p0"}.
 357      *
 358      * <li>If <i>m</i> is a {@code float} value with a
 359      * normalized representation, substrings are used to represent the
 360      * significand and exponent fields.  The significand is
 361      * represented by the characters {@code "0x1."}
 362      * followed by a lowercase hexadecimal representation of the rest
 363      * of the significand as a fraction.  Trailing zeros in the
 364      * hexadecimal representation are removed unless all the digits
 365      * are zero, in which case a single zero is used. Next, the
 366      * exponent is represented by {@code "p"} followed
 367      * by a decimal string of the unbiased exponent as if produced by
 368      * a call to {@link Integer#toString(int) Integer.toString} on the
 369      * exponent value.
 370      *
 371      * <li>If <i>m</i> is a {@code float} value with a subnormal
 372      * representation, the significand is represented by the
 373      * characters {@code "0x0."} followed by a
 374      * hexadecimal representation of the rest of the significand as a
 375      * fraction.  Trailing zeros in the hexadecimal representation are
 376      * removed. Next, the exponent is represented by
 377      * {@code "p-126"}.  Note that there must be at
 378      * least one nonzero digit in a subnormal significand.
 379      *
 380      * </ul>
 381      *
 382      * </ul>
 383      *
 384      * <table class="striped">
 385      * <caption>Examples</caption>
 386      * <thead>
 387      * <tr><th scope="col">Floating-point Value</th><th scope="col">Hexadecimal String</th>
 388      * </thead>
 389      * <tbody>
 390      * <tr><th scope="row">{@code 1.0}</th> <td>{@code 0x1.0p0}</td>
 391      * <tr><th scope="row">{@code -1.0}</th>        <td>{@code -0x1.0p0}</td>
 392      * <tr><th scope="row">{@code 2.0}</th> <td>{@code 0x1.0p1}</td>
 393      * <tr><th scope="row">{@code 3.0}</th> <td>{@code 0x1.8p1}</td>
 394      * <tr><th scope="row">{@code 0.5}</th> <td>{@code 0x1.0p-1}</td>
 395      * <tr><th scope="row">{@code 0.25}</th>        <td>{@code 0x1.0p-2}</td>
 396      * <tr><th scope="row">{@code Float.MAX_VALUE}</th>
 397      *     <td>{@code 0x1.fffffep127}</td>
 398      * <tr><th scope="row">{@code Minimum Normal Value}</th>
 399      *     <td>{@code 0x1.0p-126}</td>
 400      * <tr><th scope="row">{@code Maximum Subnormal Value}</th>
 401      *     <td>{@code 0x0.fffffep-126}</td>
 402      * <tr><th scope="row">{@code Float.MIN_VALUE}</th>
 403      *     <td>{@code 0x0.000002p-126}</td>
 404      * </tbody>
 405      * </table>
 406      *
 407      * @apiNote
 408      * This method corresponds to the convertToHexCharacter operation
 409      * defined in IEEE 754.
 410      *
 411      * @param   f   the {@code float} to be converted.
 412      * @return a hex string representation of the argument.
 413      * @since 1.5
 414      * @author Joseph D. Darcy
 415      */
 416     public static String toHexString(float f) {
 417         if (Math.abs(f) < Float.MIN_NORMAL
 418             &&  f != 0.0f ) {// float subnormal
 419             // Adjust exponent to create subnormal double, then
 420             // replace subnormal double exponent with subnormal float
 421             // exponent
 422             String s = Double.toHexString(Math.scalb((double)f,
 423                                                      /* -1022+126 */
 424                                                      Double.MIN_EXPONENT-
 425                                                      Float.MIN_EXPONENT));
 426             return s.replaceFirst("p-1022$", "p-126");
 427         }
 428         else // double string will be the same as float string
 429             return Double.toHexString(f);
 430     }
 431 
 432     /**
 433      * Returns a {@code Float} object holding the
 434      * {@code float} value represented by the argument string
 435      * {@code s}.
 436      *
 437      * <p>If {@code s} is {@code null}, then a
 438      * {@code NullPointerException} is thrown.
 439      *
 440      * <p>Leading and trailing whitespace characters in {@code s}
 441      * are ignored.  Whitespace is removed as if by the {@link
 442      * String#trim} method; that is, both ASCII space and control
 443      * characters are removed. The rest of {@code s} should
 444      * constitute a <i>FloatValue</i> as described by the lexical
 445      * syntax rules:
 446      *
 447      * <blockquote>
 448      * <dl>
 449      * <dt><i>FloatValue:</i>
 450      * <dd><i>Sign<sub>opt</sub></i> {@code NaN}
 451      * <dd><i>Sign<sub>opt</sub></i> {@code Infinity}
 452      * <dd><i>Sign<sub>opt</sub> FloatingPointLiteral</i>
 453      * <dd><i>Sign<sub>opt</sub> HexFloatingPointLiteral</i>
 454      * <dd><i>SignedInteger</i>
 455      * </dl>
 456      *
 457      * <dl>
 458      * <dt><i>HexFloatingPointLiteral</i>:
 459      * <dd> <i>HexSignificand BinaryExponent FloatTypeSuffix<sub>opt</sub></i>
 460      * </dl>
 461      *
 462      * <dl>
 463      * <dt><i>HexSignificand:</i>
 464      * <dd><i>HexNumeral</i>
 465      * <dd><i>HexNumeral</i> {@code .}
 466      * <dd>{@code 0x} <i>HexDigits<sub>opt</sub>
 467      *     </i>{@code .}<i> HexDigits</i>
 468      * <dd>{@code 0X}<i> HexDigits<sub>opt</sub>
 469      *     </i>{@code .} <i>HexDigits</i>
 470      * </dl>
 471      *
 472      * <dl>
 473      * <dt><i>BinaryExponent:</i>
 474      * <dd><i>BinaryExponentIndicator SignedInteger</i>
 475      * </dl>
 476      *
 477      * <dl>
 478      * <dt><i>BinaryExponentIndicator:</i>
 479      * <dd>{@code p}
 480      * <dd>{@code P}
 481      * </dl>
 482      *
 483      * </blockquote>
 484      *
 485      * where <i>Sign</i>, <i>FloatingPointLiteral</i>,
 486      * <i>HexNumeral</i>, <i>HexDigits</i>, <i>SignedInteger</i> and
 487      * <i>FloatTypeSuffix</i> are as defined in the lexical structure
 488      * sections of
 489      * <cite>The Java Language Specification</cite>,
 490      * except that underscores are not accepted between digits.
 491      * If {@code s} does not have the form of
 492      * a <i>FloatValue</i>, then a {@code NumberFormatException}
 493      * is thrown. Otherwise, {@code s} is regarded as
 494      * representing an exact decimal value in the usual
 495      * "computerized scientific notation" or as an exact
 496      * hexadecimal value; this exact numerical value is then
 497      * conceptually converted to an "infinitely precise"
 498      * binary value that is then rounded to type {@code float}
 499      * by the usual round-to-nearest rule of IEEE 754 floating-point
 500      * arithmetic, which includes preserving the sign of a zero
 501      * value.
 502      *
 503      * Note that the round-to-nearest rule also implies overflow and
 504      * underflow behaviour; if the exact value of {@code s} is large
 505      * enough in magnitude (greater than or equal to ({@link
 506      * #MAX_VALUE} + {@link Math#ulp(float) ulp(MAX_VALUE)}/2),
 507      * rounding to {@code float} will result in an infinity and if the
 508      * exact value of {@code s} is small enough in magnitude (less
 509      * than or equal to {@link #MIN_VALUE}/2), rounding to float will
 510      * result in a zero.
 511      *
 512      * Finally, after rounding a {@code Float} object representing
 513      * this {@code float} value is returned.
 514      *
 515      * <p>Note that trailing format specifiers, specifiers that
 516      * determine the type of a floating-point literal
 517      * ({@code 1.0f} is a {@code float} value;
 518      * {@code 1.0d} is a {@code double} value), do
 519      * <em>not</em> influence the results of this method.  In other
 520      * words, the numerical value of the input string is converted
 521      * directly to the target floating-point type.  In general, the
 522      * two-step sequence of conversions, string to {@code double}
 523      * followed by {@code double} to {@code float}, is
 524      * <em>not</em> equivalent to converting a string directly to
 525      * {@code float}.  For example, if first converted to an
 526      * intermediate {@code double} and then to
 527      * {@code float}, the string<br>
 528      * {@code "1.00000017881393421514957253748434595763683319091796875001d"}<br>
 529      * results in the {@code float} value
 530      * {@code 1.0000002f}; if the string is converted directly to
 531      * {@code float}, <code>1.000000<b>1</b>f</code> results.
 532      *
 533      * <p>To avoid calling this method on an invalid string and having
 534      * a {@code NumberFormatException} be thrown, the documentation
 535      * for {@link Double#valueOf Double.valueOf} lists a regular
 536      * expression which can be used to screen the input.
 537      *
 538      * @apiNote To interpret localized string representations of a
 539      * floating-point value, or string representations that have
 540      * non-ASCII digits, use {@link java.text.NumberFormat}. For
 541      * example,
 542      * {@snippet lang="java" :
 543      *     NumberFormat.getInstance(l).parse(s).floatValue();
 544      * }
 545      * where {@code l} is the desired locale, or
 546      * {@link java.util.Locale#ROOT} if locale insensitive.
 547      *
 548      * @apiNote
 549      * This method corresponds to the convertFromDecimalCharacter and
 550      * convertFromHexCharacter operations defined in IEEE 754.
 551      *
 552      * @param   s   the string to be parsed.
 553      * @return  a {@code Float} object holding the value
 554      *          represented by the {@code String} argument.
 555      * @throws  NumberFormatException  if the string does not contain a
 556      *          parsable number.
 557      * @see Double##decimalToBinaryConversion Decimal &harr; Binary Conversion Issues
 558      */
 559     public static Float valueOf(String s) throws NumberFormatException {
 560         return new Float(parseFloat(s));
 561     }
 562 
 563     /**
 564      * Returns a {@code Float} instance representing the specified
 565      * {@code float} value.
 566      * If a new {@code Float} instance is not required, this method
 567      * should generally be used in preference to the constructor
 568      * {@link #Float(float)}, as this method is likely to yield
 569      * significantly better space and time performance by caching
 570      * frequently requested values.
 571      *
 572      * @param  f a float value.
 573      * @return a {@code Float} instance representing {@code f}.
 574      * @since  1.5
 575      */
 576     @IntrinsicCandidate
 577     public static Float valueOf(float f) {
 578         return new Float(f);
 579     }
 580 
 581     /**
 582      * Returns a new {@code float} initialized to the value
 583      * represented by the specified {@code String}, as performed
 584      * by the {@code valueOf} method of class {@code Float}.
 585      *
 586      * @param  s the string to be parsed.
 587      * @return the {@code float} value represented by the string
 588      *         argument.
 589      * @throws NullPointerException  if the string is null
 590      * @throws NumberFormatException if the string does not contain a
 591      *               parsable {@code float}.
 592      * @see    java.lang.Float#valueOf(String)
 593      * @see    Double##decimalToBinaryConversion Decimal &harr; Binary Conversion Issues
 594      * @since 1.2
 595      */
 596     public static float parseFloat(String s) throws NumberFormatException {
 597         return FloatingDecimal.parseFloat(s);
 598     }
 599 
 600     /**
 601      * Returns {@code true} if the specified number is a
 602      * Not-a-Number (NaN) value, {@code false} otherwise.
 603      *
 604      * @apiNote
 605      * This method corresponds to the isNaN operation defined in IEEE
 606      * 754.
 607      *
 608      * @param   v   the value to be tested.
 609      * @return  {@code true} if the argument is NaN;
 610      *          {@code false} otherwise.
 611      */
 612     public static boolean isNaN(float v) {
 613         return (v != v);
 614     }
 615 
 616     /**
 617      * Returns {@code true} if the specified number is infinitely
 618      * large in magnitude, {@code false} otherwise.
 619      *
 620      * @apiNote
 621      * This method corresponds to the isInfinite operation defined in
 622      * IEEE 754.
 623      *
 624      * @param   v   the value to be tested.
 625      * @return  {@code true} if the argument is positive infinity or
 626      *          negative infinity; {@code false} otherwise.
 627      */
 628     @IntrinsicCandidate
 629     public static boolean isInfinite(float v) {
 630         return Math.abs(v) > MAX_VALUE;
 631     }
 632 
 633 
 634     /**
 635      * Returns {@code true} if the argument is a finite floating-point
 636      * value; returns {@code false} otherwise (for NaN and infinity
 637      * arguments).
 638      *
 639      * @apiNote
 640      * This method corresponds to the isFinite operation defined in
 641      * IEEE 754.
 642      *
 643      * @param f the {@code float} value to be tested
 644      * @return {@code true} if the argument is a finite
 645      * floating-point value, {@code false} otherwise.
 646      * @since 1.8
 647      */
 648      @IntrinsicCandidate
 649      public static boolean isFinite(float f) {
 650         return Math.abs(f) <= Float.MAX_VALUE;
 651     }
 652 
 653     /**
 654      * The value of the Float.
 655      *
 656      * @serial
 657      */
 658     private final float value;
 659 
 660     /**
 661      * Constructs a newly allocated {@code Float} object that
 662      * represents the primitive {@code float} argument.
 663      *
 664      * @param   value   the value to be represented by the {@code Float}.
 665      *
 666      * @deprecated
 667      * It is rarely appropriate to use this constructor. The static factory
 668      * {@link #valueOf(float)} is generally a better choice, as it is
 669      * likely to yield significantly better space and time performance.
 670      */
 671     @Deprecated(since="9", forRemoval = true)
 672     public Float(float value) {
 673         this.value = value;
 674     }
 675 
 676     /**
 677      * Constructs a newly allocated {@code Float} object that
 678      * represents the argument converted to type {@code float}.
 679      *
 680      * @param   value   the value to be represented by the {@code Float}.
 681      *
 682      * @deprecated
 683      * It is rarely appropriate to use this constructor. Instead, use the
 684      * static factory method {@link #valueOf(float)} method as follows:
 685      * {@code Float.valueOf((float)value)}.
 686      */
 687     @Deprecated(since="9", forRemoval = true)
 688     public Float(double value) {
 689         this.value = (float)value;
 690     }
 691 
 692     /**
 693      * Constructs a newly allocated {@code Float} object that
 694      * represents the floating-point value of type {@code float}
 695      * represented by the string. The string is converted to a
 696      * {@code float} value as if by the {@code valueOf} method.
 697      *
 698      * @param   s   a string to be converted to a {@code Float}.
 699      * @throws      NumberFormatException if the string does not contain a
 700      *              parsable number.
 701      *
 702      * @deprecated
 703      * It is rarely appropriate to use this constructor.
 704      * Use {@link #parseFloat(String)} to convert a string to a
 705      * {@code float} primitive, or use {@link #valueOf(String)}
 706      * to convert a string to a {@code Float} object.
 707      */
 708     @Deprecated(since="9", forRemoval = true)
 709     public Float(String s) throws NumberFormatException {
 710         value = parseFloat(s);
 711     }
 712 
 713     /**
 714      * Returns {@code true} if this {@code Float} value is a
 715      * Not-a-Number (NaN), {@code false} otherwise.
 716      *
 717      * @return  {@code true} if the value represented by this object is
 718      *          NaN; {@code false} otherwise.
 719      */
 720     public boolean isNaN() {
 721         return isNaN(value);
 722     }
 723 
 724     /**
 725      * Returns {@code true} if this {@code Float} value is
 726      * infinitely large in magnitude, {@code false} otherwise.
 727      *
 728      * @return  {@code true} if the value represented by this object is
 729      *          positive infinity or negative infinity;
 730      *          {@code false} otherwise.
 731      */
 732     public boolean isInfinite() {
 733         return isInfinite(value);
 734     }
 735 
 736     /**
 737      * Returns a string representation of this {@code Float} object.
 738      * The primitive {@code float} value represented by this object
 739      * is converted to a {@code String} exactly as if by the method
 740      * {@code toString} of one argument.
 741      *
 742      * @return  a {@code String} representation of this object.
 743      * @see java.lang.Float#toString(float)
 744      */
 745     public String toString() {
 746         return Float.toString(value);
 747     }
 748 
 749     /**
 750      * Returns the value of this {@code Float} as a {@code byte} after
 751      * a narrowing primitive conversion.
 752      *
 753      * @return  the {@code float} value represented by this object
 754      *          converted to type {@code byte}
 755      * @jls 5.1.3 Narrowing Primitive Conversion
 756      */
 757     @Override
 758     public byte byteValue() {
 759         return (byte)value;
 760     }
 761 
 762     /**
 763      * Returns the value of this {@code Float} as a {@code short}
 764      * after a narrowing primitive conversion.
 765      *
 766      * @return  the {@code float} value represented by this object
 767      *          converted to type {@code short}
 768      * @jls 5.1.3 Narrowing Primitive Conversion
 769      * @since 1.1
 770      */
 771     @Override
 772     public short shortValue() {
 773         return (short)value;
 774     }
 775 
 776     /**
 777      * Returns the value of this {@code Float} as an {@code int} after
 778      * a narrowing primitive conversion.
 779      *
 780      * @apiNote
 781      * This method corresponds to the convertToIntegerTowardZero
 782      * operation defined in IEEE 754.
 783      *
 784      * @return  the {@code float} value represented by this object
 785      *          converted to type {@code int}
 786      * @jls 5.1.3 Narrowing Primitive Conversion
 787      */
 788     @Override
 789     public int intValue() {
 790         return (int)value;
 791     }
 792 
 793     /**
 794      * Returns value of this {@code Float} as a {@code long} after a
 795      * narrowing primitive conversion.
 796      *
 797      * @apiNote
 798      * This method corresponds to the convertToIntegerTowardZero
 799      * operation defined in IEEE 754.
 800      *
 801      * @return  the {@code float} value represented by this object
 802      *          converted to type {@code long}
 803      * @jls 5.1.3 Narrowing Primitive Conversion
 804      */
 805     @Override
 806     public long longValue() {
 807         return (long)value;
 808     }
 809 
 810     /**
 811      * Returns the {@code float} value of this {@code Float} object.
 812      *
 813      * @return the {@code float} value represented by this object
 814      */
 815     @Override
 816     @IntrinsicCandidate
 817     public float floatValue() {
 818         return value;
 819     }
 820 
 821     /**
 822      * Returns the value of this {@code Float} as a {@code double}
 823      * after a widening primitive conversion.
 824      *
 825      * @apiNote
 826      * This method corresponds to the convertFormat operation defined
 827      * in IEEE 754.
 828      *
 829      * @return the {@code float} value represented by this
 830      *         object converted to type {@code double}
 831      * @jls 5.1.2 Widening Primitive Conversion
 832      */
 833     @Override
 834     public double doubleValue() {
 835         return (double)value;
 836     }
 837 
 838     /**
 839      * Returns a hash code for this {@code Float} object. The
 840      * result is the integer bit representation, exactly as produced
 841      * by the method {@link #floatToIntBits(float)}, of the primitive
 842      * {@code float} value represented by this {@code Float}
 843      * object.
 844      *
 845      * @return a hash code value for this object.
 846      */
 847     @Override
 848     public int hashCode() {
 849         return Float.hashCode(value);
 850     }
 851 
 852     /**
 853      * Returns a hash code for a {@code float} value; compatible with
 854      * {@code Float.hashCode()}.
 855      *
 856      * @param value the value to hash
 857      * @return a hash code value for a {@code float} value.
 858      * @since 1.8
 859      */
 860     public static int hashCode(float value) {
 861         return floatToIntBits(value);
 862     }
 863 
 864     /**
 865      * Compares this object against the specified object.  The result
 866      * is {@code true} if and only if the argument is not
 867      * {@code null} and is a {@code Float} object that
 868      * represents a {@code float} with the same value as the
 869      * {@code float} represented by this object. For this
 870      * purpose, two {@code float} values are considered to be the
 871      * same if and only if the method {@link #floatToIntBits(float)}
 872      * returns the identical {@code int} value when applied to
 873      * each.
 874      *
 875      * @apiNote
 876      * This method is defined in terms of {@link
 877      * #floatToIntBits(float)} rather than the {@code ==} operator on
 878      * {@code float} values since the {@code ==} operator does
 879      * <em>not</em> define an equivalence relation and to satisfy the
 880      * {@linkplain Object#equals equals contract} an equivalence
 881      * relation must be implemented; see {@linkplain Double##equivalenceRelation
 882      * this discussion for details of floating-point equality and equivalence}.
 883      *
 884      * @param obj the object to be compared
 885      * @return  {@code true} if the objects are the same;
 886      *          {@code false} otherwise.
 887      * @see java.lang.Float#floatToIntBits(float)
 888      * @jls 15.21.1 Numerical Equality Operators == and !=
 889      */
 890     public boolean equals(Object obj) {
 891         return (obj instanceof Float f) &&
 892             (floatToIntBits(f.value) == floatToIntBits(value));
 893     }
 894 
 895     /**
 896      * Returns a representation of the specified floating-point value
 897      * according to the IEEE 754 floating-point "single format" bit
 898      * layout.
 899      *
 900      * <p>Bit 31 (the bit that is selected by the mask
 901      * {@code 0x80000000}) represents the sign of the floating-point
 902      * number.
 903      * Bits 30-23 (the bits that are selected by the mask
 904      * {@code 0x7f800000}) represent the exponent.
 905      * Bits 22-0 (the bits that are selected by the mask
 906      * {@code 0x007fffff}) represent the significand (sometimes called
 907      * the mantissa) of the floating-point number.
 908      *
 909      * <p>If the argument is positive infinity, the result is
 910      * {@code 0x7f800000}.
 911      *
 912      * <p>If the argument is negative infinity, the result is
 913      * {@code 0xff800000}.
 914      *
 915      * <p>If the argument is NaN, the result is {@code 0x7fc00000}.
 916      *
 917      * <p>In all cases, the result is an integer that, when given to the
 918      * {@link #intBitsToFloat(int)} method, will produce a floating-point
 919      * value the same as the argument to {@code floatToIntBits}
 920      * (except all NaN values are collapsed to a single
 921      * "canonical" NaN value).
 922      *
 923      * @param   value   a floating-point number.
 924      * @return the bits that represent the floating-point number.
 925      */
 926     @IntrinsicCandidate
 927     public static int floatToIntBits(float value) {
 928         if (!isNaN(value)) {
 929             return floatToRawIntBits(value);
 930         }
 931         return 0x7fc00000;
 932     }
 933 
 934     /**
 935      * Returns a representation of the specified floating-point value
 936      * according to the IEEE 754 floating-point "single format" bit
 937      * layout, preserving Not-a-Number (NaN) values.
 938      *
 939      * <p>Bit 31 (the bit that is selected by the mask
 940      * {@code 0x80000000}) represents the sign of the floating-point
 941      * number.
 942      * Bits 30-23 (the bits that are selected by the mask
 943      * {@code 0x7f800000}) represent the exponent.
 944      * Bits 22-0 (the bits that are selected by the mask
 945      * {@code 0x007fffff}) represent the significand (sometimes called
 946      * the mantissa) of the floating-point number.
 947      *
 948      * <p>If the argument is positive infinity, the result is
 949      * {@code 0x7f800000}.
 950      *
 951      * <p>If the argument is negative infinity, the result is
 952      * {@code 0xff800000}.
 953      *
 954      * <p>If the argument is NaN, the result is the integer representing
 955      * the actual NaN value.  Unlike the {@code floatToIntBits}
 956      * method, {@code floatToRawIntBits} does not collapse all the
 957      * bit patterns encoding a NaN to a single "canonical"
 958      * NaN value.
 959      *
 960      * <p>In all cases, the result is an integer that, when given to the
 961      * {@link #intBitsToFloat(int)} method, will produce a
 962      * floating-point value the same as the argument to
 963      * {@code floatToRawIntBits}.
 964      *
 965      * @param   value   a floating-point number.
 966      * @return the bits that represent the floating-point number.
 967      * @since 1.3
 968      */
 969     @IntrinsicCandidate
 970     public static native int floatToRawIntBits(float value);
 971 
 972     /**
 973      * Returns the {@code float} value corresponding to a given
 974      * bit representation.
 975      * The argument is considered to be a representation of a
 976      * floating-point value according to the IEEE 754 floating-point
 977      * "single format" bit layout.
 978      *
 979      * <p>If the argument is {@code 0x7f800000}, the result is positive
 980      * infinity.
 981      *
 982      * <p>If the argument is {@code 0xff800000}, the result is negative
 983      * infinity.
 984      *
 985      * <p>If the argument is any value in the range
 986      * {@code 0x7f800001} through {@code 0x7fffffff} or in
 987      * the range {@code 0xff800001} through
 988      * {@code 0xffffffff}, the result is a NaN.  No IEEE 754
 989      * floating-point operation provided by Java can distinguish
 990      * between two NaN values of the same type with different bit
 991      * patterns.  Distinct values of NaN are only distinguishable by
 992      * use of the {@code Float.floatToRawIntBits} method.
 993      *
 994      * <p>In all other cases, let <i>s</i>, <i>e</i>, and <i>m</i> be three
 995      * values that can be computed from the argument:
 996      *
 997      * {@snippet lang="java" :
 998      * int s = ((bits >> 31) == 0) ? 1 : -1;
 999      * int e = ((bits >> 23) & 0xff);
1000      * int m = (e == 0) ?
1001      *                 (bits & 0x7fffff) << 1 :
1002      *                 (bits & 0x7fffff) | 0x800000;
1003      * }
1004      *
1005      * Then the floating-point result equals the value of the mathematical
1006      * expression <i>s</i>&middot;<i>m</i>&middot;2<sup><i>e</i>-150</sup>.
1007      *
1008      * <p>Note that this method may not be able to return a
1009      * {@code float} NaN with exactly same bit pattern as the
1010      * {@code int} argument.  IEEE 754 distinguishes between two
1011      * kinds of NaNs, quiet NaNs and <i>signaling NaNs</i>.  The
1012      * differences between the two kinds of NaN are generally not
1013      * visible in Java.  Arithmetic operations on signaling NaNs turn
1014      * them into quiet NaNs with a different, but often similar, bit
1015      * pattern.  However, on some processors merely copying a
1016      * signaling NaN also performs that conversion.  In particular,
1017      * copying a signaling NaN to return it to the calling method may
1018      * perform this conversion.  So {@code intBitsToFloat} may
1019      * not be able to return a {@code float} with a signaling NaN
1020      * bit pattern.  Consequently, for some {@code int} values,
1021      * {@code floatToRawIntBits(intBitsToFloat(start))} may
1022      * <i>not</i> equal {@code start}.  Moreover, which
1023      * particular bit patterns represent signaling NaNs is platform
1024      * dependent; although all NaN bit patterns, quiet or signaling,
1025      * must be in the NaN range identified above.
1026      *
1027      * @param   bits   an integer.
1028      * @return  the {@code float} floating-point value with the same bit
1029      *          pattern.
1030      */
1031     @IntrinsicCandidate
1032     public static native float intBitsToFloat(int bits);
1033 
1034     /**
1035      * {@return the {@code float} value closest to the numerical value
1036      * of the argument, a floating-point binary16 value encoded in a
1037      * {@code short}} The conversion is exact; all binary16 values can
1038      * be exactly represented in {@code float}.
1039      *
1040      * Special cases:
1041      * <ul>
1042      * <li> If the argument is zero, the result is a zero with the
1043      * same sign as the argument.
1044      * <li> If the argument is infinite, the result is an infinity
1045      * with the same sign as the argument.
1046      * <li> If the argument is a NaN, the result is a NaN.
1047      * </ul>
1048      *
1049      * <h4><a id=binary16Format>IEEE 754 binary16 format</a></h4>
1050      * The IEEE 754 standard defines binary16 as a 16-bit format, along
1051      * with the 32-bit binary32 format (corresponding to the {@code
1052      * float} type) and the 64-bit binary64 format (corresponding to
1053      * the {@code double} type). The binary16 format is similar to the
1054      * other IEEE 754 formats, except smaller, having all the usual
1055      * IEEE 754 values such as NaN, signed infinities, signed zeros,
1056      * and subnormals. The parameters (JLS {@jls 4.2.3}) for the
1057      * binary16 format are N = 11 precision bits, K = 5 exponent bits,
1058      * <i>E</i><sub><i>max</i></sub> = 15, and
1059      * <i>E</i><sub><i>min</i></sub> = -14.
1060      *
1061      * @apiNote
1062      * This method corresponds to the convertFormat operation defined
1063      * in IEEE 754 from the binary16 format to the binary32 format.
1064      * The operation of this method is analogous to a primitive
1065      * widening conversion (JLS {@jls 5.1.2}).
1066      *
1067      * @param floatBinary16 the binary16 value to convert to {@code float}
1068      * @since 20
1069      */
1070     @IntrinsicCandidate
1071     public static float float16ToFloat(short floatBinary16) {
1072         /*
1073          * The binary16 format has 1 sign bit, 5 exponent bits, and 10
1074          * significand bits. The exponent bias is 15.
1075          */
1076         int bin16arg = (int)floatBinary16;
1077         int bin16SignBit     = 0x8000 & bin16arg;
1078         int bin16ExpBits     = 0x7c00 & bin16arg;
1079         int bin16SignifBits  = 0x03FF & bin16arg;
1080 
1081         // Shift left difference in the number of significand bits in
1082         // the float and binary16 formats
1083         final int SIGNIF_SHIFT = (FloatConsts.SIGNIFICAND_WIDTH - 11);
1084 
1085         float sign = (bin16SignBit != 0) ? -1.0f : 1.0f;
1086 
1087         // Extract binary16 exponent, remove its bias, add in the bias
1088         // of a float exponent and shift to correct bit location
1089         // (significand width includes the implicit bit so shift one
1090         // less).
1091         int bin16Exp = (bin16ExpBits >> 10) - 15;
1092         if (bin16Exp == -15) {
1093             // For subnormal binary16 values and 0, the numerical
1094             // value is 2^24 * the significand as an integer (no
1095             // implicit bit).
1096             return sign * (0x1p-24f * bin16SignifBits);
1097         } else if (bin16Exp == 16) {
1098             return (bin16SignifBits == 0) ?
1099                 sign * Float.POSITIVE_INFINITY :
1100                 Float.intBitsToFloat((bin16SignBit << 16) |
1101                                      0x7f80_0000 |
1102                                      // Preserve NaN signif bits
1103                                      ( bin16SignifBits << SIGNIF_SHIFT ));
1104         }
1105 
1106         assert -15 < bin16Exp  && bin16Exp < 16;
1107 
1108         int floatExpBits = (bin16Exp + FloatConsts.EXP_BIAS)
1109             << (FloatConsts.SIGNIFICAND_WIDTH - 1);
1110 
1111         // Compute and combine result sign, exponent, and significand bits.
1112         return Float.intBitsToFloat((bin16SignBit << 16) |
1113                                     floatExpBits |
1114                                     (bin16SignifBits << SIGNIF_SHIFT));
1115     }
1116 
1117     /**
1118      * {@return the floating-point binary16 value, encoded in a {@code
1119      * short}, closest in value to the argument}
1120      * The conversion is computed under the {@linkplain
1121      * java.math.RoundingMode#HALF_EVEN round to nearest even rounding
1122      * mode}.
1123      *
1124      * Special cases:
1125      * <ul>
1126      * <li> If the argument is zero, the result is a zero with the
1127      * same sign as the argument.
1128      * <li> If the argument is infinite, the result is an infinity
1129      * with the same sign as the argument.
1130      * <li> If the argument is a NaN, the result is a NaN.
1131      * </ul>
1132      *
1133      * The {@linkplain ##binary16Format binary16 format} is discussed in
1134      * more detail in the {@link #float16ToFloat} method.
1135      *
1136      * @apiNote
1137      * This method corresponds to the convertFormat operation defined
1138      * in IEEE 754 from the binary32 format to the binary16 format.
1139      * The operation of this method is analogous to a primitive
1140      * narrowing conversion (JLS {@jls 5.1.3}).
1141      *
1142      * @param f the {@code float} value to convert to binary16
1143      * @since 20
1144      */
1145     @IntrinsicCandidate
1146     public static short floatToFloat16(float f) {
1147         int doppel = Float.floatToRawIntBits(f);
1148         short sign_bit = (short)((doppel & 0x8000_0000) >> 16);
1149 
1150         if (Float.isNaN(f)) {
1151             // Preserve sign and attempt to preserve significand bits
1152             return (short)(sign_bit
1153                     | 0x7c00 // max exponent + 1
1154                     // Preserve high order bit of float NaN in the
1155                     // binary16 result NaN (tenth bit); OR in remaining
1156                     // bits into lower 9 bits of binary 16 significand.
1157                     | (doppel & 0x007f_e000) >> 13 // 10 bits
1158                     | (doppel & 0x0000_1ff0) >> 4  //  9 bits
1159                     | (doppel & 0x0000_000f));     //  4 bits
1160         }
1161 
1162         float abs_f = Math.abs(f);
1163 
1164         // The overflow threshold is binary16 MAX_VALUE + 1/2 ulp
1165         if (abs_f >= (0x1.ffcp15f + 0x0.002p15f) ) {
1166             return (short)(sign_bit | 0x7c00); // Positive or negative infinity
1167         }
1168 
1169         // Smallest magnitude nonzero representable binary16 value
1170         // is equal to 0x1.0p-24; half-way and smaller rounds to zero.
1171         if (abs_f <= 0x1.0p-24f * 0.5f) { // Covers float zeros and subnormals.
1172             return sign_bit; // Positive or negative zero
1173         }
1174 
1175         // Dealing with finite values in exponent range of binary16
1176         // (when rounding is done, could still round up)
1177         int exp = Math.getExponent(f);
1178         assert -25 <= exp && exp <= 15;
1179 
1180         // For binary16 subnormals, beside forcing exp to -15, retain
1181         // the difference expdelta = E_min - exp.  This is the excess
1182         // shift value, in addition to 13, to be used in the
1183         // computations below.  Further the (hidden) msb with value 1
1184         // in f must be involved as well.
1185         int expdelta = 0;
1186         int msb = 0x0000_0000;
1187         if (exp < -14) {
1188             expdelta = -14 - exp;
1189             exp = -15;
1190             msb = 0x0080_0000;
1191         }
1192         int f_signif_bits = doppel & 0x007f_ffff | msb;
1193 
1194         // Significand bits as if using rounding to zero (truncation).
1195         short signif_bits = (short)(f_signif_bits >> (13 + expdelta));
1196 
1197         // For round to nearest even, determining whether or not to
1198         // round up (in magnitude) is a function of the least
1199         // significant bit (LSB), the next bit position (the round
1200         // position), and the sticky bit (whether there are any
1201         // nonzero bits in the exact result to the right of the round
1202         // digit). An increment occurs in three cases:
1203         //
1204         // LSB  Round Sticky
1205         // 0    1     1
1206         // 1    1     0
1207         // 1    1     1
1208         // See "Computer Arithmetic Algorithms," Koren, Table 4.9
1209 
1210         int lsb    = f_signif_bits & (1 << 13 + expdelta);
1211         int round  = f_signif_bits & (1 << 12 + expdelta);
1212         int sticky = f_signif_bits & ((1 << 12 + expdelta) - 1);
1213 
1214         if (round != 0 && ((lsb | sticky) != 0 )) {
1215             signif_bits++;
1216         }
1217 
1218         // No bits set in significand beyond the *first* exponent bit,
1219         // not just the significand; quantity is added to the exponent
1220         // to implement a carry out from rounding the significand.
1221         assert (0xf800 & signif_bits) == 0x0;
1222 
1223         return (short)(sign_bit | ( ((exp + 15) << 10) + signif_bits ) );
1224     }
1225 
1226     /**
1227      * Compares two {@code Float} objects numerically.
1228      *
1229      * This method imposes a total order on {@code Float} objects
1230      * with two differences compared to the incomplete order defined by
1231      * the Java language numerical comparison operators ({@code <, <=,
1232      * ==, >=, >}) on {@code float} values.
1233      *
1234      * <ul><li> A NaN is <em>unordered</em> with respect to other
1235      *          values and unequal to itself under the comparison
1236      *          operators.  This method chooses to define {@code
1237      *          Float.NaN} to be equal to itself and greater than all
1238      *          other {@code double} values (including {@code
1239      *          Float.POSITIVE_INFINITY}).
1240      *
1241      *      <li> Positive zero and negative zero compare equal
1242      *      numerically, but are distinct and distinguishable values.
1243      *      This method chooses to define positive zero ({@code +0.0f}),
1244      *      to be greater than negative zero ({@code -0.0f}).
1245      * </ul>
1246      *
1247      * This ensures that the <i>natural ordering</i> of {@code Float}
1248      * objects imposed by this method is <i>consistent with
1249      * equals</i>; see {@linkplain Double##equivalenceRelation this
1250      * discussion for details of floating-point comparison and
1251      * ordering}.
1252      *
1253      *
1254      * @param   anotherFloat   the {@code Float} to be compared.
1255      * @return  the value {@code 0} if {@code anotherFloat} is
1256      *          numerically equal to this {@code Float}; a value
1257      *          less than {@code 0} if this {@code Float}
1258      *          is numerically less than {@code anotherFloat};
1259      *          and a value greater than {@code 0} if this
1260      *          {@code Float} is numerically greater than
1261      *          {@code anotherFloat}.
1262      *
1263      * @jls 15.20.1 Numerical Comparison Operators {@code <}, {@code <=}, {@code >}, and {@code >=}
1264      * @since   1.2
1265      */
1266     @Override
1267     public int compareTo(Float anotherFloat) {
1268         return Float.compare(value, anotherFloat.value);
1269     }
1270 
1271     /**
1272      * Compares the two specified {@code float} values. The sign
1273      * of the integer value returned is the same as that of the
1274      * integer that would be returned by the call:
1275      * <pre>
1276      *    Float.valueOf(f1).compareTo(Float.valueOf(f2))
1277      * </pre>
1278      *
1279      * @param   f1        the first {@code float} to compare.
1280      * @param   f2        the second {@code float} to compare.
1281      * @return  the value {@code 0} if {@code f1} is
1282      *          numerically equal to {@code f2}; a value less than
1283      *          {@code 0} if {@code f1} is numerically less than
1284      *          {@code f2}; and a value greater than {@code 0}
1285      *          if {@code f1} is numerically greater than
1286      *          {@code f2}.
1287      * @since 1.4
1288      */
1289     public static int compare(float f1, float f2) {
1290         if (f1 < f2)
1291             return -1;           // Neither val is NaN, thisVal is smaller
1292         if (f1 > f2)
1293             return 1;            // Neither val is NaN, thisVal is larger
1294 
1295         // Cannot use floatToRawIntBits because of possibility of NaNs.
1296         int thisBits    = Float.floatToIntBits(f1);
1297         int anotherBits = Float.floatToIntBits(f2);
1298 
1299         return (thisBits == anotherBits ?  0 : // Values are equal
1300                 (thisBits < anotherBits ? -1 : // (-0.0, 0.0) or (!NaN, NaN)
1301                  1));                          // (0.0, -0.0) or (NaN, !NaN)
1302     }
1303 
1304     /**
1305      * Adds two {@code float} values together as per the + operator.
1306      *
1307      * @apiNote This method corresponds to the addition operation
1308      * defined in IEEE 754.
1309      *
1310      * @param a the first operand
1311      * @param b the second operand
1312      * @return the sum of {@code a} and {@code b}
1313      * @jls 4.2.4 Floating-Point Operations
1314      * @see java.util.function.BinaryOperator
1315      * @since 1.8
1316      */
1317     public static float sum(float a, float b) {
1318         return a + b;
1319     }
1320 
1321     /**
1322      * Returns the greater of two {@code float} values
1323      * as if by calling {@link Math#max(float, float) Math.max}.
1324      *
1325      * @apiNote
1326      * This method corresponds to the maximum operation defined in
1327      * IEEE 754.
1328      *
1329      * @param a the first operand
1330      * @param b the second operand
1331      * @return the greater of {@code a} and {@code b}
1332      * @see java.util.function.BinaryOperator
1333      * @since 1.8
1334      */
1335     public static float max(float a, float b) {
1336         return Math.max(a, b);
1337     }
1338 
1339     /**
1340      * Returns the smaller of two {@code float} values
1341      * as if by calling {@link Math#min(float, float) Math.min}.
1342      *
1343      * @apiNote
1344      * This method corresponds to the minimum operation defined in
1345      * IEEE 754.
1346      *
1347      * @param a the first operand
1348      * @param b the second operand
1349      * @return the smaller of {@code a} and {@code b}
1350      * @see java.util.function.BinaryOperator
1351      * @since 1.8
1352      */
1353     public static float min(float a, float b) {
1354         return Math.min(a, b);
1355     }
1356 
1357     /**
1358      * Returns an {@link Optional} containing the nominal descriptor for this
1359      * instance, which is the instance itself.
1360      *
1361      * @return an {@link Optional} describing the {@linkplain Float} instance
1362      * @since 12
1363      */
1364     @Override
1365     public Optional<Float> describeConstable() {
1366         return Optional.of(this);
1367     }
1368 
1369     /**
1370      * Resolves this instance as a {@link ConstantDesc}, the result of which is
1371      * the instance itself.
1372      *
1373      * @param lookup ignored
1374      * @return the {@linkplain Float} instance
1375      * @since 12
1376      */
1377     @Override
1378     public Float resolveConstantDesc(MethodHandles.Lookup lookup) {
1379         return this;
1380     }
1381 
1382     /** use serialVersionUID from JDK 1.0.2 for interoperability */
1383     @java.io.Serial
1384     private static final long serialVersionUID = -2671257302660747028L;
1385 }