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