# HG changeset patch # User lello # Date 1584472052 -3600 # Tue Mar 17 20:07:32 2020 +0100 # Node ID ec615c419ab236ae291651f5a810807f05fb4421 # Parent 42b62267498dbea03a82eeb073c6020b4b43d288 Patch to fix JDK-4511638 4511638: Double.toString(double) sometimes produces incorrect results Reviewed-by: TBD Contributed-by: Raffaello Giulietti diff --git a/src/java.base/share/classes/java/lang/AbstractStringBuilder.java b/src/java.base/share/classes/java/lang/AbstractStringBuilder.java --- a/src/java.base/share/classes/java/lang/AbstractStringBuilder.java +++ b/src/java.base/share/classes/java/lang/AbstractStringBuilder.java @@ -25,8 +25,10 @@ package java.lang; -import jdk.internal.math.FloatingDecimal; +import jdk.internal.math.DoubleToDecimal; +import jdk.internal.math.FloatToDecimal; +import java.io.IOException; import java.util.Arrays; import java.util.Spliterator; import java.util.stream.IntStream; @@ -880,7 +882,11 @@ * @return a reference to this object. */ public AbstractStringBuilder append(float f) { - FloatingDecimal.appendTo(f,this); + try { + FloatToDecimal.appendTo(f, this); + } catch (IOException ignored) { + assert false; + } return this; } @@ -897,7 +903,11 @@ * @return a reference to this object. */ public AbstractStringBuilder append(double d) { - FloatingDecimal.appendTo(d,this); + try { + DoubleToDecimal.appendTo(d, this); + } catch (IOException ignored) { + assert false; + } return this; } diff --git a/src/java.base/share/classes/java/lang/Double.java b/src/java.base/share/classes/java/lang/Double.java --- a/src/java.base/share/classes/java/lang/Double.java +++ b/src/java.base/share/classes/java/lang/Double.java @@ -32,6 +32,7 @@ import jdk.internal.math.FloatingDecimal; import jdk.internal.math.DoubleConsts; +import jdk.internal.math.DoubleToDecimal; import jdk.internal.HotSpotIntrinsicCandidate; /** @@ -145,69 +146,120 @@ public static final Class TYPE = (Class) Class.getPrimitiveClass("double"); /** - * Returns a string representation of the {@code double} - * argument. All characters mentioned below are ASCII characters. - *
    - *
  • If the argument is NaN, the result is the string - * "{@code NaN}". - *
  • Otherwise, the result is a string that represents the sign and - * magnitude (absolute value) of the argument. If the sign is negative, - * the first character of the result is '{@code -}' - * ({@code '\u005Cu002D'}); if the sign is positive, no sign character - * appears in the result. As for the magnitude m: - *
      - *
    • If m is infinity, it is represented by the characters - * {@code "Infinity"}; thus, positive infinity produces the result - * {@code "Infinity"} and negative infinity produces the result - * {@code "-Infinity"}. - * - *
    • If m is zero, it is represented by the characters - * {@code "0.0"}; thus, negative zero produces the result - * {@code "-0.0"} and positive zero produces the result - * {@code "0.0"}. + * Returns a string rendering of the {@code double} argument. * - *
    • If m is greater than or equal to 10-3 but less - * than 107, then it is represented as the integer part of - * m, in decimal form with no leading zeroes, followed by - * '{@code .}' ({@code '\u005Cu002E'}), followed by one or - * more decimal digits representing the fractional part of m. - * - *
    • If m is less than 10-3 or greater than or - * equal to 107, then it is represented in so-called - * "computerized scientific notation." Let n be the unique - * integer such that 10nm {@literal <} - * 10n+1; then let a be the - * mathematically exact quotient of m and - * 10n so that 1 ≤ a {@literal <} 10. The - * magnitude is then represented as the integer part of a, - * as a single decimal digit, followed by '{@code .}' - * ({@code '\u005Cu002E'}), followed by decimal digits - * representing the fractional part of a, followed by the - * letter '{@code E}' ({@code '\u005Cu0045'}), followed - * by a representation of n as a decimal integer, as - * produced by the method {@link Integer#toString(int)}. + *

      The characters of the result are all drawn from the ASCII set. + *

        + *
      • Any NaN, whether quiet or signaling, is rendered as + * {@code "NaN"}, regardless of the sign bit. + *
      • The infinities +∞ and -∞ are rendered as + * {@code "Infinity"} and {@code "-Infinity"}, respectively. + *
      • The positive and negative zeroes are rendered as + * {@code "0.0"} and {@code "-0.0"}, respectively. + *
      • A finite negative {@code v} is rendered as the sign + * '{@code -}' followed by the rendering of the magnitude -{@code v}. + *
      • A finite positive {@code v} is rendered in two stages: + *
          + *
        • Selection of a decimal: A well-defined + * decimal dv is selected + * to represent {@code v}. + *
        • Formatting as a string: The decimal + * dv is formatted as a string, + * either in plain or in computerized scientific notation, + * depending on its value. *
        *
      - * How many digits must be printed for the fractional part of - * m or a? There must be at least one digit to represent - * the fractional part, and beyond that as many, but only as many, more - * digits as are needed to uniquely distinguish the argument value from - * adjacent values of type {@code double}. That is, suppose that - * x is the exact mathematical value represented by the decimal - * representation produced by this method for a finite nonzero argument - * d. Then d must be the {@code double} value nearest - * to x; or if two {@code double} values are equally close - * to x, then d must be one of them and the least - * significant bit of the significand of d must be {@code 0}. + * + *

      A decimal is a number of the form + * d×10i + * for some (unique) integers d > 0 and i such that + * d is not a multiple of 10. + * These integers are the significand and + * the exponent, respectively, of the decimal. + * The length of the decimal is the (unique) + * integer n meeting + * 10n-1d < 10n. + * + *

      The decimal dv + * for a finite positive {@code v} is defined as follows: + *

        + *
      • Let R be the set of all decimals that round to {@code v} + * according to the usual round-to-closest rule of + * IEEE 754 floating-point arithmetic. + *
      • Let m be the minimal length over all decimals in R. + *
      • When m ≥ 2, let T be the set of all decimals + * in R with length m. + * Otherwise, let T be the set of all decimals + * in R with length 1 or 2. + *
      • Define dv as + * the decimal in T that is closest to {@code v}. + * Or if there are two such decimals in T, + * select the one with the even significand (there is exactly one). + *
      + * + *

      The (uniquely) selected decimal dv + * is then formatted. * - *

      To create localized string representations of a floating-point - * value, use subclasses of {@link java.text.NumberFormat}. + *

      Let d, i and n be the significand, exponent and + * length of dv, respectively. + * Further, let e = n + i - 1 and let + * d1dn + * be the usual decimal expansion of the significand. + * Note that d1 ≠ 0 ≠ dn. + *

        + *
      • Case -3 ≤ e < 0: + * dv is formatted as + * 0.00d1dn, + * where there are exactly -(n + i) zeroes between + * the decimal point and d1. + * For example, 123 × 10-4 is formatted as + * {@code 0.0123}. + *
      • Case 0 ≤ e < 7: + *
          + *
        • Subcase i ≥ 0: + * dv is formatted as + * d1dn00.0, + * where there are exactly i zeroes + * between dn and the decimal point. + * For example, 123 × 102 is formatted as + * {@code 12300.0}. + *
        • Subcase i < 0: + * dv is formatted as + * d1dn+i.dn+i+1dn. + * There are exactly -i digits to the right of + * the decimal point. + * For example, 123 × 10-1 is formatted as + * {@code 12.3}. + *
        + *
      • Case e < -3 or e ≥ 7: + * computerized scientific notation is used to format + * dv. + * Here e is formatted as by {@link Integer#toString(int)}. + *
          + *
        • Subcase n = 1: + * dv is formatted as + * d1.0Ee. + * For example, 1 × 1023 is formatted as + * {@code 1.0E23}. + *
        • Subcase n > 1: + * dv is formatted as + * d1.d2dnEe. + * For example, 123 × 10-21 is formatted as + * {@code 1.23E-19}. + *
        + *
      * - * @param d the {@code double} to be converted. - * @return a string representation of the argument. + * @param v the {@code double} to be rendered. + * @return a string rendering of the argument. */ - public static String toString(double d) { - return FloatingDecimal.toJavaFormatString(d); + public static String toString(double v) { + return DoubleToDecimal.toString(v); } /** diff --git a/src/java.base/share/classes/java/lang/Float.java b/src/java.base/share/classes/java/lang/Float.java --- a/src/java.base/share/classes/java/lang/Float.java +++ b/src/java.base/share/classes/java/lang/Float.java @@ -31,6 +31,7 @@ import java.util.Optional; import jdk.internal.math.FloatingDecimal; +import jdk.internal.math.FloatToDecimal; import jdk.internal.HotSpotIntrinsicCandidate; /** @@ -142,73 +143,120 @@ public static final Class TYPE = (Class) Class.getPrimitiveClass("float"); /** - * Returns a string representation of the {@code float} - * argument. All characters mentioned below are ASCII characters. - *
        - *
      • If the argument is NaN, the result is the string - * "{@code NaN}". - *
      • Otherwise, the result is a string that represents the sign and - * magnitude (absolute value) of the argument. If the sign is - * negative, the first character of the result is - * '{@code -}' ({@code '\u005Cu002D'}); if the sign is - * positive, no sign character appears in the result. As for - * the magnitude m: + * Returns a string rendering of the {@code float} argument. + * + *

        The characters of the result are all drawn from the ASCII set. *

          - *
        • If m is infinity, it is represented by the characters - * {@code "Infinity"}; thus, positive infinity produces - * the result {@code "Infinity"} and negative infinity - * produces the result {@code "-Infinity"}. - *
        • If m is zero, it is represented by the characters - * {@code "0.0"}; thus, negative zero produces the result - * {@code "-0.0"} and positive zero produces the result - * {@code "0.0"}. - *
        • If m is greater than or equal to 10-3 but - * less than 107, then it is represented as the - * integer part of m, in decimal form with no leading - * zeroes, followed by '{@code .}' - * ({@code '\u005Cu002E'}), followed by one or more - * decimal digits representing the fractional part of - * m. - *
        • If m is less than 10-3 or greater than or - * equal to 107, then it is represented in - * so-called "computerized scientific notation." Let n - * be the unique integer such that 10n ≤ - * m {@literal <} 10n+1; then let a - * be the mathematically exact quotient of m and - * 10n so that 1 ≤ a {@literal <} 10. - * The magnitude is then represented as the integer part of - * a, as a single decimal digit, followed by - * '{@code .}' ({@code '\u005Cu002E'}), followed by - * decimal digits representing the fractional part of - * a, followed by the letter '{@code E}' - * ({@code '\u005Cu0045'}), followed by a representation - * of n as a decimal integer, as produced by the - * method {@link java.lang.Integer#toString(int)}. - * + *
        • Any NaN, whether quiet or signaling, is rendered as + * {@code "NaN"}, regardless of the sign bit. + *
        • The infinities +∞ and -∞ are rendered as + * {@code "Infinity"} and {@code "-Infinity"}, respectively. + *
        • The positive and negative zeroes are rendered as + * {@code "0.0"} and {@code "-0.0"}, respectively. + *
        • A finite negative {@code v} is rendered as the sign + * '{@code -}' followed by the rendering of the magnitude -{@code v}. + *
        • A finite positive {@code v} is rendered in two stages: + *
            + *
          • Selection of a decimal: A well-defined + * decimal dv is selected + * to represent {@code v}. + *
          • Formatting as a string: The decimal + * dv is formatted as a string, + * either in plain or in computerized scientific notation, + * depending on its value. *
          *
        - * How many digits must be printed for the fractional part of - * m or a? There must be at least one digit - * to represent the fractional part, and beyond that as many, but - * only as many, more digits as are needed to uniquely distinguish - * the argument value from adjacent values of type - * {@code float}. That is, suppose that x is the - * exact mathematical value represented by the decimal - * representation produced by this method for a finite nonzero - * argument f. Then f must be the {@code float} - * value nearest to x; or, if two {@code float} values are - * equally close to x, then f must be one of - * them and the least significant bit of the significand of - * f must be {@code 0}. + * + *

        A decimal is a number of the form + * d×10i + * for some (unique) integers d > 0 and i such that + * d is not a multiple of 10. + * These integers are the significand and + * the exponent, respectively, of the decimal. + * The length of the decimal is the (unique) + * integer n meeting + * 10n-1d < 10n. + * + *

        The decimal dv + * for a finite positive {@code v} is defined as follows: + *

          + *
        • Let R be the set of all decimals that round to {@code v} + * according to the usual round-to-closest rule of + * IEEE 754 floating-point arithmetic. + *
        • Let m be the minimal length over all decimals in R. + *
        • When m ≥ 2, let T be the set of all decimals + * in R with length m. + * Otherwise, let T be the set of all decimals + * in R with length 1 or 2. + *
        • Define dv as + * the decimal in T that is closest to {@code v}. + * Or if there are two such decimals in T, + * select the one with the even significand (there is exactly one). + *
        + * + *

        The (uniquely) selected decimal dv + * is then formatted. * - *

        To create localized string representations of a floating-point - * value, use subclasses of {@link java.text.NumberFormat}. + *

        Let d, i and n be the significand, exponent and + * length of dv, respectively. + * Further, let e = n + i - 1 and let + * d1dn + * be the usual decimal expansion of the significand. + * Note that d1 ≠ 0 ≠ dn. + *

          + *
        • Case -3 ≤ e < 0: + * dv is formatted as + * 0.00d1dn, + * where there are exactly -(n + i) zeroes between + * the decimal point and d1. + * For example, 123 × 10-4 is formatted as + * {@code 0.0123}. + *
        • Case 0 ≤ e < 7: + *
            + *
          • Subcase i ≥ 0: + * dv is formatted as + * d1dn00.0, + * where there are exactly i zeroes + * between dn and the decimal point. + * For example, 123 × 102 is formatted as + * {@code 12300.0}. + *
          • Subcase i < 0: + * dv is formatted as + * d1dn+i.dn+i+1dn. + * There are exactly -i digits to the right of + * the decimal point. + * For example, 123 × 10-1 is formatted as + * {@code 12.3}. + *
          + *
        • Case e < -3 or e ≥ 7: + * computerized scientific notation is used to format + * dv. + * Here e is formatted as by {@link Integer#toString(int)}. + *
            + *
          • Subcase n = 1: + * dv is formatted as + * d1.0Ee. + * For example, 1 × 1023 is formatted as + * {@code 1.0E23}. + *
          • Subcase n > 1: + * dv is formatted as + * d1.d2dnEe. + * For example, 123 × 10-21 is formatted as + * {@code 1.23E-19}. + *
          + *
        * - * @param f the float to be converted. - * @return a string representation of the argument. + * @param v the {@code float} to be rendered. + * @return a string rendering of the argument. */ - public static String toString(float f) { - return FloatingDecimal.toJavaFormatString(f); + public static String toString(float v) { + return FloatToDecimal.toString(v); } /** diff --git a/src/java.base/share/classes/jdk/internal/math/DoubleToDecimal.java b/src/java.base/share/classes/jdk/internal/math/DoubleToDecimal.java new file mode 100644 --- /dev/null +++ b/src/java.base/share/classes/jdk/internal/math/DoubleToDecimal.java @@ -0,0 +1,644 @@ +/* + * Copyright 2018-2020 Raffaello Giulietti + * + * Permission is hereby granted, free of charge, to any person obtaining a copy + * of this software and associated documentation files (the "Software"), to deal + * in the Software without restriction, including without limitation the rights + * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell + * copies of the Software, and to permit persons to whom the Software is + * furnished to do so, subject to the following conditions: + * + * The above copyright notice and this permission notice shall be included in + * all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR + * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, + * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE + * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER + * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, + * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN + * THE SOFTWARE. + */ + +package jdk.internal.math; + +import java.io.IOException; + +import static java.lang.Double.*; +import static java.lang.Long.*; +import static java.lang.Math.multiplyHigh; +import static jdk.internal.math.MathUtils.*; + +/** + * This class exposes a method to render a {@code double} as a string. + * + * @author Raffaello Giulietti + */ +final public class DoubleToDecimal { + /* + For full details about this code see the following references: + + [1] Giulietti, "The Schubfach way to render doubles", + https://drive.google.com/open?id=1luHhyQF9zKlM8yJ1nebU0OgVYhfC6CBN + + [2] IEEE Computer Society, "IEEE Standard for Floating-Point Arithmetic" + + [3] Bouvier & Zimmermann, "Division-Free Binary-to-Decimal Conversion" + + Divisions are avoided altogether for the benefit of those architectures + that do not provide specific machine instructions or where they are slow. + This is discussed in section 10 of [1]. + */ + + // The precision in bits. + static final int P = 53; + + // Exponent width in bits. + private static final int W = (Double.SIZE - 1) - (P - 1); + + // Minimum value of the exponent: -(2^(W-1)) - P + 3. + static final int Q_MIN = (-1 << W - 1) - P + 3; + + // Maximum value of the exponent: 2^(W-1) - P. + static final int Q_MAX = (1 << W - 1) - P; + + // 10^(E_MIN - 1) <= MIN_VALUE < 10^E_MIN + static final int E_MIN = -323; + + // 10^(E_MAX - 1) <= MAX_VALUE < 10^E_MAX + static final int E_MAX = 309; + + // Threshold to detect tiny values, as in section 8.1.1 of [1] + static final long C_TINY = 3; + + // The minimum and maximum k, as in section 8 of [1] + static final int K_MIN = -324; + static final int K_MAX = 292; + + // H is as in section 8 of [1]. + static final int H = 17; + + // Minimum value of the significand of a normal value: 2^(P-1). + private static final long C_MIN = 1L << P - 1; + + // Mask to extract the biased exponent. + private static final int BQ_MASK = (1 << W) - 1; + + // Mask to extract the fraction bits. + private static final long T_MASK = (1L << P - 1) - 1; + + // Used in rop(). + private static final long MASK_63 = (1L << 63) - 1; + + // Used for left-to-tight digit extraction. + private static final int MASK_28 = (1 << 28) - 1; + + private static final int NON_SPECIAL = 0; + private static final int PLUS_ZERO = 1; + private static final int MINUS_ZERO = 2; + private static final int PLUS_INF = 3; + private static final int MINUS_INF = 4; + private static final int NAN = 5; + + // For thread-safety, each thread gets its own instance of this class. + private static final ThreadLocal threadLocal = + ThreadLocal.withInitial(DoubleToDecimal::new); + + /* + Room for the longer of the forms + -ddddd.dddddddddddd H + 2 characters + -0.00ddddddddddddddddd H + 5 characters + -d.ddddddddddddddddE-eee H + 7 characters + where there are H digits d + */ + public final int MAX_CHARS = H + 7; + + // Numerical results are created here... + private final byte[] bytes = new byte[MAX_CHARS]; + + // ... and copied here in appendTo() + private final char[] chars = new char[MAX_CHARS]; + + // Index into bytes of rightmost valid character. + private int index; + + private DoubleToDecimal() { + } + + /** + * Returns a string rendering of the {@code double} argument. + * + *

        The characters of the result are all drawn from the ASCII set. + *

          + *
        • Any NaN, whether quiet or signaling, is rendered as + * {@code "NaN"}, regardless of the sign bit. + *
        • The infinities +∞ and -∞ are rendered as + * {@code "Infinity"} and {@code "-Infinity"}, respectively. + *
        • The positive and negative zeroes are rendered as + * {@code "0.0"} and {@code "-0.0"}, respectively. + *
        • A finite negative {@code v} is rendered as the sign + * '{@code -}' followed by the rendering of the magnitude -{@code v}. + *
        • A finite positive {@code v} is rendered in two stages: + *
            + *
          • Selection of a decimal: A well-defined + * decimal dv is selected + * to represent {@code v}. + *
          • Formatting as a string: The decimal + * dv is formatted as a string, + * either in plain or in computerized scientific notation, + * depending on its value. + *
          + *
        + * + *

        A decimal is a number of the form + * d×10i + * for some (unique) integers d > 0 and i such that + * d is not a multiple of 10. + * These integers are the significand and + * the exponent, respectively, of the decimal. + * The length of the decimal is the (unique) + * integer n meeting + * 10n-1d < 10n. + * + *

        The decimal dv + * for a finite positive {@code v} is defined as follows: + *

          + *
        • Let R be the set of all decimals that round to {@code v} + * according to the usual round-to-closest rule of + * IEEE 754 floating-point arithmetic. + *
        • Let m be the minimal length over all decimals in R. + *
        • When m ≥ 2, let T be the set of all decimals + * in R with length m. + * Otherwise, let T be the set of all decimals + * in R with length 1 or 2. + *
        • Define dv as + * the decimal in T that is closest to {@code v}. + * Or if there are two such decimals in T, + * select the one with the even significand (there is exactly one). + *
        + * + *

        The (uniquely) selected decimal dv + * is then formatted. + * + *

        Let d, i and n be the significand, exponent and + * length of dv, respectively. + * Further, let e = n + i - 1 and let + * d1dn + * be the usual decimal expansion of the significand. + * Note that d1 ≠ 0 ≠ dn. + *

          + *
        • Case -3 ≤ e < 0: + * dv is formatted as + * 0.00d1dn, + * where there are exactly -(n + i) zeroes between + * the decimal point and d1. + * For example, 123 × 10-4 is formatted as + * {@code 0.0123}. + *
        • Case 0 ≤ e < 7: + *
            + *
          • Subcase i ≥ 0: + * dv is formatted as + * d1dn00.0, + * where there are exactly i zeroes + * between dn and the decimal point. + * For example, 123 × 102 is formatted as + * {@code 12300.0}. + *
          • Subcase i < 0: + * dv is formatted as + * d1dn+i.dn+i+1dn. + * There are exactly -i digits to the right of + * the decimal point. + * For example, 123 × 10-1 is formatted as + * {@code 12.3}. + *
          + *
        • Case e < -3 or e ≥ 7: + * computerized scientific notation is used to format + * dv. + * Here e is formatted as by {@link Integer#toString(int)}. + *
            + *
          • Subcase n = 1: + * dv is formatted as + * d1.0Ee. + * For example, 1 × 1023 is formatted as + * {@code 1.0E23}. + *
          • Subcase n > 1: + * dv is formatted as + * d1.d2dnEe. + * For example, 123 × 10-21 is formatted as + * {@code 1.23E-19}. + *
          + *
        + * + * @param v the {@code double} to be rendered. + * @return a string rendering of the argument. + */ + public static String toString(double v) { + return threadLocalInstance().toDecimalString(v); + } + + /** + * Appends the rendering of the {@code v} to {@code app}. + * + *

        The outcome is the same as if {@code v} were first + * {@link #toString(double) rendered} and the resulting string were then + * {@link Appendable#append(CharSequence) appended} to {@code app}. + * + * @param v the {@code double} whose rendering is appended. + * @param app the {@link Appendable} to append to. + * @throws IOException If an I/O error occurs + */ + public static Appendable appendTo(double v, Appendable app) + throws IOException { + return threadLocalInstance().appendDecimalTo(v, app); + } + + private static DoubleToDecimal threadLocalInstance() { + return threadLocal.get(); + } + + private String toDecimalString(double v) { + switch (toDecimal(v)) { + case NON_SPECIAL: return charsToString(); + case PLUS_ZERO: return "0.0"; + case MINUS_ZERO: return "-0.0"; + case PLUS_INF: return "Infinity"; + case MINUS_INF: return "-Infinity"; + default: return "NaN"; + } + } + + private Appendable appendDecimalTo(double v, Appendable app) + throws IOException { + switch (toDecimal(v)) { + case NON_SPECIAL: + for (int i = 0; i <= index; ++i) { + chars[i] = (char) bytes[i]; + } + if (app instanceof StringBuilder) { + return ((StringBuilder) app).append(chars, 0, index + 1); + } + if (app instanceof StringBuffer) { + return ((StringBuffer) app).append(chars, 0, index + 1); + } + for (int i = 0; i <= index; ++i) { + app.append(chars[i]); + } + return app; + case PLUS_ZERO: return app.append("0.0"); + case MINUS_ZERO: return app.append("-0.0"); + case PLUS_INF: return app.append("Infinity"); + case MINUS_INF: return app.append("-Infinity"); + default: return app.append("NaN"); + } + } + + /* + Returns + PLUS_ZERO iff v is 0.0 + MINUS_ZERO iff v is -0.0 + PLUS_INF iff v is POSITIVE_INFINITY + MINUS_INF iff v is NEGATIVE_INFINITY + NAN iff v is NaN + */ + private int toDecimal(double v) { + /* + For full details see references [2] and [1]. + + For finite v != 0, determine integers c and q such that + |v| = c 2^q and + Q_MIN <= q <= Q_MAX and + either 2^(P-1) <= c < 2^P (normal) + or 0 < c < 2^(P-1) and q = Q_MIN (subnormal) + */ + long bits = doubleToRawLongBits(v); + long t = bits & T_MASK; + int bq = (int) (bits >>> P - 1) & BQ_MASK; + if (bq < BQ_MASK) { + index = -1; + if (bits < 0) { + append('-'); + } + if (bq != 0) { + // normal value. Here mq = -q + int mq = -Q_MIN + 1 - bq; + long c = C_MIN | t; + // The fast path discussed in section 8.2 of [1]. + if (0 < mq & mq < P) { + long f = c >> mq; + if (f << mq == c) { + return toChars(f, 0); + } + } + return toDecimal(-mq, c, 0); + } + if (t != 0) { + // subnormal value + return t < C_TINY + ? toDecimal(Q_MIN, 10 * t, -1) + : toDecimal(Q_MIN, t, 0); + } + return bits == 0 ? PLUS_ZERO : MINUS_ZERO; + } + if (t != 0) { + return NAN; + } + return bits > 0 ? PLUS_INF : MINUS_INF; + } + + private int toDecimal(int q, long c, int dk) { + /* + The skeleton corresponds to figure 4 of [1]. + The efficient computations are those summarized in figure 7. + + Here's a correspondence between Java names and names in [1], + expressed as approximate LaTeX source code and informally. + Other names are identical. + cb: \bar{c} "c-bar" + cbr: \bar{c}_r "c-bar-r" + cbl: \bar{c}_l "c-bar-l" + + vb: \bar{v} "v-bar" + vbr: \bar{v}_r "v-bar-r" + vbl: \bar{v}_l "v-bar-l" + + rop: r_o' "r-o-prime" + */ + int out = (int) c & 0x1; + long cb = c << 2; + long cbr = cb + 2; + long cbl; + int k; + /* + flog10pow2(e) = floor(log_10(2^e)) + flog10threeQuartersPow2(e) = floor(log_10(3/4 2^e)) + flog2pow10(e) = floor(log_2(10^e)) + */ + if (c != C_MIN | q == Q_MIN) { + // regular spacing + cbl = cb - 2; + k = flog10pow2(q); + } else { + // irregular spacing + cbl = cb - 1; + k = flog10threeQuartersPow2(q); + } + int h = q + flog2pow10(-k) + 2; + + // g1 and g0 are as in section 9.9.3 of [1], so g = g1 2^63 + g0 + long g1 = g1(k); + long g0 = g0(k); + + long vb = rop(g1, g0, cb << h); + long vbl = rop(g1, g0, cbl << h); + long vbr = rop(g1, g0, cbr << h); + + long s = vb >> 2; + if (s >= 100) { + /* + For n = 17, m = 1 the table in section 10 of [1] shows + s' = floor(s / 10) = floor(s 115_292_150_460_684_698 / 2^60) + = floor(s 115_292_150_460_684_698 2^4 / 2^64) + + sp10 = 10 s' + tp10 = 10 t' + upin iff u' = sp10 10^k in Rv + wpin iff w' = tp10 10^k in Rv + See section 9.4 of [1]. + */ + long sp10 = 10 * multiplyHigh(s, 115_292_150_460_684_698L << 4); + long tp10 = sp10 + 10; + boolean upin = vbl + out <= sp10 << 2; + boolean wpin = (tp10 << 2) + out <= vbr; + if (upin != wpin) { + return toChars(upin ? sp10 : tp10, k); + } + } + + /* + 10 <= s < 100 or s >= 100 and u', w' not in Rv + uin iff u = s 10^k in Rv + win iff w = t 10^k in Rv + See section 9.4 of [1]. + */ + long t = s + 1; + boolean uin = vbl + out <= s << 2; + boolean win = (t << 2) + out <= vbr; + if (uin != win) { + // Exactly one of u or w lies in Rv. + return toChars(uin ? s : t, k + dk); + } + /* + Both u and w lie in Rv: determine the one closest to v. + See section 9.4 of [1]. + */ + long cmp = vb - (s + t << 1); + return toChars(cmp < 0 || cmp == 0 && (s & 0x1) == 0 ? s : t, k + dk); + } + + /* + Computes rop(cp g 2^(-127)), where g = g1 2^63 + g0 + See section 9.10 and figure 5 of [1]. + */ + private static long rop(long g1, long g0, long cp) { + long x1 = multiplyHigh(g0, cp); + long y0 = g1 * cp; + long y1 = multiplyHigh(g1, cp); + long z = (y0 >>> 1) + x1; + long vbp = y1 + (z >>> 63); + return vbp | (z & MASK_63) + MASK_63 >>> 63; + } + + /* + Formats the decimal f 10^e. + */ + private int toChars(long f, int e) { + /* + For details not discussed here see section 10 of [1]. + + Determine len such that + 10^(len-1) <= f < 10^len + */ + int len = flog10pow2(Long.SIZE - numberOfLeadingZeros(f)); + if (f >= pow10(len)) { + len += 1; + } + + /* + Let fp and ep be the original f and e, respectively. + Transform f and e to ensure + 10^(H-1) <= f < 10^H + fp 10^ep = f 10^(e-H) = 0.f 10^e + */ + f *= pow10(H - len); + e += len; + + /* + The toChars?() methods perform left-to-right digits extraction + using ints, provided that the arguments are limited to 8 digits. + Therefore, split the H = 17 digits of f into: + h = the most significant digit of f + m = the next 8 most significant digits of f + l = the last 8, least significant digits of f + + For n = 17, m = 8 the table in section 10 of [1] shows + floor(f / 10^8) = floor(193_428_131_138_340_668 f / 2^84) = + floor(floor(193_428_131_138_340_668 f / 2^64) / 2^20) + and for n = 9, m = 8 + floor(hm / 10^8) = floor(1_441_151_881 hm / 2^57) + */ + long hm = multiplyHigh(f, 193_428_131_138_340_668L) >>> 20; + int l = (int) (f - 100_000_000L * hm); + int h = (int) (hm * 1_441_151_881L >>> 57); + int m = (int) (hm - 100_000_000 * h); + + if (0 < e && e <= 7) { + return toChars1(h, m, l, e); + } + if (-3 < e && e <= 0) { + return toChars2(h, m, l, e); + } + return toChars3(h, m, l, e); + } + + private int toChars1(int h, int m, int l, int e) { + /* + 0 < e <= 7: plain format without leading zeroes. + Left-to-right digits extraction: + algorithm 1 in [3], with b = 10, k = 8, n = 28. + */ + appendDigit(h); + int y = y(m); + int t; + int i = 1; + for (; i < e; ++i) { + t = 10 * y; + appendDigit(t >>> 28); + y = t & MASK_28; + } + append('.'); + for (; i <= 8; ++i) { + t = 10 * y; + appendDigit(t >>> 28); + y = t & MASK_28; + } + lowDigits(l); + return NON_SPECIAL; + } + + private int toChars2(int h, int m, int l, int e) { + // -3 < e <= 0: plain format with leading zeroes. + appendDigit(0); + append('.'); + for (; e < 0; ++e) { + appendDigit(0); + } + appendDigit(h); + append8Digits(m); + lowDigits(l); + return NON_SPECIAL; + } + + private int toChars3(int h, int m, int l, int e) { + // -3 >= e | e > 7: computerized scientific notation + appendDigit(h); + append('.'); + append8Digits(m); + lowDigits(l); + exponent(e - 1); + return NON_SPECIAL; + } + + private void lowDigits(int l) { + if (l != 0) { + append8Digits(l); + } + removeTrailingZeroes(); + } + + private void append8Digits(int m) { + /* + Left-to-right digits extraction: + algorithm 1 in [3], with b = 10, k = 8, n = 28. + */ + int y = y(m); + for (int i = 0; i < 8; ++i) { + int t = 10 * y; + appendDigit(t >>> 28); + y = t & MASK_28; + } + } + + private void removeTrailingZeroes() { + while (bytes[index] == '0') { + --index; + } + // ... but do not remove the one directly to the right of '.' + if (bytes[index] == '.') { + ++index; + } + } + + private int y(int a) { + /* + Algorithm 1 in [3] needs computation of + floor((a + 1) 2^n / b^k) - 1 + with a < 10^8, b = 10, k = 8, n = 28. + Noting that + (a + 1) 2^n <= 10^8 2^28 < 10^17 + For n = 17, m = 8 the table in section 10 of [1] leads to: + */ + return (int) (multiplyHigh( + (long) (a + 1) << 28, + 193_428_131_138_340_668L) >>> 20) - 1; + } + + private void exponent(int e) { + append('E'); + if (e < 0) { + append('-'); + e = -e; + } + if (e < 10) { + appendDigit(e); + return; + } + int d; + if (e >= 100) { + /* + For n = 3, m = 2 the table in section 10 of [1] shows + floor(e / 100) = floor(1_311 e / 2^17) + */ + d = e * 1_311 >>> 17; + appendDigit(d); + e -= 100 * d; + } + /* + For n = 2, m = 1 the table in section 10 of [1] shows + floor(e / 10) = floor(103 e / 2^10) + */ + d = e * 103 >>> 10; + appendDigit(d); + appendDigit(e - 10 * d); + } + + private void append(int c) { + bytes[++index] = (byte) c; + } + + private void appendDigit(int d) { + bytes[++index] = (byte) ('0' + d); + } + + // Using the deprecated constructor enhances performance. + @SuppressWarnings("deprecation") + private String charsToString() { + return new String(bytes, 0, 0, index + 1); + } + +} diff --git a/src/java.base/share/classes/jdk/internal/math/FloatToDecimal.java b/src/java.base/share/classes/jdk/internal/math/FloatToDecimal.java new file mode 100644 --- /dev/null +++ b/src/java.base/share/classes/jdk/internal/math/FloatToDecimal.java @@ -0,0 +1,617 @@ +/* + * Copyright 2018-2020 Raffaello Giulietti + * + * Permission is hereby granted, free of charge, to any person obtaining a copy + * of this software and associated documentation files (the "Software"), to deal + * in the Software without restriction, including without limitation the rights + * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell + * copies of the Software, and to permit persons to whom the Software is + * furnished to do so, subject to the following conditions: + * + * The above copyright notice and this permission notice shall be included in + * all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR + * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, + * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE + * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER + * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, + * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN + * THE SOFTWARE. + */ + +package jdk.internal.math; + +import java.io.IOException; + +import static java.lang.Float.*; +import static java.lang.Integer.*; +import static java.lang.Math.multiplyHigh; +import static jdk.internal.math.MathUtils.*; + +/** + * This class exposes a method to render a {@code float} as a string. + * + * @author Raffaello Giulietti + */ +final public class FloatToDecimal { + /* + For full details about this code see the following references: + + [1] Giulietti, "The Schubfach way to render doubles", + https://drive.google.com/open?id=1luHhyQF9zKlM8yJ1nebU0OgVYhfC6CBN + + [2] IEEE Computer Society, "IEEE Standard for Floating-Point Arithmetic" + + [3] Bouvier & Zimmermann, "Division-Free Binary-to-Decimal Conversion" + + Divisions are avoided altogether for the benefit of those architectures + that do not provide specific machine instructions or where they are slow. + This is discussed in section 10 of [1]. + */ + + // The precision in bits. + static final int P = 24; + + // Exponent width in bits. + private static final int W = (Float.SIZE - 1) - (P - 1); + + // Minimum value of the exponent: -(2^(W-1)) - P + 3. + static final int Q_MIN = (-1 << W - 1) - P + 3; + + // Maximum value of the exponent: 2^(W-1) - P. + static final int Q_MAX = (1 << W - 1) - P; + + // 10^(E_MIN - 1) <= MIN_VALUE < 10^E_MIN + static final int E_MIN = -44; + + // 10^(E_MAX - 1) <= MAX_VALUE < 10^E_MAX + static final int E_MAX = 39; + + // Threshold to detect tiny values, as in section 8.1.1 of [1] + static final int C_TINY = 8; + + // The minimum and maximum k, as in section 8 of [1] + static final int K_MIN = -45; + static final int K_MAX = 31; + + // H is as in section 8 of [1]. + static final int H = 9; + + // Minimum value of the significand of a normal value: 2^(P-1). + private static final int C_MIN = 1 << P - 1; + + // Mask to extract the biased exponent. + private static final int BQ_MASK = (1 << W) - 1; + + // Mask to extract the fraction bits. + private static final int T_MASK = (1 << P - 1) - 1; + + // Used in rop(). + private static final long MASK_32 = (1L << 32) - 1; + + // Used for left-to-tight digit extraction. + private static final int MASK_28 = (1 << 28) - 1; + + private static final int NON_SPECIAL = 0; + private static final int PLUS_ZERO = 1; + private static final int MINUS_ZERO = 2; + private static final int PLUS_INF = 3; + private static final int MINUS_INF = 4; + private static final int NAN = 5; + + // For thread-safety, each thread gets its own instance of this class. + private static final ThreadLocal threadLocal = + ThreadLocal.withInitial(FloatToDecimal::new); + + /* + Room for the longer of the forms + -ddddd.dddd H + 2 characters + -0.00ddddddddd H + 5 characters + -d.ddddddddE-ee H + 6 characters + where there are H digits d + */ + public final int MAX_CHARS = H + 6; + + // Numerical results are created here... + private final byte[] bytes = new byte[MAX_CHARS]; + + // ... and copied here in appendTo() + private final char[] chars = new char[MAX_CHARS]; + + // Index into buf of rightmost valid character. + private int index; + + private FloatToDecimal() { + } + + /** + * Returns a string rendering of the {@code float} argument. + * + *

        The characters of the result are all drawn from the ASCII set. + *

          + *
        • Any NaN, whether quiet or signaling, is rendered as + * {@code "NaN"}, regardless of the sign bit. + *
        • The infinities +∞ and -∞ are rendered as + * {@code "Infinity"} and {@code "-Infinity"}, respectively. + *
        • The positive and negative zeroes are rendered as + * {@code "0.0"} and {@code "-0.0"}, respectively. + *
        • A finite negative {@code v} is rendered as the sign + * '{@code -}' followed by the rendering of the magnitude -{@code v}. + *
        • A finite positive {@code v} is rendered in two stages: + *
            + *
          • Selection of a decimal: A well-defined + * decimal dv is selected + * to represent {@code v}. + *
          • Formatting as a string: The decimal + * dv is formatted as a string, + * either in plain or in computerized scientific notation, + * depending on its value. + *
          + *
        + * + *

        A decimal is a number of the form + * d×10i + * for some (unique) integers d > 0 and i such that + * d is not a multiple of 10. + * These integers are the significand and + * the exponent, respectively, of the decimal. + * The length of the decimal is the (unique) + * integer n meeting + * 10n-1d < 10n. + * + *

        The decimal dv + * for a finite positive {@code v} is defined as follows: + *

          + *
        • Let R be the set of all decimals that round to {@code v} + * according to the usual round-to-closest rule of + * IEEE 754 floating-point arithmetic. + *
        • Let m be the minimal length over all decimals in R. + *
        • When m ≥ 2, let T be the set of all decimals + * in R with length m. + * Otherwise, let T be the set of all decimals + * in R with length 1 or 2. + *
        • Define dv as + * the decimal in T that is closest to {@code v}. + * Or if there are two such decimals in T, + * select the one with the even significand (there is exactly one). + *
        + * + *

        The (uniquely) selected decimal dv + * is then formatted. + * + *

        Let d, i and n be the significand, exponent and + * length of dv, respectively. + * Further, let e = n + i - 1 and let + * d1dn + * be the usual decimal expansion of the significand. + * Note that d1 ≠ 0 ≠ dn. + *

          + *
        • Case -3 ≤ e < 0: + * dv is formatted as + * 0.00d1dn, + * where there are exactly -(n + i) zeroes between + * the decimal point and d1. + * For example, 123 × 10-4 is formatted as + * {@code 0.0123}. + *
        • Case 0 ≤ e < 7: + *
            + *
          • Subcase i ≥ 0: + * dv is formatted as + * d1dn00.0, + * where there are exactly i zeroes + * between dn and the decimal point. + * For example, 123 × 102 is formatted as + * {@code 12300.0}. + *
          • Subcase i < 0: + * dv is formatted as + * d1dn+i.dn+i+1dn. + * There are exactly -i digits to the right of + * the decimal point. + * For example, 123 × 10-1 is formatted as + * {@code 12.3}. + *
          + *
        • Case e < -3 or e ≥ 7: + * computerized scientific notation is used to format + * dv. + * Here e is formatted as by {@link Integer#toString(int)}. + *
            + *
          • Subcase n = 1: + * dv is formatted as + * d1.0Ee. + * For example, 1 × 1023 is formatted as + * {@code 1.0E23}. + *
          • Subcase n > 1: + * dv is formatted as + * d1.d2dnEe. + * For example, 123 × 10-21 is formatted as + * {@code 1.23E-19}. + *
          + *
        + * + * @param v the {@code float} to be rendered. + * @return a string rendering of the argument. + */ + public static String toString(float v) { + return threadLocalInstance().toDecimalString(v); + } + + /** + * Appends the rendering of the {@code v} to {@code app}. + * + *

        The outcome is the same as if {@code v} were first + * {@link #toString(float) rendered} and the resulting string were then + * {@link Appendable#append(CharSequence) appended} to {@code app}. + * + * @param v the {@code float} whose rendering is appended. + * @param app the {@link Appendable} to append to. + * @throws IOException If an I/O error occurs + */ + public static Appendable appendTo(float v, Appendable app) + throws IOException { + return threadLocalInstance().appendDecimalTo(v, app); + } + + private static FloatToDecimal threadLocalInstance() { + return threadLocal.get(); + } + + private String toDecimalString(float v) { + switch (toDecimal(v)) { + case NON_SPECIAL: return charsToString(); + case PLUS_ZERO: return "0.0"; + case MINUS_ZERO: return "-0.0"; + case PLUS_INF: return "Infinity"; + case MINUS_INF: return "-Infinity"; + default: return "NaN"; + } + } + + private Appendable appendDecimalTo(float v, Appendable app) + throws IOException { + switch (toDecimal(v)) { + case NON_SPECIAL: + for (int i = 0; i <= index; ++i) { + chars[i] = (char) bytes[i]; + } + if (app instanceof StringBuilder) { + return ((StringBuilder) app).append(chars, 0, index + 1); + } + if (app instanceof StringBuffer) { + return ((StringBuffer) app).append(chars, 0, index + 1); + } + for (int i = 0; i <= index; ++i) { + app.append(chars[i]); + } + return app; + case PLUS_ZERO: return app.append("0.0"); + case MINUS_ZERO: return app.append("-0.0"); + case PLUS_INF: return app.append("Infinity"); + case MINUS_INF: return app.append("-Infinity"); + default: return app.append("NaN"); + } + } + + /* + Returns + PLUS_ZERO iff v is 0.0 + MINUS_ZERO iff v is -0.0 + PLUS_INF iff v is POSITIVE_INFINITY + MINUS_INF iff v is NEGATIVE_INFINITY + NAN iff v is NaN + */ + private int toDecimal(float v) { + /* + For full details see references [2] and [1]. + + For finite v != 0, determine integers c and q such that + |v| = c 2^q and + Q_MIN <= q <= Q_MAX and + either 2^(P-1) <= c < 2^P (normal) + or 0 < c < 2^(P-1) and q = Q_MIN (subnormal) + */ + int bits = floatToRawIntBits(v); + int t = bits & T_MASK; + int bq = (bits >>> P - 1) & BQ_MASK; + if (bq < BQ_MASK) { + index = -1; + if (bits < 0) { + append('-'); + } + if (bq != 0) { + // normal value. Here mq = -q + int mq = -Q_MIN + 1 - bq; + int c = C_MIN | t; + // The fast path discussed in section 8.2 of [1]. + if (0 < mq & mq < P) { + int f = c >> mq; + if (f << mq == c) { + return toChars(f, 0); + } + } + return toDecimal(-mq, c, 0); + } + if (t != 0) { + // subnormal value + return t < C_TINY + ? toDecimal(Q_MIN, 10 * t, -1) + : toDecimal(Q_MIN, t, 0); + } + return bits == 0 ? PLUS_ZERO : MINUS_ZERO; + } + if (t != 0) { + return NAN; + } + return bits > 0 ? PLUS_INF : MINUS_INF; + } + + private int toDecimal(int q, int c, int dk) { + /* + The skeleton corresponds to figure 4 of [1]. + The efficient computations are those summarized in figure 7. + Also check the appendix. + + Here's a correspondence between Java names and names in [1], + expressed as approximate LaTeX source code and informally. + Other names are identical. + cb: \bar{c} "c-bar" + cbr: \bar{c}_r "c-bar-r" + cbl: \bar{c}_l "c-bar-l" + + vb: \bar{v} "v-bar" + vbr: \bar{v}_r "v-bar-r" + vbl: \bar{v}_l "v-bar-l" + + rop: r_o' "r-o-prime" + */ + int out = c & 0x1; + long cb = c << 2; + long cbr = cb + 2; + long cbl; + int k; + /* + flog10pow2(e) = floor(log_10(2^e)) + flog10threeQuartersPow2(e) = floor(log_10(3/4 2^e)) + flog2pow10(e) = floor(log_2(10^e)) + */ + if (c != C_MIN | q == Q_MIN) { + // regular spacing + cbl = cb - 2; + k = flog10pow2(q); + } else { + // irregular spacing0 + cbl = cb - 1; + k = flog10threeQuartersPow2(q); + } + int h = q + flog2pow10(-k) + 33; + + // g is as in the appendix + long g = g1(k) + 1; + + int vb = rop(g, cb << h); + int vbl = rop(g, cbl << h); + int vbr = rop(g, cbr << h); + + int s = vb >> 2; + if (s >= 100) { + /* + For n = 9, m = 1 the table in section 10 of [1] shows + s' = floor(s / 10) = floor(s 1_717_986_919 / 2^34) + + sp10 = 10 s' + tp10 = 10 t' + upin iff u' = sp10 10^k in Rv + wpin iff w' = tp10 10^k in Rv + See section 9.4 of [1]. + */ + int sp10 = 10 * (int) (s * 1_717_986_919L >>> 34); + int tp10 = sp10 + 10; + boolean upin = vbl + out <= sp10 << 2; + boolean wpin = (tp10 << 2) + out <= vbr; + if (upin != wpin) { + return toChars(upin ? sp10 : tp10, k); + } + } + + /* + 10 <= s < 100 or s >= 100 and u', w' not in Rv + uin iff u = s 10^k in Rv + win iff w = t 10^k in Rv + See section 9.4 of [1]. + */ + int t = s + 1; + boolean uin = vbl + out <= s << 2; + boolean win = (t << 2) + out <= vbr; + if (uin != win) { + // Exactly one of u or w lies in Rv. + return toChars(uin ? s : t, k + dk); + } + /* + Both u and w lie in Rv: determine the one closest to v. + See section 9.4 of [1]. + */ + int cmp = vb - (s + t << 1); + return toChars(cmp < 0 || cmp == 0 && (s & 0x1) == 0 ? s : t, k + dk); + } + + /* + Computes rop(cp g 2^(-95)) + See appendix and figure 8 of [1]. + */ + private static int rop(long g, long cp) { + long x1 = multiplyHigh(g, cp); + long vbp = x1 >>> 31; + return (int) (vbp | (x1 & MASK_32) + MASK_32 >>> 32); + } + + /* + Formats the decimal f 10^e. + */ + private int toChars(int f, int e) { + /* + For details not discussed here see section 10 of [1]. + + Determine len such that + 10^(len-1) <= f < 10^len + */ + int len = flog10pow2(Integer.SIZE - numberOfLeadingZeros(f)); + if (f >= pow10(len)) { + len += 1; + } + + /* + Let fp and ep be the original f and e, respectively. + Transform f and e to ensure + 10^(H-1) <= f < 10^H + fp 10^ep = f 10^(e-H) = 0.f 10^e + */ + f *= pow10(H - len); + e += len; + + /* + The toChars?() methods perform left-to-right digits extraction + using ints, provided that the arguments are limited to 8 digits. + Therefore, split the H = 9 digits of f into: + h = the most significant digit of f + l = the last 8, least significant digits of f + + For n = 9, m = 8 the table in section 10 of [1] shows + floor(f / 10^8) = floor(1_441_151_881 f / 2^57) + */ + int h = (int) (f * 1_441_151_881L >>> 57); + int l = f - 100_000_000 * h; + + if (0 < e && e <= 7) { + return toChars1(h, l, e); + } + if (-3 < e && e <= 0) { + return toChars2(h, l, e); + } + return toChars3(h, l, e); + } + + private int toChars1(int h, int l, int e) { + /* + 0 < e <= 7: plain format without leading zeroes. + Left-to-right digits extraction: + algorithm 1 in [3], with b = 10, k = 8, n = 28. + */ + appendDigit(h); + int y = y(l); + int t; + int i = 1; + for (; i < e; ++i) { + t = 10 * y; + appendDigit(t >>> 28); + y = t & MASK_28; + } + append('.'); + for (; i <= 8; ++i) { + t = 10 * y; + appendDigit(t >>> 28); + y = t & MASK_28; + } + removeTrailingZeroes(); + return NON_SPECIAL; + } + + private int toChars2(int h, int l, int e) { + // -3 < e <= 0: plain format with leading zeroes. + appendDigit(0); + append('.'); + for (; e < 0; ++e) { + appendDigit(0); + } + appendDigit(h); + append8Digits(l); + removeTrailingZeroes(); + return NON_SPECIAL; + } + + private int toChars3(int h, int l, int e) { + // -3 >= e | e > 7: computerized scientific notation + appendDigit(h); + append('.'); + append8Digits(l); + removeTrailingZeroes(); + exponent(e - 1); + return NON_SPECIAL; + } + + private void append8Digits(int m) { + /* + Left-to-right digits extraction: + algorithm 1 in [3], with b = 10, k = 8, n = 28. + */ + int y = y(m); + for (int i = 0; i < 8; ++i) { + int t = 10 * y; + appendDigit(t >>> 28); + y = t & MASK_28; + } + } + + private void removeTrailingZeroes() { + while (bytes[index] == '0') { + --index; + } + // ... but do not remove the one directly to the right of '.' + if (bytes[index] == '.') { + ++index; + } + } + + private int y(int a) { + /* + Algorithm 1 in [3] needs computation of + floor((a + 1) 2^n / b^k) - 1 + with a < 10^8, b = 10, k = 8, n = 28. + Noting that + (a + 1) 2^n <= 10^8 2^28 < 10^17 + For n = 17, m = 8 the table in section 10 of [1] leads to: + */ + return (int) (multiplyHigh( + (long) (a + 1) << 28, + 193_428_131_138_340_668L) >>> 20) - 1; + } + + private void exponent(int e) { + append('E'); + if (e < 0) { + append('-'); + e = -e; + } + if (e < 10) { + appendDigit(e); + return; + } + /* + For n = 2, m = 1 the table in section 10 of [1] shows + floor(e / 10) = floor(103 e / 2^10) + */ + int d = e * 103 >>> 10; + appendDigit(d); + appendDigit(e - 10 * d); + } + + private void append(int c) { + bytes[++index] = (byte) c; + } + + private void appendDigit(int d) { + bytes[++index] = (byte) ('0' + d); + } + + // Using the deprecated constructor enhances performance. + @SuppressWarnings("deprecation") + private String charsToString() { + return new String(bytes, 0, 0, index + 1); + } + +} diff --git a/src/java.base/share/classes/jdk/internal/math/MathUtils.java b/src/java.base/share/classes/jdk/internal/math/MathUtils.java new file mode 100644 --- /dev/null +++ b/src/java.base/share/classes/jdk/internal/math/MathUtils.java @@ -0,0 +1,811 @@ +/* + * Copyright 2018-2020 Raffaello Giulietti + * + * Permission is hereby granted, free of charge, to any person obtaining a copy + * of this software and associated documentation files (the "Software"), to deal + * in the Software without restriction, including without limitation the rights + * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell + * copies of the Software, and to permit persons to whom the Software is + * furnished to do so, subject to the following conditions: + * + * The above copyright notice and this permission notice shall be included in + * all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR + * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, + * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE + * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER + * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, + * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN + * THE SOFTWARE. + */ + +package jdk.internal.math; + +/** + * This class exposes package private utilities for other classes. + * Thus, all methods are assumed to be invoked with correct arguments, + * so these are not checked at all. + * + * @author Raffaello Giulietti + */ +final class MathUtils { + /* + For full details about this code see the following reference: + + Giulietti, "The Schubfach way to render doubles", + https://drive.google.com/open?id=1luHhyQF9zKlM8yJ1nebU0OgVYhfC6CBN + */ + + /* + The boundaries for k in g0(int) and g1(int). + K_MIN must be DoubleToDecimal.K_MIN or less. + K_MAX must be DoubleToDecimal.K_MAX or more. + */ + static final int K_MIN = -324; + static final int K_MAX = 292; + + // Must be DoubleToDecimal.H or more + static final int H = 17; + + // C_10 = floor(log10(2) * 2^Q_10), A_10 = floor(log10(3/4) * 2^Q_10) + private static final int Q_10 = 41; + private static final long C_10 = 661_971_961_083L; + private static final long A_10 = -274_743_187_321L; + + // C_2 = floor(log2(10) * 2^Q_2) + private static final int Q_2 = 38; + private static final long C_2 = 913_124_641_741L; + + private MathUtils() { + } + + // The first powers of 10. The last entry must be 10^H. + private static final long[] pow10 = { + 1L, + 10L, + 100L, + 1_000L, + 10_000L, + 100_000L, + 1_000_000L, + 10_000_000L, + 100_000_000L, + 1_000_000_000L, + 10_000_000_000L, + 100_000_000_000L, + 1_000_000_000_000L, + 10_000_000_000_000L, + 100_000_000_000_000L, + 1_000_000_000_000_000L, + 10_000_000_000_000_000L, + 100_000_000_000_000_000L, + }; + + /** + * Returns 10{@code e}. + * + * @param e The exponent which must meet + * 0 ≤ {@code e} ≤ {@link #H}. + * @return 10{@code e}. + */ + static long pow10(int e) { + return pow10[e]; + } + + /** + * Returns the unique integer k such that + * 10k ≤ 2{@code e} + * < 10k+1. + *

        + * The result is correct when |{@code e}| ≤ 5_456_721. + * Otherwise the result is undefined. + * + * @param e The exponent of 2, which should meet + * |{@code e}| ≤ 5_456_721 for safe results. + * @return ⌊log102{@code e}⌋. + */ + static int flog10pow2(int e) { + return (int) (e * C_10 >> Q_10); + } + + /** + * Returns the unique integer k such that + * 10k ≤ 3/4 · 2{@code e} + * < 10k+1. + *

        + * The result is correct when + * -2_956_395 ≤ {@code e} ≤ 2_500_325. + * Otherwise the result is undefined. + * + * @param e The exponent of 2, which should meet + * -2_956_395 ≤ {@code e} ≤ 2_500_325 for safe results. + * @return ⌊log10(3/4 · + * 2{@code e})⌋. + */ + static int flog10threeQuartersPow2(int e) { + return (int) (e * C_10 + A_10 >> Q_10); + } + + /** + * Returns the unique integer k such that + * 2k ≤ 10{@code e} + * < 2k+1. + *

        + * The result is correct when |{@code e}| ≤ 1_838_394. + * Otherwise the result is undefined. + * + * @param e The exponent of 10, which should meet + * |{@code e}| ≤ 1_838_394 for safe results. + * @return ⌊log210{@code e}⌋. + */ + static int flog2pow10(int e) { + return (int) (e * C_2 >> Q_2); + } + + /** + * Let 10-{@code k} = β 2r, + * for the unique pair of integer r and real β meeting + * 2125β < 2126. + * Further, let g = ⌊β⌋ + 1. + * Split g into the higher 63 bits g1 and + * the lower 63 bits g0. Thus, + * g1 = + * ⌊g 2-63⌋ + * and + * g0 = + * g - g1 263. + *

        + * This method returns g1 while + * {@link #g0(int)} returns g0. + *

        + * If needed, the exponent r can be computed as + * r = {@code flog2pow10(-k)} - 125 (see {@link #flog2pow10(int)}). + * + * @param k The exponent of 10, which must meet + * {@link #K_MIN} ≤ {@code e} ≤ {@link #K_MAX}. + * @return g1 as described above. + */ + static long g1(int k) { + return g[k - K_MIN << 1]; + } + + /** + * Returns g0 as described in + * {@link #g1(int)}. + * + * @param k The exponent of 10, which must meet + * {@link #K_MIN} ≤ {@code e} ≤ {@link #K_MAX}. + * @return g0 as described in + * {@link #g1(int)}. + */ + static long g0(int k) { + return g[k - K_MIN << 1 | 1]; + } + + /* + The precomputed values for g1(int) and g0(int). + The first entry must be for an exponent of K_MIN or less. + The last entry must be for an exponent of K_MAX or more. + */ + private static final long[] g = { + /* -324 */ 0x4F0C_EDC9_5A71_8DD4L, 0x5B01_E8B0_9AA0_D1B5L, + /* -323 */ 0x7E7B_160E_F71C_1621L, 0x119C_A780_F767_B5EEL, + /* -322 */ 0x652F_44D8_C5B0_11B4L, 0x0E16_EC67_2C52_F7F2L, + /* -321 */ 0x50F2_9D7A_37C0_0E29L, 0x5812_56B8_F042_5FF5L, + /* -320 */ 0x40C2_1794_F966_71BAL, 0x79A8_4560_C035_1991L, + /* -319 */ 0x679C_F287_F570_B5F7L, 0x75DA_089A_CD21_C281L, + /* -318 */ 0x52E3_F539_9126_F7F9L, 0x44AE_6D48_A41B_0201L, + /* -317 */ 0x424F_F761_40EB_F994L, 0x36F1_F106_E9AF_34CDL, + /* -316 */ 0x6A19_8BCE_CE46_5C20L, 0x57E9_81A4_A918_547BL, + /* -315 */ 0x54E1_3CA5_71D1_E34DL, 0x2CBA_CE1D_5413_76C9L, + /* -314 */ 0x43E7_63B7_8E41_82A4L, 0x23C8_A4E4_4342_C56EL, + /* -313 */ 0x6CA5_6C58_E39C_043AL, 0x060D_D4A0_6B9E_08B0L, + /* -312 */ 0x56EA_BD13_E949_9CFBL, 0x1E71_76E6_BC7E_6D59L, + /* -311 */ 0x4588_9743_2107_B0C8L, 0x7EC1_2BEB_C9FE_BDE1L, + /* -310 */ 0x6F40_F205_01A5_E7A7L, 0x7E01_DFDF_A997_9635L, + /* -309 */ 0x5900_C19D_9AEB_1FB9L, 0x4B34_B319_5479_44F7L, + /* -308 */ 0x4733_CE17_AF22_7FC7L, 0x55C3_C27A_A9FA_9D93L, + /* -307 */ 0x71EC_7CF2_B1D0_CC72L, 0x5606_03F7_765D_C8EAL, + /* -306 */ 0x5B23_9728_8E40_A38EL, 0x7804_CFF9_2B7E_3A55L, + /* -305 */ 0x48E9_45BA_0B66_E93FL, 0x1337_0CC7_55FE_9511L, + /* -304 */ 0x74A8_6F90_123E_41FEL, 0x51F1_AE0B_BCCA_881BL, + /* -303 */ 0x5D53_8C73_41CB_67FEL, 0x74C1_5809_63D5_39AFL, + /* -302 */ 0x4AA9_3D29_016F_8665L, 0x43CD_E007_8310_FAF3L, + /* -301 */ 0x7775_2EA8_024C_0A3CL, 0x0616_333F_381B_2B1EL, + /* -300 */ 0x5F90_F220_01D6_6E96L, 0x3811_C298_F9AF_55B1L, + /* -299 */ 0x4C73_F4E6_67DE_BEDEL, 0x600E_3547_2E25_DE28L, + /* -298 */ 0x7A53_2170_A631_3164L, 0x3349_EED8_49D6_303FL, + /* -297 */ 0x61DC_1AC0_84F4_2783L, 0x42A1_8BE0_3B11_C033L, + /* -296 */ 0x4E49_AF00_6A5C_EC69L, 0x1BB4_6FE6_95A7_CCF5L, + /* -295 */ 0x7D42_B19A_43C7_E0A8L, 0x2C53_E63D_BC3F_AE55L, + /* -294 */ 0x6435_5AE1_CFD3_1A20L, 0x2376_51CA_FCFF_BEAAL, + /* -293 */ 0x502A_AF1B_0CA8_E1B3L, 0x35F8_416F_30CC_9888L, + /* -292 */ 0x4022_25AF_3D53_E7C2L, 0x5E60_3458_F3D6_E06DL, + /* -291 */ 0x669D_0918_621F_D937L, 0x4A33_86F4_B957_CD7BL, + /* -290 */ 0x5217_3A79_E819_7A92L, 0x6E8F_9F2A_2DDF_D796L, + /* -289 */ 0x41AC_2EC7_ECE1_2EDBL, 0x720C_7F54_F17F_DFABL, + /* -288 */ 0x6913_7E0C_AE35_17C6L, 0x1CE0_CBBB_1BFF_CC45L, + /* -287 */ 0x540F_980A_24F7_4638L, 0x171A_3C95_AFFF_D69EL, + /* -286 */ 0x433F_ACD4_EA5F_6B60L, 0x127B_63AA_F333_1218L, + /* -285 */ 0x6B99_1487_DD65_7899L, 0x6A5F_05DE_51EB_5026L, + /* -284 */ 0x5614_106C_B11D_FA14L, 0x5518_D17E_A7EF_7352L, + /* -283 */ 0x44DC_D9F0_8DB1_94DDL, 0x2A7A_4132_1FF2_C2A8L, + /* -282 */ 0x6E2E_2980_E2B5_BAFBL, 0x5D90_6850_331E_043FL, + /* -281 */ 0x5824_EE00_B55E_2F2FL, 0x6473_86A6_8F4B_3699L, + /* -280 */ 0x4683_F19A_2AB1_BF59L, 0x36C2_D21E_D908_F87BL, + /* -279 */ 0x70D3_1C29_DDE9_3228L, 0x579E_1CFE_280E_5A5DL, + /* -278 */ 0x5A42_7CEE_4B20_F4EDL, 0x2C7E_7D98_200B_7B7EL, + /* -277 */ 0x4835_30BE_A280_C3F1L, 0x09FE_CAE0_19A2_C932L, + /* -276 */ 0x7388_4DFD_D0CE_064EL, 0x4331_4499_C29E_0EB6L, + /* -275 */ 0x5C6D_0B31_73D8_050BL, 0x4F5A_9D47_CEE4_D891L, + /* -274 */ 0x49F0_D5C1_2979_9DA2L, 0x72AE_E439_7250_AD41L, + /* -273 */ 0x764E_22CE_A8C2_95D1L, 0x377E_39F5_83B4_4868L, + /* -272 */ 0x5EA4_E8A5_53CE_DE41L, 0x12CB_6191_3629_D387L, + /* -271 */ 0x4BB7_2084_430B_E500L, 0x756F_8140_F821_7605L, + /* -270 */ 0x7925_00D3_9E79_6E67L, 0x6F18_CECE_59CF_233CL, + /* -269 */ 0x60EA_670F_B1FA_BEB9L, 0x3F47_0BD8_47D8_E8FDL, + /* -268 */ 0x4D88_5272_F4C8_9894L, 0x329F_3CAD_0647_20CAL, + /* -267 */ 0x7C0D_50B7_EE0D_C0EDL, 0x3765_2DE1_A3A5_0143L, + /* -266 */ 0x633D_DA2C_BE71_6724L, 0x2C50_F181_4FB7_3436L, + /* -265 */ 0x4F64_AE8A_31F4_5283L, 0x3D0D_8E01_0C92_902BL, + /* -264 */ 0x7F07_7DA9_E986_EA6BL, 0x7B48_E334_E0EA_8045L, + /* -263 */ 0x659F_97BB_2138_BB89L, 0x4907_1C2A_4D88_669DL, + /* -262 */ 0x514C_7962_80FA_2FA1L, 0x20D2_7CEE_A46D_1EE4L, + /* -261 */ 0x4109_FAB5_33FB_594DL, 0x670E_CA58_838A_7F1DL, + /* -260 */ 0x680F_F788_532B_C216L, 0x0B4A_DD5A_6C10_CB62L, + /* -259 */ 0x533F_F939_DC23_01ABL, 0x22A2_4AAE_BCDA_3C4EL, + /* -258 */ 0x4299_942E_49B5_9AEFL, 0x354E_A225_63E1_C9D8L, + /* -257 */ 0x6A8F_537D_42BC_2B18L, 0x554A_9D08_9FCF_A95AL, + /* -256 */ 0x553F_75FD_CEFC_EF46L, 0x776E_E406_E63F_BAAEL, + /* -255 */ 0x4432_C4CB_0BFD_8C38L, 0x5F8B_E99F_1E99_6225L, + /* -254 */ 0x6D1E_07AB_4662_79F4L, 0x3279_75CB_6428_9D08L, + /* -253 */ 0x574B_3955_D1E8_6190L, 0x2861_2B09_1CED_4A6DL, + /* -252 */ 0x45D5_C777_DB20_4E0DL, 0x06B4_226D_B0BD_D524L, + /* -251 */ 0x6FBC_7259_5E9A_167BL, 0x2453_6A49_1AC9_5506L, + /* -250 */ 0x5963_8EAD_E548_11FCL, 0x1D0F_883A_7BD4_4405L, + /* -249 */ 0x4782_D88B_1DD3_4196L, 0x4A72_D361_FCA9_D004L, + /* -248 */ 0x726A_F411_C952_028AL, 0x43EA_EBCF_FAA9_4CD3L, + /* -247 */ 0x5B88_C341_6DDB_353BL, 0x4FEF_230C_C887_70A9L, + /* -246 */ 0x493A_35CD_F17C_2A96L, 0x0CBF_4F3D_6D39_26EEL, + /* -245 */ 0x7529_EFAF_E8C6_AA89L, 0x6132_1862_485B_717CL, + /* -244 */ 0x5DBB_2626_53D2_2207L, 0x675B_46B5_06AF_8DFDL, + /* -243 */ 0x4AFC_1E85_0FDB_4E6CL, 0x52AF_6BC4_0559_3E64L, + /* -242 */ 0x77F9_CA6E_7FC5_4A47L, 0x377F_12D3_3BC1_FD6DL, + /* -241 */ 0x5FFB_0858_6637_6E9FL, 0x45FF_4242_9634_CABDL, + /* -240 */ 0x4CC8_D379_EB5F_8BB2L, 0x6B32_9B68_782A_3BCBL, + /* -239 */ 0x7ADA_EBF6_4565_AC51L, 0x2B84_2BDA_59DD_2C77L, + /* -238 */ 0x6248_BCC5_0451_56A7L, 0x3C69_BCAE_AE4A_89F9L, + /* -237 */ 0x4EA0_9704_0374_4552L, 0x6387_CA25_583B_A194L, + /* -236 */ 0x7DCD_BE6C_D253_A21EL, 0x05A6_103B_C05F_68EDL, + /* -235 */ 0x64A4_9857_0EA9_4E7EL, 0x37B8_0CFC_99E5_ED8AL, + /* -234 */ 0x5083_AD12_7221_0B98L, 0x2C93_3D96_E184_BE08L, + /* -233 */ 0x4069_5741_F4E7_3C79L, 0x7075_CADF_1AD0_9807L, + /* -232 */ 0x670E_F203_2171_FA5CL, 0x4D89_4498_2AE7_59A4L, + /* -231 */ 0x5272_5B35_B45B_2EB0L, 0x3E07_6A13_5585_E150L, + /* -230 */ 0x41F5_15C4_9048_F226L, 0x64D2_BB42_AAD1_810DL, + /* -229 */ 0x6988_22D4_1A0E_503EL, 0x07B7_9204_4482_6815L, + /* -228 */ 0x546C_E8A9_AE71_D9CBL, 0x1FC6_0E69_D068_5344L, + /* -227 */ 0x438A_53BA_F1F4_AE3CL, 0x196B_3EBB_0D20_429DL, + /* -226 */ 0x6C10_85F7_E987_7D2DL, 0x0F11_FDF8_1500_6A94L, + /* -225 */ 0x5673_9E5F_EE05_FDBDL, 0x58DB_3193_4400_5543L, + /* -224 */ 0x4529_4B7F_F19E_6497L, 0x60AF_5ADC_3666_AA9CL, + /* -223 */ 0x6EA8_78CC_B5CA_3A8CL, 0x344B_C493_8A3D_DDC7L, + /* -222 */ 0x5886_C70A_2B08_2ED6L, 0x5D09_6A0F_A1CB_17D2L, + /* -221 */ 0x46D2_38D4_EF39_BF12L, 0x173A_BB3F_B4A2_7975L, + /* -220 */ 0x7150_5AEE_4B8F_981DL, 0x0B91_2B99_2103_F588L, + /* -219 */ 0x5AA6_AF25_093F_ACE4L, 0x0940_EFAD_B403_2AD3L, + /* -218 */ 0x4885_58EA_6DCC_8A50L, 0x0767_2624_9002_88A9L, + /* -217 */ 0x7408_8E43_E2E0_DD4CL, 0x723E_A36D_B337_410EL, + /* -216 */ 0x5CD3_A503_1BE7_1770L, 0x5B65_4F8A_F5C5_CDA5L, + /* -215 */ 0x4A42_EA68_E31F_45F3L, 0x62B7_72D5_916B_0AEBL, + /* -214 */ 0x76D1_770E_3832_0986L, 0x0458_B7BC_1BDE_77DDL, + /* -213 */ 0x5F0D_F8D8_2CF4_D46BL, 0x1D13_C630_164B_9318L, + /* -212 */ 0x4C0B_2D79_BD90_A9EFL, 0x30DC_9E8C_DEA2_DC13L, + /* -211 */ 0x79AB_7BF5_FC1A_A97FL, 0x0160_FDAE_3104_9351L, + /* -210 */ 0x6155_FCC4_C9AE_EDFFL, 0x1AB3_FE24_F403_A90EL, + /* -209 */ 0x4DDE_63D0_A158_BE65L, 0x6229_981D_9002_EDA5L, + /* -208 */ 0x7C97_061A_9BC1_30A2L, 0x69DC_2695_B337_E2A1L, + /* -207 */ 0x63AC_04E2_1634_26E8L, 0x54B0_1EDE_28F9_821BL, + /* -206 */ 0x4FBC_D0B4_DE90_1F20L, 0x43C0_18B1_BA61_34E2L, + /* -205 */ 0x7F94_8121_6419_CB67L, 0x1F99_C11C_5D68_549DL, + /* -204 */ 0x6610_674D_E9AE_3C52L, 0x4C7B_00E3_7DED_107EL, + /* -203 */ 0x51A6_B90B_2158_3042L, 0x09FC_00B5_FE57_4065L, + /* -202 */ 0x4152_2DA2_8113_59CEL, 0x3B30_0091_9845_CD1DL, + /* -201 */ 0x6883_7C37_34EB_C2E3L, 0x784C_CDB5_C06F_AE95L, + /* -200 */ 0x539C_635F_5D89_68B6L, 0x2D0A_3E2B_0059_5877L, + /* -199 */ 0x42E3_82B2_B13A_BA2BL, 0x3DA1_CB55_99E1_1393L, + /* -198 */ 0x6B05_9DEA_B52A_C378L, 0x629C_7888_F634_EC1EL, + /* -197 */ 0x559E_17EE_F755_692DL, 0x3549_FA07_2B5D_89B1L, + /* -196 */ 0x447E_798B_F911_20F1L, 0x1107_FB38_EF7E_07C1L, + /* -195 */ 0x6D97_28DF_F4E8_34B5L, 0x01A6_5EC1_7F30_0C68L, + /* -194 */ 0x57AC_20B3_2A53_5D5DL, 0x4E1E_B234_65C0_09EDL, + /* -193 */ 0x4623_4D5C_21DC_4AB1L, 0x24E5_5B5D_1E33_3B24L, + /* -192 */ 0x7038_7BC6_9C93_AAB5L, 0x216E_F894_FD1E_C506L, + /* -191 */ 0x59C6_C96B_B076_222AL, 0x4DF2_6077_30E5_6A6CL, + /* -190 */ 0x47D2_3ABC_8D2B_4E88L, 0x3E5B_805F_5A51_21F0L, + /* -189 */ 0x72E9_F794_1512_1740L, 0x63C5_9A32_2A1B_697FL, + /* -188 */ 0x5BEE_5FA9_AA74_DF67L, 0x0304_7B5B_54E2_BACCL, + /* -187 */ 0x498B_7FBA_EEC3_E5ECL, 0x0269_FC49_10B5_623DL, + /* -186 */ 0x75AB_FF91_7E06_3CACL, 0x6A43_2D41_B455_69FBL, + /* -185 */ 0x5E23_32DA_CB38_308AL, 0x21CF_5767_C377_87FCL, + /* -184 */ 0x4B4F_5BE2_3C2C_F3A1L, 0x67D9_12B9_692C_6CCAL, + /* -183 */ 0x787E_F969_F9E1_85CFL, 0x595B_5128_A847_1476L, + /* -182 */ 0x6065_9454_C7E7_9E3FL, 0x6115_DA86_ED05_A9F8L, + /* -181 */ 0x4D1E_1043_D31F_B1CCL, 0x4DAB_1538_BD9E_2193L, + /* -180 */ 0x7B63_4D39_51CC_4FADL, 0x62AB_5527_95C9_CF52L, + /* -179 */ 0x62B5_D761_0E3D_0C8BL, 0x0222_AA86_116E_3F75L, + /* -178 */ 0x4EF7_DF80_D830_D6D5L, 0x4E82_2204_DABE_992AL, + /* -177 */ 0x7E59_659A_F381_57BCL, 0x1736_9CD4_9130_F510L, + /* -176 */ 0x6514_5148_C2CD_DFC9L, 0x5F5E_E3DD_40F3_F740L, + /* -175 */ 0x50DD_0DD3_CF0B_196EL, 0x1918_B64A_9A5C_C5CDL, + /* -174 */ 0x40B0_D7DC_A5A2_7ABEL, 0x4746_F83B_AEB0_9E3EL, + /* -173 */ 0x6781_5961_0903_F797L, 0x253E_59F9_1780_FD2FL, + /* -172 */ 0x52CD_E11A_6D9C_C612L, 0x50FE_AE60_DF9A_6426L, + /* -171 */ 0x423E_4DAE_BE17_04DBL, 0x5A65_584D_7FAE_B685L, + /* -170 */ 0x69FD_4917_968B_3AF9L, 0x10A2_26E2_65E4_573BL, + /* -169 */ 0x54CA_A0DF_ABA2_9594L, 0x0D4E_8581_EB1D_1295L, + /* -168 */ 0x43D5_4D7F_BC82_1143L, 0x243E_D134_BC17_4211L, + /* -167 */ 0x6C88_7BFF_9403_4ED2L, 0x06CA_E854_6025_3682L, + /* -166 */ 0x56D3_9666_1002_A574L, 0x6BD5_86A9_E684_2B9BL, + /* -165 */ 0x4576_11EB_4002_1DF7L, 0x0977_9EEE_5203_5616L, + /* -164 */ 0x6F23_4FDE_CCD0_2FF1L, 0x5BF2_97E3_B66B_BCEFL, + /* -163 */ 0x58E9_0CB2_3D73_598EL, 0x165B_ACB6_2B89_63F3L, + /* -162 */ 0x4720_D6F4_FDF5_E13EL, 0x4516_23C4_EFA1_1CC2L, + /* -161 */ 0x71CE_24BB_2FEF_CECAL, 0x3B56_9FA1_7F68_2E03L, + /* -160 */ 0x5B0B_5095_BFF3_0BD5L, 0x15DE_E61A_CC53_5803L, + /* -159 */ 0x48D5_DA11_665C_0977L, 0x2B18_B815_7042_ACCFL, + /* -158 */ 0x7489_5CE8_A3C6_758BL, 0x5E8D_F355_806A_AE18L, + /* -157 */ 0x5D3A_B0BA_1C9E_C46FL, 0x653E_5C44_66BB_BE7AL, + /* -156 */ 0x4A95_5A2E_7D4B_D059L, 0x3765_169D_1EFC_9861L, + /* -155 */ 0x7755_5D17_2EDF_B3C2L, 0x256E_8A94_FE60_F3CFL, + /* -154 */ 0x5F77_7DAC_257F_C301L, 0x6ABE_D543_FEB3_F63FL, + /* -153 */ 0x4C5F_97BC_EACC_9C01L, 0x3BCB_DDCF_FEF6_5E99L, + /* -152 */ 0x7A32_8C61_77AD_C668L, 0x5FAC_9619_97F0_975BL, + /* -151 */ 0x61C2_09E7_92F1_6B86L, 0x7FBD_44E1_465A_12AFL, + /* -150 */ 0x4E34_D4B9_425A_BC6BL, 0x7FCA_9D81_0514_DBBFL, + /* -149 */ 0x7D21_545B_9D5D_FA46L, 0x32DD_C8CE_6E87_C5FFL, + /* -148 */ 0x641A_A9E2_E44B_2E9EL, 0x5BE4_A0A5_2539_6B32L, + /* -147 */ 0x5015_54B5_836F_587EL, 0x7CB6_E6EA_842D_EF5CL, + /* -146 */ 0x4011_1091_35F2_AD32L, 0x3092_5255_368B_25E3L, + /* -145 */ 0x6681_B41B_8984_4850L, 0x4DB6_EA21_F0DE_A304L, + /* -144 */ 0x5201_5CE2_D469_D373L, 0x57C5_881B_2718_826AL, + /* -143 */ 0x419A_B0B5_76BB_0F8FL, 0x5FD1_39AF_527A_01EFL, + /* -142 */ 0x68F7_8122_5791_B27FL, 0x4C81_F5E5_50C3_364AL, + /* -141 */ 0x53F9_341B_7941_5B99L, 0x239B_2B1D_DA35_C508L, + /* -140 */ 0x432D_C349_2DCD_E2E1L, 0x02E2_88E4_AE91_6A6DL, + /* -139 */ 0x6B7C_6BA8_4949_6B01L, 0x516A_74A1_174F_10AEL, + /* -138 */ 0x55FD_22ED_076D_EF34L, 0x4121_F6E7_45D8_DA25L, + /* -137 */ 0x44CA_8257_3924_BF5DL, 0x1A81_9252_9E47_14EBL, + /* -136 */ 0x6E10_D08B_8EA1_322EL, 0x5D9C_1D50_FD3E_87DDL, + /* -135 */ 0x580D_73A2_D880_F4F2L, 0x17B0_1773_FDCB_9FE4L, + /* -134 */ 0x4671_294F_139A_5D8EL, 0x4626_7929_97D6_1984L, + /* -133 */ 0x70B5_0EE4_EC2A_2F4AL, 0x3D0A_5B75_BFBC_F59FL, + /* -132 */ 0x5A2A_7250_BCEE_8C3BL, 0x4A6E_AF91_6630_C47FL, + /* -131 */ 0x4821_F50D_63F2_09C9L, 0x21F2_260D_EB5A_36CCL, + /* -130 */ 0x7369_8815_6CB6_760EL, 0x6983_7016_455D_247AL, + /* -129 */ 0x5C54_6CDD_F091_F80BL, 0x6E02_C011_D117_5062L, + /* -128 */ 0x49DD_23E4_C074_C66FL, 0x719B_CCDB_0DAC_404EL, + /* -127 */ 0x762E_9FD4_6721_3D7FL, 0x68F9_47C4_E2AD_33B0L, + /* -126 */ 0x5E8B_B310_5280_FDFFL, 0x6D94_396A_4EF0_F627L, + /* -125 */ 0x4BA2_F5A6_A867_3199L, 0x3E10_2DEE_A58D_91B9L, + /* -124 */ 0x7904_BC3D_DA3E_B5C2L, 0x3019_E317_6F48_E927L, + /* -123 */ 0x60D0_9697_E1CB_C49BL, 0x4014_B5AC_5907_20ECL, + /* -122 */ 0x4D73_ABAC_B4A3_03AFL, 0x4CDD_5E23_7A6C_1A57L, + /* -121 */ 0x7BEC_45E1_2104_D2B2L, 0x47C8_969F_2A46_908AL, + /* -120 */ 0x6323_6B1A_80D0_A88EL, 0x6CA0_787F_5505_406FL, + /* -119 */ 0x4F4F_88E2_00A6_ED3FL, 0x0A19_F9FF_7737_66BFL, + /* -118 */ 0x7EE5_A7D0_010B_1531L, 0x5CF6_5CCB_F1F2_3DFEL, + /* -117 */ 0x6584_8640_00D5_AA8EL, 0x172B_7D6F_F4C1_CB32L, + /* -116 */ 0x5136_D1CC_CD77_BBA4L, 0x78EF_978C_C3CE_3C28L, + /* -115 */ 0x40F8_A7D7_0AC6_2FB7L, 0x13F2_DFA3_CFD8_3020L, + /* -114 */ 0x67F4_3FBE_77A3_7F8BL, 0x3984_9906_1959_E699L, + /* -113 */ 0x5329_CC98_5FB5_FFA2L, 0x6136_E0D1_ADE1_8548L, + /* -112 */ 0x4287_D6E0_4C91_994FL, 0x00F8_B3DA_F181_376DL, + /* -111 */ 0x6A72_F166_E0E8_F54BL, 0x1B27_862B_1C01_F247L, + /* -110 */ 0x5528_C11F_1A53_F76FL, 0x2F52_D1BC_1667_F506L, + /* -109 */ 0x4420_9A7F_4843_2C59L, 0x0C42_4163_451F_F738L, + /* -108 */ 0x6D00_F732_0D38_46F4L, 0x7A03_9BD2_0833_2526L, + /* -107 */ 0x5733_F8F4_D760_38C3L, 0x7B36_1641_A028_EA85L, + /* -106 */ 0x45C3_2D90_AC4C_FA36L, 0x2F5E_7834_8020_BB9EL, + /* -105 */ 0x6F9E_AF4D_E07B_29F0L, 0x4BCA_59ED_99CD_F8FCL, + /* -104 */ 0x594B_BF71_8062_87F3L, 0x563B_7B24_7B0B_2D96L, + /* -103 */ 0x476F_CC5A_CD1B_9FF6L, 0x11C9_2F50_626F_57ACL, + /* -102 */ 0x724C_7A2A_E1C5_CCBDL, 0x02DB_7EE7_03E5_5912L, + /* -101 */ 0x5B70_61BB_E7D1_7097L, 0x1BE2_CBEC_031D_E0DCL, + /* -100 */ 0x4926_B496_530D_F3ACL, 0x164F_0989_9C17_E716L, + /* -99 */ 0x750A_BA8A_1E7C_B913L, 0x3D4B_4275_C68C_A4F0L, + /* -98 */ 0x5DA2_2ED4_E530_940FL, 0x4AA2_9B91_6BA3_B726L, + /* -97 */ 0x4AE8_2577_1DC0_7672L, 0x6EE8_7C74_561C_9285L, + /* -96 */ 0x77D9_D58B_62CD_8A51L, 0x3173_FA53_BCFA_8408L, + /* -95 */ 0x5FE1_77A2_B571_3B74L, 0x278F_FB76_30C8_69A0L, + /* -94 */ 0x4CB4_5FB5_5DF4_2F90L, 0x1FA6_62C4_F3D3_87B3L, + /* -93 */ 0x7ABA_32BB_C986_B280L, 0x32A3_D13B_1FB8_D91FL, + /* -92 */ 0x622E_8EFC_A138_8ECDL, 0x0EE9_742F_4C93_E0E6L, + /* -91 */ 0x4E8B_A596_E760_723DL, 0x58BA_C359_0A0F_E71EL, + /* -90 */ 0x7DAC_3C24_A567_1D2FL, 0x412A_D228_1019_71C9L, + /* -89 */ 0x6489_C9B6_EAB8_E426L, 0x00EF_0E86_7347_8E3BL, + /* -88 */ 0x506E_3AF8_BBC7_1CEBL, 0x1A58_D86B_8F6C_71C9L, + /* -87 */ 0x4058_2F2D_6305_B0BCL, 0x1513_E056_0C56_C16EL, + /* -86 */ 0x66F3_7EAF_04D5_E793L, 0x3B53_0089_AD57_9BE2L, + /* -85 */ 0x525C_6558_D0AB_1FA9L, 0x15DC_006E_2446_164FL, + /* -84 */ 0x41E3_8447_0D55_B2EDL, 0x5E49_99F1_B69E_783FL, + /* -83 */ 0x696C_06D8_1555_EB15L, 0x7D42_8FE9_2430_C065L, + /* -82 */ 0x5456_6BE0_1111_88DEL, 0x3102_0CBA_835A_3384L, + /* -81 */ 0x4378_564C_DA74_6D7EL, 0x5A68_0A2E_CF7B_5C69L, + /* -80 */ 0x6BF3_BD47_C3ED_7BFDL, 0x770C_DD17_B25E_FA42L, + /* -79 */ 0x565C_976C_9CBD_FCCBL, 0x1270_B0DF_C1E5_9502L, + /* -78 */ 0x4516_DF8A_16FE_63D5L, 0x5B8D_5A4C_9B1E_10CEL, + /* -77 */ 0x6E8A_FF43_57FD_6C89L, 0x127B_C3AD_C4FC_E7B0L, + /* -76 */ 0x586F_329C_4664_56D4L, 0x0EC9_6957_D0CA_52F3L, + /* -75 */ 0x46BF_5BB0_3850_4576L, 0x3F07_8779_73D5_0F29L, + /* -74 */ 0x7132_2C4D_26E6_D58AL, 0x31A5_A58F_1FBB_4B75L, + /* -73 */ 0x5A8E_89D7_5252_446EL, 0x5AEA_EAD8_E62F_6F91L, + /* -72 */ 0x4872_07DF_750E_9D25L, 0x2F22_557A_51BF_8C74L, + /* -71 */ 0x73E9_A632_54E4_2EA2L, 0x1836_EF2A_1C65_AD86L, + /* -70 */ 0x5CBA_EB5B_771C_F21BL, 0x2CF8_BF54_E384_8AD2L, + /* -69 */ 0x4A2F_22AF_927D_8E7CL, 0x23FA_32AA_4F9D_3BDBL, + /* -68 */ 0x76B1_D118_EA62_7D93L, 0x5329_EAAA_18FB_92F8L, + /* -67 */ 0x5EF4_A747_21E8_6476L, 0x0F54_BBBB_472F_A8C6L, + /* -66 */ 0x4BF6_EC38_E7ED_1D2BL, 0x25DD_62FC_38F2_ED6CL, + /* -65 */ 0x798B_138E_3FE1_C845L, 0x22FB_D193_8E51_7BDFL, + /* -64 */ 0x613C_0FA4_FFE7_D36AL, 0x4F2F_DADC_71DA_C97FL, + /* -63 */ 0x4DC9_A61D_9986_42BBL, 0x58F3_157D_27E2_3ACCL, + /* -62 */ 0x7C75_D695_C270_6AC5L, 0x74B8_2261_D969_F7ADL, + /* -61 */ 0x6391_7877_CEC0_556BL, 0x1093_4EB4_ADEE_5FBEL, + /* -60 */ 0x4FA7_9393_0BCD_1122L, 0x4075_D890_8B25_1965L, + /* -59 */ 0x7F72_85B8_12E1_B504L, 0x00BC_8DB4_11D4_F56EL, + /* -58 */ 0x65F5_37C6_7581_5D9CL, 0x66FD_3E29_A7DD_9125L, + /* -57 */ 0x5190_F96B_9134_4AE3L, 0x6BFD_CB54_864A_DA84L, + /* -56 */ 0x4140_C789_40F6_A24FL, 0x6FFE_3C43_9EA2_486AL, + /* -55 */ 0x6867_A5A8_67F1_03B2L, 0x7FFD_2D38_FDD0_73DCL, + /* -54 */ 0x5386_1E20_5327_3628L, 0x6664_242D_97D9_F64AL, + /* -53 */ 0x42D1_B1B3_75B8_F820L, 0x51E9_B68A_DFE1_91D5L, + /* -52 */ 0x6AE9_1C52_55F4_C034L, 0x1CA9_2411_6635_B621L, + /* -51 */ 0x5587_49DB_77F7_0029L, 0x63BA_8341_1E91_5E81L, + /* -50 */ 0x446C_3B15_F992_6687L, 0x6962_029A_7EDA_B201L, + /* -49 */ 0x6D79_F823_28EA_3DA6L, 0x0F03_375D_97C4_5001L, + /* -48 */ 0x5794_C682_8721_CAEBL, 0x259C_2C4A_DFD0_4001L, + /* -47 */ 0x4610_9ECE_D281_6F22L, 0x5149_BD08_B30D_0001L, + /* -46 */ 0x701A_97B1_50CF_1837L, 0x3542_C80D_EB48_0001L, + /* -45 */ 0x59AE_DFC1_0D72_79C5L, 0x7768_A00B_22A0_0001L, + /* -44 */ 0x47BF_1967_3DF5_2E37L, 0x7920_8008_E880_0001L, + /* -43 */ 0x72CB_5BD8_6321_E38CL, 0x5B67_3341_7400_0001L, + /* -42 */ 0x5BD5_E313_8281_82D6L, 0x7C52_8F67_9000_0001L, + /* -41 */ 0x4977_E8DC_6867_9BDFL, 0x16A8_72B9_4000_0001L, + /* -40 */ 0x758C_A7C7_0D72_92FEL, 0x5773_EAC2_0000_0001L, + /* -39 */ 0x5E0A_1FD2_7128_7598L, 0x45F6_5568_0000_0001L, + /* -38 */ 0x4B3B_4CA8_5A86_C47AL, 0x04C5_1120_0000_0001L, + /* -37 */ 0x785E_E10D_5DA4_6D90L, 0x07A1_B500_0000_0001L, + /* -36 */ 0x604B_E73D_E483_8AD9L, 0x52E7_C400_0000_0001L, + /* -35 */ 0x4D09_85CB_1D36_08AEL, 0x0F1F_D000_0000_0001L, + /* -34 */ 0x7B42_6FAB_61F0_0DE3L, 0x31CC_8000_0000_0001L, + /* -33 */ 0x629B_8C89_1B26_7182L, 0x5B0A_0000_0000_0001L, + /* -32 */ 0x4EE2_D6D4_15B8_5ACEL, 0x7C08_0000_0000_0001L, + /* -31 */ 0x7E37_BE20_22C0_914BL, 0x1340_0000_0000_0001L, + /* -30 */ 0x64F9_64E6_8233_A76FL, 0x2900_0000_0000_0001L, + /* -29 */ 0x50C7_83EB_9B5C_85F2L, 0x5400_0000_0000_0001L, + /* -28 */ 0x409F_9CBC_7C4A_04C2L, 0x1000_0000_0000_0001L, + /* -27 */ 0x6765_C793_FA10_079DL, 0x0000_0000_0000_0001L, + /* -26 */ 0x52B7_D2DC_C80C_D2E4L, 0x0000_0000_0000_0001L, + /* -25 */ 0x422C_A8B0_A00A_4250L, 0x0000_0000_0000_0001L, + /* -24 */ 0x69E1_0DE7_6676_D080L, 0x0000_0000_0000_0001L, + /* -23 */ 0x54B4_0B1F_852B_DA00L, 0x0000_0000_0000_0001L, + /* -22 */ 0x43C3_3C19_3756_4800L, 0x0000_0000_0000_0001L, + /* -21 */ 0x6C6B_935B_8BBD_4000L, 0x0000_0000_0000_0001L, + /* -20 */ 0x56BC_75E2_D631_0000L, 0x0000_0000_0000_0001L, + /* -19 */ 0x4563_9182_44F4_0000L, 0x0000_0000_0000_0001L, + /* -18 */ 0x6F05_B59D_3B20_0000L, 0x0000_0000_0000_0001L, + /* -17 */ 0x58D1_5E17_6280_0000L, 0x0000_0000_0000_0001L, + /* -16 */ 0x470D_E4DF_8200_0000L, 0x0000_0000_0000_0001L, + /* -15 */ 0x71AF_D498_D000_0000L, 0x0000_0000_0000_0001L, + /* -14 */ 0x5AF3_107A_4000_0000L, 0x0000_0000_0000_0001L, + /* -13 */ 0x48C2_7395_0000_0000L, 0x0000_0000_0000_0001L, + /* -12 */ 0x746A_5288_0000_0000L, 0x0000_0000_0000_0001L, + /* -11 */ 0x5D21_DBA0_0000_0000L, 0x0000_0000_0000_0001L, + /* -10 */ 0x4A81_7C80_0000_0000L, 0x0000_0000_0000_0001L, + /* -9 */ 0x7735_9400_0000_0000L, 0x0000_0000_0000_0001L, + /* -8 */ 0x5F5E_1000_0000_0000L, 0x0000_0000_0000_0001L, + /* -7 */ 0x4C4B_4000_0000_0000L, 0x0000_0000_0000_0001L, + /* -6 */ 0x7A12_0000_0000_0000L, 0x0000_0000_0000_0001L, + /* -5 */ 0x61A8_0000_0000_0000L, 0x0000_0000_0000_0001L, + /* -4 */ 0x4E20_0000_0000_0000L, 0x0000_0000_0000_0001L, + /* -3 */ 0x7D00_0000_0000_0000L, 0x0000_0000_0000_0001L, + /* -2 */ 0x6400_0000_0000_0000L, 0x0000_0000_0000_0001L, + /* -1 */ 0x5000_0000_0000_0000L, 0x0000_0000_0000_0001L, + /* 0 */ 0x4000_0000_0000_0000L, 0x0000_0000_0000_0001L, + /* 1 */ 0x6666_6666_6666_6666L, 0x3333_3333_3333_3334L, + /* 2 */ 0x51EB_851E_B851_EB85L, 0x0F5C_28F5_C28F_5C29L, + /* 3 */ 0x4189_374B_C6A7_EF9DL, 0x5916_872B_020C_49BBL, + /* 4 */ 0x68DB_8BAC_710C_B295L, 0x74F0_D844_D013_A92BL, + /* 5 */ 0x53E2_D623_8DA3_C211L, 0x43F3_E037_0CDC_8755L, + /* 6 */ 0x431B_DE82_D7B6_34DAL, 0x698F_E692_70B0_6C44L, + /* 7 */ 0x6B5F_CA6A_F2BD_215EL, 0x0F4C_A41D_811A_46D4L, + /* 8 */ 0x55E6_3B88_C230_E77EL, 0x3F70_834A_CDAE_9F10L, + /* 9 */ 0x44B8_2FA0_9B5A_52CBL, 0x4C5A_02A2_3E25_4C0DL, + /* 10 */ 0x6DF3_7F67_5EF6_EADFL, 0x2D5C_D103_96A2_1347L, + /* 11 */ 0x57F5_FF85_E592_557FL, 0x3DE3_DA69_454E_75D3L, + /* 12 */ 0x465E_6604_B7A8_4465L, 0x7E4F_E1ED_D10B_9175L, + /* 13 */ 0x7097_09A1_25DA_0709L, 0x4A19_697C_81AC_1BEFL, + /* 14 */ 0x5A12_6E1A_84AE_6C07L, 0x54E1_2130_67BC_E326L, + /* 15 */ 0x480E_BE7B_9D58_566CL, 0x43E7_4DC0_52FD_8285L, + /* 16 */ 0x734A_CA5F_6226_F0ADL, 0x530B_AF9A_1E62_6A6DL, + /* 17 */ 0x5C3B_D519_1B52_5A24L, 0x426F_BFAE_7EB5_21F1L, + /* 18 */ 0x49C9_7747_490E_AE83L, 0x4EBF_CC8B_9890_E7F4L, + /* 19 */ 0x760F_253E_DB4A_B0D2L, 0x4ACC_7A78_F41B_0CBAL, + /* 20 */ 0x5E72_8432_4908_8D75L, 0x223D_2EC7_29AF_3D62L, + /* 21 */ 0x4B8E_D028_3A6D_3DF7L, 0x34FD_BF05_BAF2_9781L, + /* 22 */ 0x78E4_8040_5D7B_9658L, 0x54C9_31A2_C4B7_58CFL, + /* 23 */ 0x60B6_CD00_4AC9_4513L, 0x5D6D_C14F_03C5_E0A5L, + /* 24 */ 0x4D5F_0A66_A23A_9DA9L, 0x3124_9AA5_9C9E_4D51L, + /* 25 */ 0x7BCB_43D7_69F7_62A8L, 0x4EA0_F76F_60FD_4882L, + /* 26 */ 0x6309_0312_BB2C_4EEDL, 0x254D_92BF_80CA_A068L, + /* 27 */ 0x4F3A_68DB_C8F0_3F24L, 0x1DD7_A899_33D5_4D20L, + /* 28 */ 0x7EC3_DAF9_4180_6506L, 0x62F2_A75B_8622_1500L, + /* 29 */ 0x6569_7BFA_9ACD_1D9FL, 0x025B_B916_04E8_10CDL, + /* 30 */ 0x5121_2FFB_AF0A_7E18L, 0x6849_60DE_6A53_40A4L, + /* 31 */ 0x40E7_5996_25A1_FE7AL, 0x203A_B3E5_21DC_33B6L, + /* 32 */ 0x67D8_8F56_A29C_CA5DL, 0x19F7_863B_6960_52BDL, + /* 33 */ 0x5313_A5DE_E87D_6EB0L, 0x7B2C_6B62_BAB3_7564L, + /* 34 */ 0x4276_1E4B_ED31_255AL, 0x2F56_BC4E_FBC2_C450L, + /* 35 */ 0x6A56_96DF_E1E8_3BC3L, 0x6557_93B1_92D1_3A1AL, + /* 36 */ 0x5512_124C_B4B9_C969L, 0x3779_42F4_7574_2E7BL, + /* 37 */ 0x440E_750A_2A2E_3ABAL, 0x5F94_3590_5DF6_8B96L, + /* 38 */ 0x6CE3_EE76_A9E3_912AL, 0x65B9_EF4D_6324_1289L, + /* 39 */ 0x571C_BEC5_54B6_0DBBL, 0x6AFB_25D7_8283_4207L, + /* 40 */ 0x45B0_989D_DD5E_7163L, 0x08C8_EB12_CECF_6806L, + /* 41 */ 0x6F80_F42F_C897_1BD1L, 0x5ADB_11B7_B14B_D9A3L, + /* 42 */ 0x5933_F68C_A078_E30EL, 0x157C_0E2C_8DD6_47B5L, + /* 43 */ 0x475C_C53D_4D2D_8271L, 0x5DFC_D823_A4AB_6C91L, + /* 44 */ 0x722E_0862_1515_9D82L, 0x632E_269F_6DDF_141BL, + /* 45 */ 0x5B58_06B4_DDAA_E468L, 0x4F58_1EE5_F17F_4349L, + /* 46 */ 0x4913_3890_B155_8386L, 0x72AC_E584_C132_9C3BL, + /* 47 */ 0x74EB_8DB4_4EEF_38D7L, 0x6AAE_3C07_9B84_2D2AL, + /* 48 */ 0x5D89_3E29_D8BF_60ACL, 0x5558_3006_1603_5755L, + /* 49 */ 0x4AD4_31BB_13CC_4D56L, 0x7779_C004_DE69_12ABL, + /* 50 */ 0x77B9_E92B_52E0_7BBEL, 0x258F_99A1_63DB_5111L, + /* 51 */ 0x5FC7_EDBC_424D_2FCBL, 0x37A6_1481_1CAF_740DL, + /* 52 */ 0x4C9F_F163_683D_BFD5L, 0x7951_AA00_E3BF_900BL, + /* 53 */ 0x7A99_8238_A6C9_32EFL, 0x754F_7667_D2CC_19ABL, + /* 54 */ 0x6214_682D_523A_8F26L, 0x2AA5_F853_0F09_AE22L, + /* 55 */ 0x4E76_B9BD_DB62_0C1EL, 0x5551_9375_A5A1_581BL, + /* 56 */ 0x7D8A_C2C9_5F03_4697L, 0x3BB5_B8BC_3C35_59C5L, + /* 57 */ 0x646F_023A_B269_0545L, 0x7C91_6096_9691_149EL, + /* 58 */ 0x5058_CE95_5B87_376BL, 0x16DA_B3AB_ABA7_43B2L, + /* 59 */ 0x4047_0BAA_AF9F_5F88L, 0x78AE_F622_EFB9_02F5L, + /* 60 */ 0x66D8_12AA_B298_98DBL, 0x0DE4_BD04_B2C1_9E54L, + /* 61 */ 0x5246_7555_5BAD_4715L, 0x57EA_30D0_8F01_4B76L, + /* 62 */ 0x41D1_F777_7C8A_9F44L, 0x4654_F3DA_0C01_092CL, + /* 63 */ 0x694F_F258_C744_3207L, 0x23BB_1FC3_4668_0EACL, + /* 64 */ 0x543F_F513_D29C_F4D2L, 0x4FC8_E635_D1EC_D88AL, + /* 65 */ 0x4366_5DA9_754A_5D75L, 0x263A_51C4_A7F0_AD3BL, + /* 66 */ 0x6BD6_FC42_5543_C8BBL, 0x56C3_B607_731A_AEC4L, + /* 67 */ 0x5645_969B_7769_6D62L, 0x789C_919F_8F48_8BD0L, + /* 68 */ 0x4504_787C_5F87_8AB5L, 0x46E3_A7B2_D906_D640L, + /* 69 */ 0x6E6D_8D93_CC0C_1122L, 0x3E39_0C51_5B3E_239AL, + /* 70 */ 0x5857_A476_3CD6_741BL, 0x4B60_D6A7_7C31_B615L, + /* 71 */ 0x46AC_8391_CA45_29AFL, 0x55E7_121F_968E_2B44L, + /* 72 */ 0x7114_05B6_106E_A919L, 0x0971_B698_F0E3_786DL, + /* 73 */ 0x5A76_6AF8_0D25_5414L, 0x078E_2BAD_8D82_C6BDL, + /* 74 */ 0x485E_BBF9_A41D_DCDCL, 0x6C71_BC8A_D79B_D231L, + /* 75 */ 0x73CA_C65C_39C9_6161L, 0x2D82_C744_8C2C_8382L, + /* 76 */ 0x5CA2_3849_C7D4_4DE7L, 0x3E02_3903_A356_CF9BL, + /* 77 */ 0x4A1B_603B_0643_7185L, 0x7E68_2D9C_82AB_D949L, + /* 78 */ 0x7692_3391_A39F_1C09L, 0x4A40_48FA_6AAC_8EDBL, + /* 79 */ 0x5EDB_5C74_82E5_B007L, 0x5500_3A61_EEF0_7249L, + /* 80 */ 0x4BE2_B05D_3584_8CD2L, 0x7733_61E7_F259_F507L, + /* 81 */ 0x796A_B3C8_55A0_E151L, 0x3EB8_9CA6_508F_EE71L, + /* 82 */ 0x6122_296D_114D_810DL, 0x7EFA_16EB_73A6_585BL, + /* 83 */ 0x4DB4_EDF0_DAA4_673EL, 0x3261_ABEF_8FB8_46AFL, + /* 84 */ 0x7C54_AFE7_C43A_3ECAL, 0x1D69_1318_E5F3_A44BL, + /* 85 */ 0x6376_F31F_D02E_98A1L, 0x6454_0F47_1E5C_836FL, + /* 86 */ 0x4F92_5C19_7358_7A1BL, 0x0376_729F_4B7D_35F3L, + /* 87 */ 0x7F50_935B_EBC0_C35EL, 0x38BD_8432_1261_EFEBL, + /* 88 */ 0x65DA_0F7C_BC9A_35E5L, 0x13CA_D028_0EB4_BFEFL, + /* 89 */ 0x517B_3F96_FD48_2B1DL, 0x5CA2_4020_0BC3_CCBFL, + /* 90 */ 0x412F_6612_6439_BC17L, 0x63B5_0019_A303_0A33L, + /* 91 */ 0x684B_D683_D38F_9359L, 0x1F88_0029_04D1_A9EAL, + /* 92 */ 0x536F_DECF_DC72_DC47L, 0x32D3_3354_03DA_EE55L, + /* 93 */ 0x42BF_E573_16C2_49D2L, 0x5BDC_2910_0315_8B77L, + /* 94 */ 0x6ACC_A251_BE03_A951L, 0x12F9_DB4C_D1BC_1258L, + /* 95 */ 0x5570_81DA_FE69_5440L, 0x7594_AF70_A7C9_A847L, + /* 96 */ 0x445A_017B_FEBA_A9CDL, 0x4476_F2C0_863A_ED06L, + /* 97 */ 0x6D5C_CF2C_CAC4_42E2L, 0x3A57_EACD_A391_7B3CL, + /* 98 */ 0x577D_728A_3BD0_3581L, 0x7B79_88A4_82DA_C8FDL, + /* 99 */ 0x45FD_F53B_630C_F79BL, 0x15FA_D3B6_CF15_6D97L, + /* 100 */ 0x6FFC_BB92_3814_BF5EL, 0x565E_1F8A_E4EF_15BEL, + /* 101 */ 0x5996_FC74_F9AA_32B2L, 0x11E4_E608_B725_AAFFL, + /* 102 */ 0x47AB_FD2A_6154_F55BL, 0x27EA_51A0_9284_88CCL, + /* 103 */ 0x72AC_C843_CEEE_555EL, 0x7310_829A_8407_4146L, + /* 104 */ 0x5BBD_6D03_0BF1_DDE5L, 0x4273_9BAE_D005_CDD2L, + /* 105 */ 0x4964_5735_A327_E4B7L, 0x4EC2_E2F2_4004_A4A8L, + /* 106 */ 0x756D_5855_D1D9_6DF2L, 0x4AD1_6B1D_333A_A10CL, + /* 107 */ 0x5DF1_1377_DB14_57F5L, 0x2241_227D_C295_4DA3L, + /* 108 */ 0x4B27_42C6_48DD_132AL, 0x4E9A_81FE_3544_3E1CL, + /* 109 */ 0x783E_D13D_4161_B844L, 0x175D_9CC9_EED3_9694L, + /* 110 */ 0x6032_40FD_CDE7_C69CL, 0x7917_B0A1_8BDC_7876L, + /* 111 */ 0x4CF5_00CB_0B1F_D217L, 0x1412_F3B4_6FE3_9392L, + /* 112 */ 0x7B21_9ADE_7832_E9BEL, 0x5351_85ED_7FD2_85B6L, + /* 113 */ 0x6281_48B1_F9C2_5498L, 0x42A7_9E57_9975_37C5L, + /* 114 */ 0x4ECD_D3C1_949B_76E0L, 0x3552_E512_E12A_9304L, + /* 115 */ 0x7E16_1F9C_20F8_BE33L, 0x6EEB_081E_3510_EB39L, + /* 116 */ 0x64DE_7FB0_1A60_9829L, 0x3F22_6CE4_F740_BC2EL, + /* 117 */ 0x50B1_FFC0_151A_1354L, 0x3281_F0B7_2C33_C9BEL, + /* 118 */ 0x408E_6633_4414_DC43L, 0x4201_8D5F_568F_D498L, + /* 119 */ 0x674A_3D1E_D354_939FL, 0x1CCF_4898_8A7F_BA8DL, + /* 120 */ 0x52A1_CA7F_0F76_DC7FL, 0x30A5_D3AD_3B99_620BL, + /* 121 */ 0x421B_0865_A5F8_B065L, 0x73B7_DC8A_9614_4E6FL, + /* 122 */ 0x69C4_DA3C_3CC1_1A3CL, 0x52BF_C744_2353_B0B1L, + /* 123 */ 0x549D_7B63_63CD_AE96L, 0x7566_3903_4F76_26F4L, + /* 124 */ 0x43B1_2F82_B63E_2545L, 0x4451_C735_D92B_525DL, + /* 125 */ 0x6C4E_B26A_BD30_3BA2L, 0x3A1C_71EF_C1DE_EA2EL, + /* 126 */ 0x56A5_5B88_9759_C94EL, 0x61B0_5B26_34B2_54F2L, + /* 127 */ 0x4551_1606_DF7B_0772L, 0x1AF3_7C1E_908E_AA5BL, + /* 128 */ 0x6EE8_233E_325E_7250L, 0x2B1F_2CFD_B417_76F8L, + /* 129 */ 0x58B9_B5CB_5B7E_C1D9L, 0x6F4C_23FE_29AC_5F2DL, + /* 130 */ 0x46FA_F7D5_E2CB_CE47L, 0x72A3_4FFE_87BD_18F1L, + /* 131 */ 0x7191_8C89_6ADF_B073L, 0x0438_7FFD_A5FB_5B1BL, + /* 132 */ 0x5ADA_D6D4_557F_C05CL, 0x0360_6664_84C9_15AFL, + /* 133 */ 0x48AF_1243_7799_66B0L, 0x02B3_851D_3707_448CL, + /* 134 */ 0x744B_506B_F28F_0AB3L, 0x1DEC_082E_BE72_0746L, + /* 135 */ 0x5D09_0D23_2872_6EF5L, 0x64BC_D358_985B_3905L, + /* 136 */ 0x4A6D_A41C_205B_8BF7L, 0x6A30_A913_AD15_C738L, + /* 137 */ 0x7715_D360_33C5_ACBFL, 0x5D1A_A81F_7B56_0B8CL, + /* 138 */ 0x5F44_A919_C304_8A32L, 0x7DAE_ECE5_FC44_D609L, + /* 139 */ 0x4C36_EDAE_359D_3B5BL, 0x7E25_8A51_969D_7808L, + /* 140 */ 0x79F1_7C49_EF61_F893L, 0x16A2_76E8_F0FB_F33FL, + /* 141 */ 0x618D_FD07_F2B4_C6DCL, 0x121B_9253_F3FC_C299L, + /* 142 */ 0x4E0B_30D3_2890_9F16L, 0x41AF_A843_2997_0214L, + /* 143 */ 0x7CDE_B485_0DB4_31BDL, 0x4F7F_739E_A8F1_9CEDL, + /* 144 */ 0x63E5_5D37_3E29_C164L, 0x3F99_294B_BA5A_E3F1L, + /* 145 */ 0x4FEA_B0F8_FE87_CDE9L, 0x7FAD_BAA2_FB7B_E98DL, + /* 146 */ 0x7FDD_E7F4_CA72_E30FL, 0x7F7C_5DD1_925F_DC15L, + /* 147 */ 0x664B_1FF7_085B_E8D9L, 0x4C63_7E41_41E6_49ABL, + /* 148 */ 0x51D5_B32C_06AF_ED7AL, 0x704F_9834_34B8_3AEFL, + /* 149 */ 0x4177_C289_9EF3_2462L, 0x26A6_135C_F6F9_C8BFL, + /* 150 */ 0x68BF_9DA8_FE51_D3D0L, 0x3DD6_8561_8B29_4132L, + /* 151 */ 0x53CC_7E20_CB74_A973L, 0x4B12_044E_08ED_CDC2L, + /* 152 */ 0x4309_FE80_A2C3_BAC2L, 0x6F41_9D0B_3A57_D7CEL, + /* 153 */ 0x6B43_30CD_D139_2AD1L, 0x3202_94DE_C3BF_BFB0L, + /* 154 */ 0x55CF_5A3E_40FA_88A7L, 0x419B_AA4B_CFCC_995AL, + /* 155 */ 0x44A5_E1CB_672E_D3B9L, 0x1AE2_EEA3_0CA3_ADE1L, + /* 156 */ 0x6DD6_3612_3EB1_52C1L, 0x77D1_7DD1_ADD2_AFCFL, + /* 157 */ 0x57DE_91A8_3227_7567L, 0x7974_64A7_BE42_263FL, + /* 158 */ 0x464B_A7B9_C1B9_2AB9L, 0x4790_5086_31CE_84FFL, + /* 159 */ 0x7079_0C5C_6928_445CL, 0x0C1A_1A70_4FB0_D4CCL, + /* 160 */ 0x59FA_7049_EDB9_D049L, 0x567B_4859_D95A_43D6L, + /* 161 */ 0x47FB_8D07_F161_736EL, 0x11FC_39E1_7AAE_9CABL, + /* 162 */ 0x732C_14D9_8235_857DL, 0x032D_2968_C44A_9445L, + /* 163 */ 0x5C23_43E1_34F7_9DFDL, 0x4F57_5453_D03B_A9D1L, + /* 164 */ 0x49B5_CFE7_5D92_E4CAL, 0x72AC_4376_402F_BB0EL, + /* 165 */ 0x75EF_B30B_C8EB_07ABL, 0x0446_D256_CD19_2B49L, + /* 166 */ 0x5E59_5C09_6D88_D2EFL, 0x1D05_7512_3DAD_BC3AL, + /* 167 */ 0x4B7A_B007_8AD3_DBF2L, 0x4A6A_C40E_97BE_302FL, + /* 168 */ 0x78C4_4CD8_DE1F_C650L, 0x7711_39B0_F2C9_E6B1L, + /* 169 */ 0x609D_0A47_1819_6B73L, 0x78DA_948D_8F07_EBC1L, + /* 170 */ 0x4D4A_6E9F_467A_BC5CL, 0x60AE_DD3E_0C06_5634L, + /* 171 */ 0x7BAA_4A98_70C4_6094L, 0x344A_FB96_79A3_BD20L, + /* 172 */ 0x62EE_A213_8D69_E6DDL, 0x103B_FC78_614F_CA80L, + /* 173 */ 0x4F25_4E76_0ABB_1F17L, 0x2696_6393_810C_A200L, + /* 174 */ 0x7EA2_1723_445E_9825L, 0x2423_D285_9B47_6999L, + /* 175 */ 0x654E_78E9_037E_E01DL, 0x69B6_4204_7C39_2148L, + /* 176 */ 0x510B_93ED_9C65_8017L, 0x6E2B_6803_9694_1AA0L, + /* 177 */ 0x40D6_0FF1_49EA_CCDFL, 0x71BC_5336_1210_154DL, + /* 178 */ 0x67BC_E64E_DCAA_E166L, 0x1C60_8523_5019_BBAEL, + /* 179 */ 0x52FD_850B_E3BB_E784L, 0x7D1A_041C_4014_9625L, + /* 180 */ 0x4264_6A6F_E963_1F9DL, 0x4A7B_367D_0010_781DL, + /* 181 */ 0x6A3A_43E6_4238_3295L, 0x5D91_F0C8_001A_59C8L, + /* 182 */ 0x54FB_6985_01C6_8EDEL, 0x17A7_F3D3_3348_47D4L, + /* 183 */ 0x43FC_546A_67D2_0BE4L, 0x7953_2975_C2A0_3976L, + /* 184 */ 0x6CC6_ED77_0C83_463BL, 0x0EEB_7589_3766_C256L, + /* 185 */ 0x5705_8AC5_A39C_382FL, 0x2589_2AD4_2C52_3512L, + /* 186 */ 0x459E_089E_1C7C_F9BFL, 0x37A0_EF10_2374_F742L, + /* 187 */ 0x6F63_40FC_FA61_8F98L, 0x5901_7E80_38BB_2536L, + /* 188 */ 0x591C_33FD_951A_D946L, 0x7A67_9866_93C8_EA91L, + /* 189 */ 0x4749_C331_4415_7A9FL, 0x151F_AD1E_DCA0_BBA8L, + /* 190 */ 0x720F_9EB5_39BB_F765L, 0x0832_AE97_C767_92A5L, + /* 191 */ 0x5B3F_B22A_9496_5F84L, 0x068E_F213_05EC_7551L, + /* 192 */ 0x48FF_C1BB_AA11_E603L, 0x1ED8_C1A8_D189_F774L, + /* 193 */ 0x74CC_692C_434F_D66BL, 0x4AF4_690E_1C0F_F253L, + /* 194 */ 0x5D70_5423_690C_AB89L, 0x225D_20D8_1673_2843L, + /* 195 */ 0x4AC0_434F_873D_5607L, 0x3517_4D79_AB8F_5369L, + /* 196 */ 0x779A_054C_0B95_5672L, 0x21BE_E25C_45B2_1F0EL, + /* 197 */ 0x5FAE_6AA3_3C77_785BL, 0x3498_B516_9E28_18D8L, + /* 198 */ 0x4C8B_8882_96C5_F9E2L, 0x5D46_F745_4B53_4713L, + /* 199 */ 0x7A78_DA6A_8AD6_5C9DL, 0x7BA4_BED5_4552_0B52L, + /* 200 */ 0x61FA_4855_3BDE_B07EL, 0x2FB6_FF11_0441_A2A8L, + /* 201 */ 0x4E61_D377_6318_8D31L, 0x72F8_CC0D_9D01_4EEDL, + /* 202 */ 0x7D69_5258_9E8D_AEB6L, 0x1E5A_E015_C802_17E1L, + /* 203 */ 0x6454_41E0_7ED7_BEF8L, 0x1848_B344_A001_ACB4L, + /* 204 */ 0x5043_67E6_CBDF_CBF9L, 0x603A_2903_B334_8A2AL, + /* 205 */ 0x4035_ECB8_A319_6FFBL, 0x002E_8736_28F6_D4EEL, + /* 206 */ 0x66BC_ADF4_3828_B32BL, 0x19E4_0B89_DB24_87E3L, + /* 207 */ 0x5230_8B29_C686_F5BCL, 0x14B6_6FA1_7C1D_3983L, + /* 208 */ 0x41C0_6F54_9ED2_5E30L, 0x1091_F2E7_967D_C79CL, + /* 209 */ 0x6933_E554_3150_96B3L, 0x341C_B7D8_F0C9_3F5FL, + /* 210 */ 0x5429_8443_5AA6_DEF5L, 0x767D_5FE0_C0A0_FF80L, + /* 211 */ 0x4354_69CF_7BB8_B25EL, 0x2B97_7FE7_0080_CC66L, + /* 212 */ 0x6BBA_42E5_92C1_1D63L, 0x5F58_CCA4_CD9A_E0A3L, + /* 213 */ 0x562E_9BEA_DBCD_B11CL, 0x4C47_0A1D_7148_B3B6L, + /* 214 */ 0x44F2_1655_7CA4_8DB0L, 0x3D05_A1B1_276D_5C92L, + /* 215 */ 0x6E50_23BB_FAA0_E2B3L, 0x7B3C_35E8_3F15_60E9L, + /* 216 */ 0x5840_1C96_621A_4EF6L, 0x2F63_5E53_65AA_B3EDL, + /* 217 */ 0x4699_B078_4E7B_725EL, 0x591C_4B75_EAEE_F658L, + /* 218 */ 0x70F5_E726_E3F8_B6FDL, 0x74FA_1256_44B1_8A26L, + /* 219 */ 0x5A5E_5285_832D_5F31L, 0x43FB_41DE_9D5A_D4EBL, + /* 220 */ 0x484B_7537_9C24_4C27L, 0x4FFC_34B2_177B_DD89L, + /* 221 */ 0x73AB_EEBF_603A_1372L, 0x4CC6_BAB6_8BF9_6274L, + /* 222 */ 0x5C89_8BCC_4CFB_42C2L, 0x0A38_955E_D661_1B90L, + /* 223 */ 0x4A07_A309_D72F_689BL, 0x21C6_DDE5_784D_AFA7L, + /* 224 */ 0x7672_9E76_2518_A75EL, 0x693E_2FD5_8D49_190BL, + /* 225 */ 0x5EC2_185E_8413_B918L, 0x5431_BFDE_0AA0_E0D5L, + /* 226 */ 0x4BCE_79E5_3676_2DADL, 0x29C1_664B_3BB3_E711L, + /* 227 */ 0x794A_5CA1_F0BD_15E2L, 0x0F9B_D6DE_C5EC_A4E8L, + /* 228 */ 0x6108_4A1B_26FD_AB1BL, 0x2616_457F_04BD_50BAL, + /* 229 */ 0x4DA0_3B48_EBFE_227CL, 0x1E78_3798_D097_73C8L, + /* 230 */ 0x7C33_920E_4663_6A60L, 0x30C0_58F4_80F2_52D9L, + /* 231 */ 0x635C_74D8_384F_884DL, 0x0D66_AD90_6728_4247L, + /* 232 */ 0x4F7D_2A46_9372_D370L, 0x711E_F140_5286_9B6CL, + /* 233 */ 0x7F2E_AA0A_8584_8581L, 0x34FE_4ECD_50D7_5F14L, + /* 234 */ 0x65BE_EE6E_D136_D134L, 0x2A65_0BD7_73DF_7F43L, + /* 235 */ 0x5165_8B8B_DA92_40F6L, 0x551D_A312_C319_329CL, + /* 236 */ 0x411E_093C_AEDB_672BL, 0x5DB1_4F42_35AD_C217L, + /* 237 */ 0x6830_0EC7_7E2B_D845L, 0x7C4E_E536_BC49_368AL, + /* 238 */ 0x5359_A56C_64EF_E037L, 0x7D0B_EA92_303A_9208L, + /* 239 */ 0x42AE_1DF0_50BF_E693L, 0x173C_BBA8_2695_41A0L, + /* 240 */ 0x6AB0_2FE6_E799_70EBL, 0x3EC7_92A6_A422_029AL, + /* 241 */ 0x5559_BFEB_EC7A_C0BCL, 0x3239_421E_E9B4_CEE1L, + /* 242 */ 0x4447_CCBC_BD2F_0096L, 0x5B61_01B2_5490_A581L, + /* 243 */ 0x6D3F_ADFA_C84B_3424L, 0x2BCE_691D_541A_A268L, + /* 244 */ 0x5766_24C8_A03C_29B6L, 0x563E_BA7D_DCE2_1B87L, + /* 245 */ 0x45EB_50A0_8030_215EL, 0x7832_2ECB_171B_4939L, + /* 246 */ 0x6FDE_E767_3380_3564L, 0x59E9_E478_24F8_7527L, + /* 247 */ 0x597F_1F85_C2CC_F783L, 0x6187_E9F9_B72D_2A86L, + /* 248 */ 0x4798_E604_9BD7_2C69L, 0x346C_BB2E_2C24_2205L, + /* 249 */ 0x728E_3CD4_2C8B_7A42L, 0x20AD_F849_E039_D007L, + /* 250 */ 0x5BA4_FD76_8A09_2E9BL, 0x33BE_603B_19C7_D99FL, + /* 251 */ 0x4950_CAC5_3B3A_8BAFL, 0x42FE_B362_7B06_47B3L, + /* 252 */ 0x754E_113B_91F7_45E5L, 0x5197_856A_5E70_72B8L, + /* 253 */ 0x5DD8_0DC9_4192_9E51L, 0x27AC_6ABB_7EC0_5BC6L, + /* 254 */ 0x4B13_3E3A_9ADB_B1DAL, 0x52F0_5562_CBCD_1638L, + /* 255 */ 0x781E_C9F7_5E2C_4FC4L, 0x1E4D_556A_DFAE_89F3L, + /* 256 */ 0x6018_A192_B1BD_0C9CL, 0x7EA4_4455_7FBE_D4C3L, + /* 257 */ 0x4CE0_8142_27CA_707DL, 0x4BB6_9D11_32FF_109CL, + /* 258 */ 0x7B00_CED0_3FAA_4D95L, 0x5F8A_94E8_5198_1A93L, + /* 259 */ 0x6267_0BD9_CC88_3E11L, 0x32D5_43ED_0E13_4875L, + /* 260 */ 0x4EB8_D647_D6D3_64DAL, 0x5BDD_CFF0_D80F_6D2BL, + /* 261 */ 0x7DF4_8A0C_8AEB_D491L, 0x12FC_7FE7_C018_AEABL, + /* 262 */ 0x64C3_A1A3_A256_43A7L, 0x28C9_FFEC_99AD_5889L, + /* 263 */ 0x509C_814F_B511_CFB9L, 0x0707_FFF0_7AF1_13A1L, + /* 264 */ 0x407D_343F_C40E_3FC7L, 0x1F39_998D_2F27_42E7L, + /* 265 */ 0x672E_B9FF_A016_CC71L, 0x7EC2_8F48_4B72_04A4L, + /* 266 */ 0x528B_C7FF_B345_705BL, 0x189B_A5D3_6F8E_6A1DL, + /* 267 */ 0x4209_6CCC_8F6A_C048L, 0x7A16_1E42_BFA5_21B1L, + /* 268 */ 0x69A8_AE14_18AA_CD41L, 0x4356_96D1_32A1_CF81L, + /* 269 */ 0x5486_F1A9_AD55_7101L, 0x1C45_4574_2881_72CEL, + /* 270 */ 0x439F_27BA_F111_2734L, 0x169D_D129_BA01_28A5L, + /* 271 */ 0x6C31_D92B_1B4E_A520L, 0x242F_B50F_9001_DAA1L, + /* 272 */ 0x568E_4755_AF72_1DB3L, 0x368C_90D9_4001_7BB4L, + /* 273 */ 0x453E_9F77_BF8E_7E29L, 0x120A_0D7A_999A_C95DL, + /* 274 */ 0x6ECA_98BF_98E3_FD0EL, 0x5010_1590_F5C4_7561L, + /* 275 */ 0x58A2_13CC_7A4F_FDA5L, 0x2673_4473_F7D0_5DE8L, + /* 276 */ 0x46E8_0FD6_C83F_FE1DL, 0x6B8F_69F6_5FD9_E4B9L, + /* 277 */ 0x7173_4C8A_D9FF_FCFCL, 0x45B2_4323_CC8F_D45CL, + /* 278 */ 0x5AC2_A3A2_47FF_FD96L, 0x6AF5_0283_0A0C_A9E3L, + /* 279 */ 0x489B_B61B_6CCC_CADFL, 0x08C4_0202_6E70_87E9L, + /* 280 */ 0x742C_5692_47AE_1164L, 0x746C_D003_E3E7_3FDBL, + /* 281 */ 0x5CF0_4541_D2F1_A783L, 0x76BD_7336_4FEC_3315L, + /* 282 */ 0x4A59_D101_758E_1F9CL, 0x5EFD_F5C5_0CBC_F5ABL, + /* 283 */ 0x76F6_1B35_88E3_65C7L, 0x4B2F_EFA1_ADFB_22ABL, + /* 284 */ 0x5F2B_48F7_A0B5_EB06L, 0x08F3_261A_F195_B555L, + /* 285 */ 0x4C22_A0C6_1A2B_226BL, 0x20C2_84E2_5ADE_2AABL, + /* 286 */ 0x79D1_013C_F6AB_6A45L, 0x1AD0_D49D_5E30_4444L, + /* 287 */ 0x6174_00FD_9222_BB6AL, 0x48A7_107D_E4F3_69D0L, + /* 288 */ 0x4DF6_6731_41B5_62BBL, 0x53B8_D9FE_50C2_BB0DL, + /* 289 */ 0x7CBD_71E8_6922_3792L, 0x52C1_5CCA_1AD1_2B48L, + /* 290 */ 0x63CA_C186_BA81_C60EL, 0x7567_7D6E_7BDA_8906L, + /* 291 */ 0x4FD5_679E_FB9B_04D8L, 0x5DEC_6458_6315_3A6CL, + /* 292 */ 0x7FBB_D8FE_5F5E_6E27L, 0x497A_3A27_04EE_C3DFL, + }; + +} diff --git a/test/jdk/java/lang/String/concat/ImplicitStringConcatBoundaries.java b/test/jdk/java/lang/String/concat/ImplicitStringConcatBoundaries.java --- a/test/jdk/java/lang/String/concat/ImplicitStringConcatBoundaries.java +++ b/test/jdk/java/lang/String/concat/ImplicitStringConcatBoundaries.java @@ -169,8 +169,8 @@ test("foo-2147483648", "foo" + INT_MIN_1); test("foo-2147483648", "foo" + INT_MIN_2); - test("foo1.17549435E-38", "foo" + FLOAT_MIN_NORM_1); - test("foo1.17549435E-38", "foo" + FLOAT_MIN_NORM_2); + test("foo1.1754944E-38", "foo" + FLOAT_MIN_NORM_1); + test("foo1.1754944E-38", "foo" + FLOAT_MIN_NORM_2); test("foo-126.0", "foo" + FLOAT_MIN_EXP_1); test("foo-126.0", "foo" + FLOAT_MIN_EXP_2); test("foo1.4E-45", "foo" + FLOAT_MIN_1); diff --git a/test/jdk/jdk/internal/math/ToDecimal/DoubleToDecimalTest.java b/test/jdk/jdk/internal/math/ToDecimal/DoubleToDecimalTest.java new file mode 100644 --- /dev/null +++ b/test/jdk/jdk/internal/math/ToDecimal/DoubleToDecimalTest.java @@ -0,0 +1,56 @@ +/* + * Copyright 2019 Raffaello Giulietti + * + * Permission is hereby granted, free of charge, to any person obtaining a copy + * of this software and associated documentation files (the "Software"), to deal + * in the Software without restriction, including without limitation the rights + * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell + * copies of the Software, and to permit persons to whom the Software is + * furnished to do so, subject to the following conditions: + * + * The above copyright notice and this permission notice shall be included in + * all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR + * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, + * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE + * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER + * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, + * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN + * THE SOFTWARE. + */ + +import jdk.internal.math.DoubleToDecimalChecker; +import jdk.test.lib.RandomFactory; + +/* + * @test + * @bug 8202555 + * @author Raffaello Giulietti + * @key randomness + * + * @modules java.base/jdk.internal.math + * @library /test/lib + * @library java.base + * @build jdk.test.lib.RandomFactory + * @build java.base/jdk.internal.math.* + * @run main DoubleToDecimalTest 1_000_000 + */ +public class DoubleToDecimalTest { + + private static final int RANDOM_COUNT = 100_000; + + public static void main(String[] args) { + int count = RANDOM_COUNT; + if (args.length == 0) { + DoubleToDecimalChecker.test(count, RandomFactory.getRandom()); + return; + } + try { + count = Integer.parseInt(args[0].replace("_", "")); + } catch (NumberFormatException ignored) { + } + DoubleToDecimalChecker.test(count, RandomFactory.getRandom()); + } + +} diff --git a/test/jdk/jdk/internal/math/ToDecimal/FloatToDecimalTest.java b/test/jdk/jdk/internal/math/ToDecimal/FloatToDecimalTest.java new file mode 100644 --- /dev/null +++ b/test/jdk/jdk/internal/math/ToDecimal/FloatToDecimalTest.java @@ -0,0 +1,63 @@ +/* + * Copyright 2019 Raffaello Giulietti + * + * Permission is hereby granted, free of charge, to any person obtaining a copy + * of this software and associated documentation files (the "Software"), to deal + * in the Software without restriction, including without limitation the rights + * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell + * copies of the Software, and to permit persons to whom the Software is + * furnished to do so, subject to the following conditions: + * + * The above copyright notice and this permission notice shall be included in + * all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR + * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, + * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE + * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER + * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, + * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN + * THE SOFTWARE. + */ + +import jdk.internal.math.FloatToDecimalChecker; +import jdk.test.lib.RandomFactory; + +/* + * @test + * @author Raffaello Giulietti + * @key randomness + * + * @modules java.base/jdk.internal.math + * @library /test/lib + * @library java.base + * @build jdk.test.lib.RandomFactory + * @build java.base/jdk.internal.math.* + * @run main FloatToDecimalTest 1_000_000 + */ +public class FloatToDecimalTest { + + private static final int RANDOM_COUNT = 100_000; + + public static void main(String[] args) { + int count = RANDOM_COUNT; + if (args.length == 0) { + FloatToDecimalChecker.test(count, RandomFactory.getRandom()); + return; + } + if (args[0].equals("all")) { + FloatToDecimalChecker.testAll(); + return; + } + if (args[0].equals("positive")) { + FloatToDecimalChecker.testPositive(); + return; + } + try { + count = Integer.parseInt(args[0].replace("_", "")); + } catch (NumberFormatException ignored) { + } + FloatToDecimalChecker.test(count, RandomFactory.getRandom()); + } + +} diff --git a/test/jdk/jdk/internal/math/ToDecimal/MathUtilsTest.java b/test/jdk/jdk/internal/math/ToDecimal/MathUtilsTest.java new file mode 100644 --- /dev/null +++ b/test/jdk/jdk/internal/math/ToDecimal/MathUtilsTest.java @@ -0,0 +1,40 @@ +/* + * Copyright 2019 Raffaello Giulietti + * + * Permission is hereby granted, free of charge, to any person obtaining a copy + * of this software and associated documentation files (the "Software"), to deal + * in the Software without restriction, including without limitation the rights + * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell + * copies of the Software, and to permit persons to whom the Software is + * furnished to do so, subject to the following conditions: + * + * The above copyright notice and this permission notice shall be included in + * all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR + * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, + * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE + * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER + * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, + * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN + * THE SOFTWARE. + */ + +import jdk.internal.math.MathUtilsChecker; + +/* + * @test + * @author Raffaello Giulietti + * + * @modules java.base/jdk.internal.math + * @library java.base + * @build java.base/jdk.internal.math.* + * @run main MathUtilsTest + */ +public class MathUtilsTest { + + public static void main(String[] args) { + MathUtilsChecker.test(); + } + +} diff --git a/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/BasicChecker.java b/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/BasicChecker.java new file mode 100644 --- /dev/null +++ b/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/BasicChecker.java @@ -0,0 +1,40 @@ +/* + * Copyright 2018-2020 Raffaello Giulietti + * + * Permission is hereby granted, free of charge, to any person obtaining a copy + * of this software and associated documentation files (the "Software"), to deal + * in the Software without restriction, including without limitation the rights + * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell + * copies of the Software, and to permit persons to whom the Software is + * furnished to do so, subject to the following conditions: + * + * The above copyright notice and this permission notice shall be included in + * all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR + * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, + * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE + * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER + * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, + * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN + * THE SOFTWARE. + */ + +package jdk.internal.math; + +class BasicChecker { + + static final boolean FAILURE_THROWS_EXCEPTION = true; + + static void assertTrue(boolean ok, String valueName) { + if (ok) { + return; + } + String msg = valueName + " is not correct"; + if (FAILURE_THROWS_EXCEPTION) { + throw new RuntimeException(msg); + } + System.err.println(msg); + } + +} diff --git a/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/DoubleToDecimalChecker.java b/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/DoubleToDecimalChecker.java new file mode 100644 --- /dev/null +++ b/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/DoubleToDecimalChecker.java @@ -0,0 +1,419 @@ +/* + * Copyright 2018-2020 Raffaello Giulietti + * + * Permission is hereby granted, free of charge, to any person obtaining a copy + * of this software and associated documentation files (the "Software"), to deal + * in the Software without restriction, including without limitation the rights + * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell + * copies of the Software, and to permit persons to whom the Software is + * furnished to do so, subject to the following conditions: + * + * The above copyright notice and this permission notice shall be included in + * all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR + * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, + * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE + * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER + * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, + * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN + * THE SOFTWARE. + */ + +package jdk.internal.math; + +import java.math.BigDecimal; +import java.util.Random; + +import static java.lang.Double.*; +import static java.lang.Long.numberOfTrailingZeros; +import static java.lang.StrictMath.scalb; +import static jdk.internal.math.MathUtils.flog10pow2; + +public class DoubleToDecimalChecker extends ToDecimalChecker { + + private static final int P = + numberOfTrailingZeros(doubleToRawLongBits(3)) + 2; + private static final int W = (SIZE - 1) - (P - 1); + private static final int Q_MIN = (-1 << W - 1) - P + 3; + private static final int Q_MAX = (1 << W - 1) - P; + private static final long C_MIN = 1L << P - 1; + private static final long C_MAX = (1L << P) - 1; + + private static final int K_MIN = flog10pow2(Q_MIN); + private static final int K_MAX = flog10pow2(Q_MAX); + private static final int H = flog10pow2(P) + 2; + + private static final double MIN_VALUE = scalb(1.0, Q_MIN); + private static final double MIN_NORMAL = scalb((double) C_MIN, Q_MIN); + private static final double MAX_VALUE = scalb((double) C_MAX, Q_MAX); + + private static final int E_MIN = e(MIN_VALUE); + private static final int E_MAX = e(MAX_VALUE); + + private static final long C_TINY = cTiny(Q_MIN, K_MIN); + + private double v; + private final long originalBits; + + private DoubleToDecimalChecker(double v, String s) { + super(s); + this.v = v; + originalBits = doubleToRawLongBits(v); + } + + @Override + BigDecimal toBigDecimal() { + return new BigDecimal(v); + } + + @Override + boolean recovers(BigDecimal b) { + return b.doubleValue() == v; + } + + @Override + boolean recovers(String s) { + return parseDouble(s) == v; + } + + @Override + String hexBits() { + return String.format("0x%01X__%03X__%01X_%04X_%04X_%04X", + (int) (originalBits >>> 63) & 0x1, + (int) (originalBits >>> 52) & 0x7FF, + (int) (originalBits >>> 48) & 0xF, + (int) (originalBits >>> 32) & 0xFFFF, + (int) (originalBits >>> 16) & 0xFFFF, + (int) originalBits & 0xFFFF); + } + + @Override + int minExp() { + return E_MIN; + } + + @Override + int maxExp() { + return E_MAX; + } + + @Override + int maxLen10() { + return H; + } + + @Override + boolean isZero() { + return v == 0; + } + + @Override + boolean isInfinity() { + return v == POSITIVE_INFINITY; + } + + @Override + void negate() { + v = -v; + } + + @Override + boolean isNegative() { + return originalBits < 0; + } + + @Override + boolean isNaN() { + return Double.isNaN(v); + } + + private static void toDec(double v) { +// String s = Double.toString(v); + String s = DoubleToDecimal.toString(v); + new DoubleToDecimalChecker(v, s).assertTrue(); + } + + private static void testExtremeValues() { + toDec(NEGATIVE_INFINITY); + toDec(-MAX_VALUE); + toDec(-MIN_NORMAL); + toDec(-MIN_VALUE); + toDec(-0.0); + toDec(0.0); + toDec(MIN_VALUE); + toDec(MIN_NORMAL); + toDec(MAX_VALUE); + toDec(POSITIVE_INFINITY); + toDec(NaN); + + /* + Quiet NaNs have the most significant bit of the mantissa as 1, + while signaling NaNs have it as 0. + Exercise 4 combinations of quiet/signaling NaNs and + "positive/negative" NaNs + */ + toDec(longBitsToDouble(0x7FF8_0000_0000_0001L)); + toDec(longBitsToDouble(0x7FF0_0000_0000_0001L)); + toDec(longBitsToDouble(0xFFF8_0000_0000_0001L)); + toDec(longBitsToDouble(0xFFF0_0000_0000_0001L)); + + /* + All values treated specially by Schubfach + */ + for (int c = 1; c < C_TINY; ++c) { + toDec(c * MIN_VALUE); + } + } + + /* + A few "powers of 10" are incorrectly rendered by the JDK. + The rendering is either too long or it is not the closest decimal. + */ + private static void testPowersOf10() { + for (int e = E_MIN; e <= E_MAX; ++e) { + toDec(parseDouble("1e" + e)); + } + } + + /* + Many powers of 2 are incorrectly rendered by the JDK. + The rendering is either too long or it is not the closest decimal. + */ + private static void testPowersOf2() { + for (double v = MIN_VALUE; v <= MAX_VALUE; v *= 2) { + toDec(v); + } + } + + /* + There are tons of doubles that are rendered incorrectly by the JDK. + While the renderings correctly round back to the original value, + they are longer than needed or are not the closest decimal to the double. + Here are just a very few examples. + */ + private static final String[] Anomalies = { + // JDK renders these, and others, with 18 digits! + "2.82879384806159E17", "1.387364135037754E18", + "1.45800632428665E17", + + // JDK renders these longer than needed. + "1.6E-322", "6.3E-322", + "7.3879E20", "2.0E23", "7.0E22", "9.2E22", + "9.5E21", "3.1E22", "5.63E21", "8.41E21", + + // JDK does not render these, and many others, as the closest. + "9.9E-324", "9.9E-323", + "1.9400994884341945E25", "3.6131332396758635E25", + "2.5138990223946153E25", + }; + + private static void testSomeAnomalies() { + for (String dec : Anomalies) { + toDec(parseDouble(dec)); + } + } + + /* + Values are from + Paxson V, "A Program for Testing IEEE Decimal-Binary Conversion" + tables 3 and 4 + */ + private static final double[] PaxsonSignificands = { + 8_511_030_020_275_656L, + 5_201_988_407_066_741L, + 6_406_892_948_269_899L, + 8_431_154_198_732_492L, + 6_475_049_196_144_587L, + 8_274_307_542_972_842L, + 5_381_065_484_265_332L, + 6_761_728_585_499_734L, + 7_976_538_478_610_756L, + 5_982_403_858_958_067L, + 5_536_995_190_630_837L, + 7_225_450_889_282_194L, + 7_225_450_889_282_194L, + 8_703_372_741_147_379L, + 8_944_262_675_275_217L, + 7_459_803_696_087_692L, + 6_080_469_016_670_379L, + 8_385_515_147_034_757L, + 7_514_216_811_389_786L, + 8_397_297_803_260_511L, + 6_733_459_239_310_543L, + 8_091_450_587_292_794L, + + 6_567_258_882_077_402L, + 6_712_731_423_444_934L, + 6_712_731_423_444_934L, + 5_298_405_411_573_037L, + 5_137_311_167_659_507L, + 6_722_280_709_661_868L, + 5_344_436_398_034_927L, + 8_369_123_604_277_281L, + 8_995_822_108_487_663L, + 8_942_832_835_564_782L, + 8_942_832_835_564_782L, + 8_942_832_835_564_782L, + 6_965_949_469_487_146L, + 6_965_949_469_487_146L, + 6_965_949_469_487_146L, + 7_487_252_720_986_826L, + 5_592_117_679_628_511L, + 8_887_055_249_355_788L, + 6_994_187_472_632_449L, + 8_797_576_579_012_143L, + 7_363_326_733_505_337L, + 8_549_497_411_294_502L, + }; + + private static final int[] PaxsonExponents = { + -342, + -824, + 237, + 72, + 99, + 726, + -456, + -57, + 376, + 377, + 93, + 710, + 709, + 117, + -1, + -707, + -381, + 721, + -828, + -345, + 202, + -473, + + 952, + 535, + 534, + -957, + -144, + 363, + -169, + -853, + -780, + -383, + -384, + -385, + -249, + -250, + -251, + 548, + 164, + 665, + 690, + 588, + 272, + -448, + }; + + private static void testPaxson() { + for (int i = 0; i < PaxsonSignificands.length; ++i) { + toDec(scalb(PaxsonSignificands[i], PaxsonExponents[i])); + } + } + + /* + Tests all integers of the form yx_xxx_000_000_000_000_000, y != 0. + These are all exact doubles. + */ + private static void testLongs() { + for (int i = 10_000; i < 100_000; ++i) { + toDec(i * 1e15); + } + } + + /* + Tests all integers up to 1_000_000. + These are all exact doubles and exercise a fast path. + */ + private static void testInts() { + for (int i = 0; i <= 1_000_000; ++i) { + toDec(i); + } + } + + /* + Random doubles over the whole range + */ + private static void testRandom(int randomCount, Random r) { + for (int i = 0; i < randomCount; ++i) { + toDec(longBitsToDouble(r.nextLong())); + } + } + + /* + Random doubles over the integer range [0, 2^52). + These are all exact doubles and exercise the fast path (except 0). + */ + private static void testRandomUnit(int randomCount, Random r) { + for (int i = 0; i < randomCount; ++i) { + toDec(r.nextLong() & (1L << P - 1)); + } + } + + /* + Random doubles over the range [0, 10^15) as "multiples" of 1e-3 + */ + private static void testRandomMilli(int randomCount, Random r) { + for (int i = 0; i < randomCount; ++i) { + toDec(r.nextLong() % 1_000_000_000_000_000_000L / 1e3); + } + } + + /* + Random doubles over the range [0, 10^15) as "multiples" of 1e-6 + */ + private static void testRandomMicro(int randomCount, Random r) { + for (int i = 0; i < randomCount; ++i) { + toDec((r.nextLong() & 0x7FFF_FFFF_FFFF_FFFFL) / 1e6); + } + } + + private static void testConstants() { + assertTrue(P == DoubleToDecimal.P, "P"); + assertTrue((long) (double) C_MIN == C_MIN, "C_MIN"); + assertTrue((long) (double) C_MAX == C_MAX, "C_MAX"); + assertTrue(MIN_VALUE == Double.MIN_VALUE, "MIN_VALUE"); + assertTrue(MIN_NORMAL == Double.MIN_NORMAL, "MIN_NORMAL"); + assertTrue(MAX_VALUE == Double.MAX_VALUE, "MAX_VALUE"); + + assertTrue(Q_MIN == DoubleToDecimal.Q_MIN, "Q_MIN"); + assertTrue(Q_MAX == DoubleToDecimal.Q_MAX, "Q_MAX"); + + assertTrue(K_MIN == DoubleToDecimal.K_MIN, "K_MIN"); + assertTrue(K_MAX == DoubleToDecimal.K_MAX, "K_MAX"); + assertTrue(H == DoubleToDecimal.H, "H"); + + assertTrue(E_MIN == DoubleToDecimal.E_MIN, "E_MIN"); + assertTrue(E_MAX == DoubleToDecimal.E_MAX, "E_MAX"); + assertTrue(C_TINY == DoubleToDecimal.C_TINY, "C_TINY"); + } + + public static void test(int randomCount, Random r) { + testConstants(); + testExtremeValues(); + testSomeAnomalies(); + testPowersOf2(); + testPowersOf10(); + testPaxson(); + testInts(); + testLongs(); + testRandom(randomCount, r); + testRandomUnit(randomCount, r); + testRandomMilli(randomCount, r); + testRandomMicro(randomCount, r); + } + + public static void main(String[] args) { + test(1_000_000, new Random()); + } +} diff --git a/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/FloatToDecimalChecker.java b/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/FloatToDecimalChecker.java new file mode 100644 --- /dev/null +++ b/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/FloatToDecimalChecker.java @@ -0,0 +1,365 @@ +/* + * Copyright 2018-2020 Raffaello Giulietti + * + * Permission is hereby granted, free of charge, to any person obtaining a copy + * of this software and associated documentation files (the "Software"), to deal + * in the Software without restriction, including without limitation the rights + * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell + * copies of the Software, and to permit persons to whom the Software is + * furnished to do so, subject to the following conditions: + * + * The above copyright notice and this permission notice shall be included in + * all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR + * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, + * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE + * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER + * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, + * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN + * THE SOFTWARE. + */ + +package jdk.internal.math; + +import java.math.BigDecimal; +import java.util.Random; + +import static java.lang.Float.*; +import static java.lang.Integer.numberOfTrailingZeros; +import static java.lang.StrictMath.scalb; +import static jdk.internal.math.MathUtils.flog10pow2; + +public class FloatToDecimalChecker extends ToDecimalChecker { + + private static final int P = + numberOfTrailingZeros(floatToRawIntBits(3)) + 2; + private static final int W = (SIZE - 1) - (P - 1); + private static final int Q_MIN = (-1 << W - 1) - P + 3; + private static final int Q_MAX = (1 << W - 1) - P; + private static final int C_MIN = 1 << P - 1; + private static final int C_MAX = (1 << P) - 1; + + private static final int K_MIN = flog10pow2(Q_MIN); + private static final int K_MAX = flog10pow2(Q_MAX); + private static final int H = flog10pow2(P) + 2; + + private static final float MIN_VALUE = scalb(1.0f, Q_MIN); + private static final float MIN_NORMAL = scalb((float) C_MIN, Q_MIN); + private static final float MAX_VALUE = scalb((float) C_MAX, Q_MAX); + + private static final int E_MIN = e(MIN_VALUE); + private static final int E_MAX = e(MAX_VALUE); + + private static final long C_TINY = cTiny(Q_MIN, K_MIN); + + private float v; + private final int originalBits; + + private FloatToDecimalChecker(float v, String s) { + super(s); + this.v = v; + originalBits = floatToRawIntBits(v); + } + + @Override + BigDecimal toBigDecimal() { + return new BigDecimal(v); + } + + @Override + boolean recovers(BigDecimal b) { + return b.floatValue() == v; + } + + @Override + String hexBits() { + return String.format("0x%01X__%02X__%02X_%04X", + (originalBits >>> 31) & 0x1, + (originalBits >>> 23) & 0xFF, + (originalBits >>> 16) & 0x7F, + originalBits & 0xFFFF); + } + + @Override + boolean recovers(String s) { + return parseFloat(s) == v; + } + + @Override + int minExp() { + return E_MIN; + } + + @Override + int maxExp() { + return E_MAX; + } + + @Override + int maxLen10() { + return H; + } + + @Override + boolean isZero() { + return v == 0; + } + + @Override + boolean isInfinity() { + return v == POSITIVE_INFINITY; + } + + @Override + void negate() { + v = -v; + } + + @Override + boolean isNegative() { + return originalBits < 0; + } + + @Override + boolean isNaN() { + return Float.isNaN(v); + } + + private static void toDec(float v) { +// String s = Float.toString(v); + String s = FloatToDecimal.toString(v); + new FloatToDecimalChecker(v, s).assertTrue(); + } + + /* + MIN_NORMAL is incorrectly rendered by the JDK. + */ + private static void testExtremeValues() { + toDec(NEGATIVE_INFINITY); + toDec(-MAX_VALUE); + toDec(-MIN_NORMAL); + toDec(-MIN_VALUE); + toDec(-0.0f); + toDec(0.0f); + toDec(MIN_VALUE); + toDec(MIN_NORMAL); + toDec(MAX_VALUE); + toDec(POSITIVE_INFINITY); + toDec(NaN); + + /* + Quiet NaNs have the most significant bit of the mantissa as 1, + while signaling NaNs have it as 0. + Exercise 4 combinations of quiet/signaling NaNs and + "positive/negative" NaNs. + */ + toDec(intBitsToFloat(0x7FC0_0001)); + toDec(intBitsToFloat(0x7F80_0001)); + toDec(intBitsToFloat(0xFFC0_0001)); + toDec(intBitsToFloat(0xFF80_0001)); + + /* + All values treated specially by Schubfach + */ + for (int c = 1; c < C_TINY; ++c) { + toDec(c * MIN_VALUE); + } + } + + /* + Some "powers of 10" are incorrectly rendered by the JDK. + The rendering is either too long or it is not the closest decimal. + */ + private static void testPowersOf10() { + for (int e = E_MIN; e <= E_MAX; ++e) { + toDec(parseFloat("1e" + e)); + } + } + + /* + Many powers of 2 are incorrectly rendered by the JDK. + The rendering is either too long or it is not the closest decimal. + */ + private static void testPowersOf2() { + for (float v = MIN_VALUE; v <= MAX_VALUE; v *= 2) { + toDec(v); + } + } + + /* + There are tons of doubles that are rendered incorrectly by the JDK. + While the renderings correctly round back to the original value, + they are longer than needed or are not the closest decimal to the double. + Here are just a very few examples. + */ + private static final String[] Anomalies = { + // JDK renders these longer than needed. + "1.1754944E-38", "2.2E-44", + "1.0E16", "2.0E16", "3.0E16", "5.0E16", "3.0E17", + "3.2E18", "3.7E18", "3.7E16", "3.72E17", + + // JDK does not render this as the closest. + "9.9E-44", + }; + + private static void testSomeAnomalies() { + for (String dec : Anomalies) { + toDec(parseFloat(dec)); + } + } + + /* + Values are from + Paxson V, "A Program for Testing IEEE Decimal-Binary Conversion" + tables 16 and 17 + */ + private static final float[] PaxsonSignificands = { + 12_676_506, + 15_445_013, + 13_734_123, + 12_428_269, + 12_676_506, + 15_334_037, + 11_518_287, + 12_584_953, + 15_961_084, + 14_915_817, + 10_845_484, + 16_431_059, + + 16_093_626, + 9_983_778, + 12_745_034, + 12_706_553, + 11_005_028, + 15_059_547, + 16_015_691, + 8_667_859, + 14_855_922, + 14_855_922, + 10_144_164, + 13_248_074, + }; + + private static final int[] PaxsonExponents = { + -102, + -103, + 86, + -138, + -130, + -146, + -41, + -145, + -125, + -146, + -102, + -61, + + 69, + 25, + 104, + 72, + 45, + 71, + -99, + 56, + -82, + -83, + -110, + 95, + }; + + private static void testPaxson() { + for (int i = 0; i < PaxsonSignificands.length; ++i) { + toDec(scalb(PaxsonSignificands[i], PaxsonExponents[i])); + } + } + + /* + Tests all positive integers below 2^23. + These are all exact floats and exercise the fast path. + */ + private static void testInts() { + for (int i = 1; i < 1 << P - 1; ++i) { + toDec(i); + } + } + + /* + Random floats over the whole range. + */ + private static void testRandom(int randomCount, Random r) { + for (int i = 0; i < randomCount; ++i) { + toDec(intBitsToFloat(r.nextInt())); + } + } + + /* + All, really all, 2^32 possible floats. Takes between 90 and 120 minutes. + */ + public static void testAll() { + // Avoid wrapping around Integer.MAX_VALUE + int bits = Integer.MIN_VALUE; + for (; bits < Integer.MAX_VALUE; ++bits) { + toDec(intBitsToFloat(bits)); + } + toDec(intBitsToFloat(bits)); + } + + /* + All positive 2^31 floats. + */ + public static void testPositive() { + // Avoid wrapping around Integer.MAX_VALUE + int bits = 0; + for (; bits < Integer.MAX_VALUE; ++bits) { + toDec(intBitsToFloat(bits)); + } + toDec(intBitsToFloat(bits)); + } + + private static void testConstants() { + assertTrue(P == FloatToDecimal.P, "P"); + assertTrue((long) (float) C_MIN == C_MIN, "C_MIN"); + assertTrue((long) (float) C_MAX == C_MAX, "C_MAX"); + assertTrue(MIN_VALUE == Float.MIN_VALUE, "MIN_VALUE"); + assertTrue(MIN_NORMAL == Float.MIN_NORMAL, "MIN_NORMAL"); + assertTrue(MAX_VALUE == Float.MAX_VALUE, "MAX_VALUE"); + + assertTrue(Q_MIN == FloatToDecimal.Q_MIN, "Q_MIN"); + assertTrue(Q_MAX == FloatToDecimal.Q_MAX, "Q_MAX"); + + assertTrue(K_MIN == FloatToDecimal.K_MIN, "K_MIN"); + assertTrue(K_MAX == FloatToDecimal.K_MAX, "K_MAX"); + assertTrue(H == FloatToDecimal.H, "H"); + + assertTrue(E_MIN == FloatToDecimal.E_MIN, "E_MIN"); + assertTrue(E_MAX == FloatToDecimal.E_MAX, "E_MAX"); + assertTrue(C_TINY == FloatToDecimal.C_TINY, "C_TINY"); + } + + public static void test(int randomCount, Random r) { + testConstants(); + testExtremeValues(); + testSomeAnomalies(); + testPowersOf2(); + testPowersOf10(); + testPaxson(); + testInts(); + testRandom(randomCount, r); + } + + public static void main(String[] args) { + if (args.length > 0 && args[0].equals("all")) { + testAll(); + return; + } + if (args.length > 0 && args[0].equals("positive")) { + testPositive(); + return; + } + test(1_000_000, new Random()); + } + +} diff --git a/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/MathUtilsChecker.java b/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/MathUtilsChecker.java new file mode 100644 --- /dev/null +++ b/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/MathUtilsChecker.java @@ -0,0 +1,471 @@ +/* + * Copyright 2018-2020 Raffaello Giulietti + * + * Permission is hereby granted, free of charge, to any person obtaining a copy + * of this software and associated documentation files (the "Software"), to deal + * in the Software without restriction, including without limitation the rights + * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell + * copies of the Software, and to permit persons to whom the Software is + * furnished to do so, subject to the following conditions: + * + * The above copyright notice and this permission notice shall be included in + * all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR + * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, + * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE + * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER + * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, + * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN + * THE SOFTWARE. + */ + +package jdk.internal.math; + +import java.math.BigInteger; + +import static java.lang.Double.*; +import static java.lang.Long.numberOfTrailingZeros; +import static java.lang.StrictMath.scalb; +import static java.math.BigInteger.*; +import static jdk.internal.math.MathUtils.*; + +public class MathUtilsChecker extends BasicChecker { + + private static final BigInteger THREE = valueOf(3); + + // binary constants + private static final int P = + numberOfTrailingZeros(doubleToRawLongBits(3)) + 2; + private static final int W = (SIZE - 1) - (P - 1); + private static final int Q_MIN = (-1 << W - 1) - P + 3; + private static final int Q_MAX = (1 << W - 1) - P; + private static final long C_MIN = 1L << P - 1; + private static final long C_MAX = (1L << P) - 1; + + // decimal constants + private static final int K_MIN = flog10pow2(Q_MIN); + private static final int K_MAX = flog10pow2(Q_MAX); + private static final int H = flog10pow2(P) + 2; + + /* + Let + 10^(-k) = beta 2^r + for the unique integer r and real beta meeting + 2^125 <= beta < 2^126 + Further, let g = g1 2^63 + g0. + Checks that: + 2^62 <= g1 < 2^63, + 0 <= g0 < 2^63, + g - 1 <= beta < g, (that is, g = floor(beta) + 1) + The last predicate, after multiplying by 2^r, is equivalent to + (g - 1) 2^r <= 10^(-k) < g 2^r + This is the predicate that will be checked in various forms. + */ + private static void testG(int k, long g1, long g0) { + // 2^62 <= g1 < 2^63, 0 <= g0 < 2^63 + assertTrue(g1 << 1 < 0 && g1 >= 0 && g0 >= 0, "g"); + + BigInteger g = valueOf(g1).shiftLeft(63).or(valueOf(g0)); + // double check that 2^125 <= g < 2^126 + assertTrue(g.signum() > 0 && g.bitLength() == 126, "g"); + + // see javadoc of MathUtils.g1(int) + int r = flog2pow10(-k) - 125; + + /* + The predicate + (g - 1) 2^r <= 10^(-k) < g 2^r + is equivalent to + g - 1 <= 10^(-k) 2^(-r) < g + When + k <= 0 & r < 0 + all numerical subexpressions are integer-valued. This is the same as + g - 1 = 10^(-k) 2^(-r) + */ + if (k <= 0 && r < 0) { + assertTrue( + g.subtract(ONE).compareTo(TEN.pow(-k).shiftLeft(-r)) == 0, + "g"); + return; + } + + /* + The predicate + (g - 1) 2^r <= 10^(-k) < g 2^r + is equivalent to + g 10^k - 10^k <= 2^(-r) < g 10^k + When + k > 0 & r < 0 + all numerical subexpressions are integer-valued. + */ + if (k > 0 && r < 0) { + BigInteger pow5 = TEN.pow(k); + BigInteger mhs = ONE.shiftLeft(-r); + BigInteger rhs = g.multiply(pow5); + assertTrue(rhs.subtract(pow5).compareTo(mhs) <= 0 + && mhs.compareTo(rhs) < 0, + "g"); + return; + } + + /* + Finally, when + k <= 0 & r >= 0 + the predicate + (g - 1) 2^r <= 10^(-k) < g 2^r + can be used straightforwardly as all numerical subexpressions are + already integer-valued. + */ + if (k <= 0) { + BigInteger mhs = TEN.pow(-k); + assertTrue(g.subtract(ONE).shiftLeft(r).compareTo(mhs) <= 0 && + mhs.compareTo(g.shiftLeft(r)) < 0, + "g"); + return; + } + + /* + For combinatorial reasons, the only remaining case is + k > 0 & r >= 0 + which, however, cannot arise. Indeed, the predicate + (g - 1) 2^r <= 10^(-k) < g 2^r + has a positive integer left-hand side and a middle side < 1, + which cannot hold. + */ + assertTrue(false, "g"); + } + + /* + Verifies the soundness of the values returned by g1() and g0(). + */ + private static void testG() { + for (int k = MathUtils.K_MIN; k <= MathUtils.K_MAX; ++k) { + testG(k, g1(k), g0(k)); + } + } + + /* + Let + k = floor(log10(3/4 2^e)) + The method verifies that + k = flog10threeQuartersPow2(e), Q_MIN <= e <= Q_MAX + This range covers all binary exponents of doubles and floats. + + The first equation above is equivalent to + 10^k <= 3 2^(e-2) < 10^(k+1) + Equality never holds. Henceforth, the predicate to check is + 10^k < 3 2^(e-2) < 10^(k+1) + This will be transformed in various ways for checking purposes. + + For integer n > 0, let further + b = len2(n) + denote its length in bits. This means exactly the same as + 2^(b-1) <= n < 2^b + */ + private static void testFlog10threeQuartersPow2() { + // First check the case e = 1 + assertTrue(flog10threeQuartersPow2(1) == 0, + "flog10threeQuartersPow2"); + + /* + Now check the range Q_MIN <= e <= 0. + By rewriting, the predicate to check is equivalent to + 3 10^(-k-1) < 2^(2-e) < 3 10^(-k) + As e <= 0, it follows that 2^(2-e) >= 4 and the right inequality + implies k < 0, so the powers of 10 are integers. + + The left inequality is equivalent to + len2(3 10^(-k-1)) <= 2 - e + and the right inequality to + 2 - e < len2(3 10^(-k)) + The original predicate is therefore equivalent to + len2(3 10^(-k-1)) <= 2 - e < len2(3 10^(-k)) + + Starting with e = 0 and decrementing until the lower bound, the code + keeps track of the two powers of 10 to avoid recomputing them. + This is easy because at each iteration k changes at most by 1. A simple + multiplication by 10 computes the next power of 10 when needed. + */ + int e = 0; + int k0 = flog10threeQuartersPow2(e); + assertTrue(k0 < 0, "flog10threeQuartersPow2"); + BigInteger l = THREE.multiply(TEN.pow(-k0 - 1)); + BigInteger u = l.multiply(TEN); + for (;;) { + assertTrue(l.bitLength() <= 2 - e & 2 - e < u.bitLength(), + "flog10threeQuartersPow2"); + --e; + if (e < Q_MIN) { + break; + } + int kp = flog10threeQuartersPow2(e); + assertTrue(kp <= k0, "flog10threeQuartersPow2"); + if (kp < k0) { + // k changes at most by 1 at each iteration, hence: + assertTrue(k0 - kp == 1, "flog10threeQuartersPow2"); + k0 = kp; + l = u; + u = u.multiply(TEN); + } + } + + /* + Finally, check the range 2 <= e <= Q_MAX. + In predicate + 10^k < 3 2^(e-2) < 10^(k+1) + the right inequality shows that k >= 0 as soon as e >= 2. + It is equivalent to + 10^k / 3 < 2^(e-2) < 10^(k+1) / 3 + Both the powers of 10 and the powers of 2 are integers. + The left inequality is therefore equivalent to + floor(10^k / 3) < 2^(e-2) + and thus to + len2(floor(10^k / 3)) <= e - 2 + while the right inequality is equivalent to + 2^(e-2) <= floor(10^(k+1) / 3) + and hence to + e - 2 < len2(floor(10^(k+1) / 3)) + These are summarized as + len2(floor(10^k / 3)) <= e - 2 < len2(floor(10^(k+1) / 3)) + */ + e = 2; + k0 = flog10threeQuartersPow2(e); + assertTrue(k0 >= 0, "flog10threeQuartersPow2"); + BigInteger l10 = TEN.pow(k0); + BigInteger u10 = l10.multiply(TEN); + l = l10.divide(THREE); + u = u10.divide(THREE); + for (;;) { + assertTrue(l.bitLength() <= e - 2 & e - 2 < u.bitLength(), + "flog10threeQuartersPow2"); + ++e; + if (e > Q_MAX) { + break; + } + int kp = flog10threeQuartersPow2(e); + assertTrue(kp >= k0, "flog10threeQuartersPow2"); + if (kp > k0) { + // k changes at most by 1 at each iteration, hence: + assertTrue(kp - k0 == 1, "flog10threeQuartersPow2"); + k0 = kp; + u10 = u10.multiply(TEN); + l = u; + u = u10.divide(THREE); + } + } + } + + /* + Let + k = floor(log10(2^e)) + The method verifies that + k = flog10pow2(e), Q_MIN <= e <= Q_MAX + This range covers all binary exponents of doubles and floats. + + The first equation above is equivalent to + 10^k <= 2^e < 10^(k+1) + Equality holds iff e = k = 0. + Henceforth, the predicates to check are equivalent to + k = 0, if e = 0 + 10^k < 2^e < 10^(k+1), otherwise + The latter will be transformed in various ways for checking purposes. + + For integer n > 0, let further + b = len2(n) + denote its length in bits. This means exactly the same as + 2^(b-1) <= n < 2^b + */ + private static void testFlog10pow2() { + // First check the case e = 0 + assertTrue(flog10pow2(0) == 0, "flog10pow2"); + + /* + Now check the range F * Q_MIN <= e < 0. + By inverting all quantities, the predicate to check is equivalent to + 10^(-k-1) < 2^(-e) < 10^(-k) + As e < 0, it follows that 2^(-e) >= 2 and the right inequality + implies k < 0. + The left inequality means exactly the same as + len2(10^(-k-1)) <= -e + Similarly, the right inequality is equivalent to + -e < len2(10^(-k)) + The original predicate is therefore equivalent to + len2(10^(-k-1)) <= -e < len2(10^(-k)) + The powers of 10 are integers because k < 0. + + Starting with e = -1 and decrementing towards the lower bound, the code + keeps track of the two powers of 10 so as to avoid recomputing them. + This is easy because at each iteration k changes at most by 1. A simple + multiplication by 10 computes the next power of 10 when needed. + */ + int e = -1; + int k = flog10pow2(e); + assertTrue(k < 0, "flog10pow2"); + BigInteger l = TEN.pow(-k - 1); + BigInteger u = l.multiply(TEN); + for (;;) { + assertTrue(l.bitLength() <= -e & -e < u.bitLength(), + "flog10pow2"); + --e; + if (e < Q_MIN) { + break; + } + int kp = flog10pow2(e); + assertTrue(kp <= k, "flog10pow2"); + if (kp < k) { + // k changes at most by 1 at each iteration, hence: + assertTrue(k - kp == 1, "flog10pow2"); + k = kp; + l = u; + u = u.multiply(TEN); + } + } + + /* + Finally, in a similar vein, check the range 0 <= e <= Q_MAX. + In predicate + 10^k < 2^e < 10^(k+1) + the right inequality shows that k >= 0. + The left inequality means the same as + len2(10^k) <= e + and the right inequality holds iff + e < len2(10^(k+1)) + The original predicate is thus equivalent to + len2(10^k) <= e < len2(10^(k+1)) + As k >= 0, the powers of 10 are integers. + */ + e = 1; + k = flog10pow2(e); + assertTrue(k >= 0, "flog10pow2"); + l = TEN.pow(k); + u = l.multiply(TEN); + for (;;) { + assertTrue(l.bitLength() <= e & e < u.bitLength(), + "flog10pow2"); + ++e; + if (e > Q_MAX) { + break; + } + int kp = flog10pow2(e); + assertTrue(kp >= k, "flog10pow2"); + if (kp > k) { + // k changes at most by 1 at each iteration, hence: + assertTrue(kp - k == 1, "flog10pow2"); + k = kp; + l = u; + u = u.multiply(TEN); + } + } + } + + /* + Let + k = floor(log2(10^e)) + The method verifies that + k = flog2pow10(e), -K_MAX <= e <= -K_MIN + This range covers all decimal exponents of doubles and floats. + + The first equation above is equivalent to + 2^k <= 10^e < 2^(k+1) + Equality holds iff e = 0, implying k = 0. + Henceforth, the equivalent predicates to check are + k = 0, if e = 0 + 2^k < 10^e < 2^(k+1), otherwise + The latter will be transformed in various ways for checking purposes. + + For integer n > 0, let further + b = len2(n) + denote its length in bits. This means exactly the same as + 2^(b-1) <= n < 2^b + */ + private static void testFlog2pow10() { + // First check the case e = 0 + assertTrue(flog2pow10(0) == 0, "flog2pow10"); + + /* + Now check the range K_MIN <= e < 0. + By inverting all quantities, the predicate to check is equivalent to + 2^(-k-1) < 10^(-e) < 2^(-k) + As e < 0, this leads to 10^(-e) >= 10 and the right inequality implies + k <= -4. + The above means the same as + len2(10^(-e)) = -k + The powers of 10 are integer values since e < 0. + */ + int e = -1; + int k0 = flog2pow10(e); + assertTrue(k0 <= -4, "flog2pow10"); + BigInteger l = TEN; + for (;;) { + assertTrue(l.bitLength() == -k0, "flog2pow10"); + --e; + if (e < -K_MAX) { + break; + } + k0 = flog2pow10(e); + l = l.multiply(TEN); + } + + /* + Finally check the range 0 < e <= K_MAX. + From the predicate + 2^k < 10^e < 2^(k+1) + as e > 0, it follows that 10^e >= 10 and the right inequality implies + k >= 3. + The above means the same as + len2(10^e) = k + 1 + The powers of 10 are all integer valued, as e > 0. + */ + e = 1; + k0 = flog2pow10(e); + assertTrue(k0 >= 3, "flog2pow10"); + l = TEN; + for (;;) { + assertTrue(l.bitLength() == k0 + 1, "flog2pow10"); + ++e; + if (e > -K_MIN) { + break; + } + k0 = flog2pow10(e); + l = l.multiply(TEN); + } + } + + private static void testBinaryConstants() { + assertTrue((long) (double) C_MIN == C_MIN, "C_MIN"); + assertTrue((long) (double) C_MAX == C_MAX, "C_MAX"); + assertTrue(scalb(1.0, Q_MIN) == MIN_VALUE, "MIN_VALUE"); + assertTrue(scalb((double) C_MIN, Q_MIN) == MIN_NORMAL, "MIN_NORMAL"); + assertTrue(scalb((double) C_MAX, Q_MAX) == MAX_VALUE, "MAX_VALUE"); + } + + private static void testDecimalConstants() { + assertTrue(K_MIN == MathUtils.K_MIN, "K_MIN"); + assertTrue(K_MAX == MathUtils.K_MAX, "K_MAX"); + assertTrue(H == MathUtils.H, "H"); + } + + private static void testPow10() { + int e = 0; + long pow = 1; + for (; e <= H; e += 1, pow *= 10) { + assertTrue(pow == pow10(e), "pow10"); + } + } + + public static void test() { + testBinaryConstants(); + testFlog10pow2(); + testFlog10threeQuartersPow2(); + testDecimalConstants(); + testFlog2pow10(); + testPow10(); + testG(); + } + + public static void main(String[] args) { + test(); + } + +} diff --git a/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/ToDecimalChecker.java b/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/ToDecimalChecker.java new file mode 100644 --- /dev/null +++ b/test/jdk/jdk/internal/math/ToDecimal/java.base/jdk/internal/math/ToDecimalChecker.java @@ -0,0 +1,409 @@ +/* + * Copyright 2018-2020 Raffaello Giulietti + * + * Permission is hereby granted, free of charge, to any person obtaining a copy + * of this software and associated documentation files (the "Software"), to deal + * in the Software without restriction, including without limitation the rights + * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell + * copies of the Software, and to permit persons to whom the Software is + * furnished to do so, subject to the following conditions: + * + * The above copyright notice and this permission notice shall be included in + * all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR + * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, + * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE + * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER + * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, + * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN + * THE SOFTWARE. + */ + +package jdk.internal.math; + +import java.io.IOException; +import java.io.StringReader; +import java.math.BigDecimal; +import java.math.BigInteger; + +import static java.math.BigInteger.*; + +/* +A checker for the Javadoc specification. +It just relies on straightforward use of (expensive) BigDecimal arithmetic, +not optimized at all. + */ +abstract class ToDecimalChecker extends BasicChecker { + + // The string to check + private final String s; + + // The decimal parsed from s is c 10^q + private long c; + private int q; + + // The number of digits parsed from s: 10^(len10-1) <= c < 10^len10 + private int len10; + + ToDecimalChecker(String s) { + this.s = s; + } + + /* + Returns e be such that 10^(e-1) <= v < 10^e. + */ + static int e(double v) { + // log10(v) + 1 is a first good approximation of e + int e = (int) Math.floor(Math.log10(v)) + 1; + + // Full precision search for e such that 10^(e-1) <= c 2^q < 10^e. + BigDecimal vp = new BigDecimal(v); + BigDecimal low = new BigDecimal(BigInteger.ONE, -(e - 1)); + while (low.compareTo(vp) > 0) { + e -= 1; + low = new BigDecimal(BigInteger.ONE, -(e - 1)); + } + BigDecimal high = new BigDecimal(BigInteger.ONE, -e); + while (vp.compareTo(high) >= 0) { + e += 1; + high = new BigDecimal(BigInteger.ONE, -e); + } + return e; + } + + static long cTiny(int qMin, int kMin) { + BigInteger[] qr = ONE.shiftLeft(-qMin) + .divideAndRemainder(TEN.pow(-(kMin + 1))); + BigInteger cTiny = qr[1].signum() > 0 ? qr[0].add(ONE) : qr[0]; + assertTrue(cTiny.bitLength() < Long.SIZE, "C_TINY"); + return cTiny.longValue(); + } + + void assertTrue() { + if (isOK()) { + return; + } + String msg = "toString applied to the bits " + + hexBits() + + " returns " + + "\"" + s + "\"" + + ", which is not correct according to the specification."; + if (FAILURE_THROWS_EXCEPTION) { + throw new RuntimeException(msg); + } + System.err.println(msg); + } + + /* + Returns whether s syntactically meets the expected output of + toString. It is restricted to finite positive outputs. + It is an unusually long method but rather straightforward, too. + Many conditionals could be merged, but KISS here. + */ + private boolean parse(String t) { + try { + // first determine interesting boundaries in the string + StringReader r = new StringReader(t); + int ch = r.read(); + + int i = 0; + while (ch == '0') { + ++i; + ch = r.read(); + } + // i is just after zeroes starting the integer + + int p = i; + while ('0' <= ch && ch <= '9') { + c = 10 * c + (ch - '0'); + if (c < 0) { + return false; + } + ++len10; + ++p; + ch = r.read(); + } + // p is just after digits ending the integer + + int fz = p; + if (ch == '.') { + ++fz; + ch = r.read(); + } + // fz is just after a decimal '.' + + int f = fz; + while (ch == '0') { + c = 10 * c + (ch - '0'); + if (c < 0) { + return false; + } + ++len10; + ++f; + ch = r.read(); + } + // f is just after zeroes starting the fraction + + if (c == 0) { + len10 = 0; + } + int x = f; + while ('0' <= ch && ch <= '9') { + c = 10 * c + (ch - '0'); + if (c < 0) { + return false; + } + ++len10; + ++x; + ch = r.read(); + } + // x is just after digits ending the fraction + + int g = x; + if (ch == 'E') { + ++g; + ch = r.read(); + } + // g is just after an exponent indicator 'E' + + int ez = g; + if (ch == '-') { + ++ez; + ch = r.read(); + } + // ez is just after a '-' sign in the exponent + + int e = ez; + while (ch == '0') { + ++e; + ch = r.read(); + } + // e is just after zeroes starting the exponent + + int z = e; + while ('0' <= ch && ch <= '9') { + q = 10 * q + (ch - '0'); + if (q < 0) { + return false; + } + ++z; + ch = r.read(); + } + // z is just after digits ending the exponent + + // No other char after the number + if (z != t.length()) { + return false; + } + + // The integer must be present + if (p == 0) { + return false; + } + + // The decimal '.' must be present + if (fz == p) { + return false; + } + + // The fraction must be present + if (x == fz) { + return false; + } + + // The fraction is not 0 or it consists of exactly one 0 + if (f == x && f - fz > 1) { + return false; + } + + // Plain notation, no exponent + if (x == z) { + // At most one 0 starting the integer + if (i > 1) { + return false; + } + + // If the integer is 0, at most 2 zeroes start the fraction + if (i == 1 && f - fz > 2) { + return false; + } + + // The integer cannot have more than 7 digits + if (p > 7) { + return false; + } + + q = fz - x; + + // OK for plain notation + return true; + } + + // Computerized scientific notation + + // The integer has exactly one nonzero digit + if (i != 0 || p != 1) { + return false; + } + + // + // There must be an exponent indicator + if (x == g) { + return false; + } + + // There must be an exponent + if (ez == z) { + return false; + } + + // The exponent must not start with zeroes + if (ez != e) { + return false; + } + + if (g != ez) { + q = -q; + } + + // The exponent must not lie in [-3, 7) + if (-3 <= q && q < 7) { + return false; + } + + q += fz - x; + + // OK for computerized scientific notation + return true; + } catch (IOException ex) { + // An IOException on a StringReader??? Please... + return false; + } + } + + private boolean isOK() { + if (isNaN()) { + return s.equals("NaN"); + } + String t = s; + if (isNegative()) { + if (s.isEmpty() || s.charAt(0) != '-') { + return false; + } + negate(); + t = s.substring(1); + } + if (isInfinity()) { + return t.equals("Infinity"); + } + if (isZero()) { + return t.equals("0.0"); + } + if (!parse(t)) { + return false; + } + if (len10 < 2) { + c *= 10; + q -= 1; + len10 += 1; + } + if (2 > len10 || len10 > maxLen10()) { + return false; + } + + // The exponent is bounded + if (minExp() > q + len10 || q + len10 > maxExp()) { + return false; + } + + // s must recover v + try { + if (!recovers(t)) { + return false; + } + } catch (NumberFormatException e) { + return false; + } + + // Get rid of trailing zeroes, still ensuring at least 2 digits + while (len10 > 2 && c % 10 == 0) { + c /= 10; + q += 1; + len10 -= 1; + } + + if (len10 > 2) { + // Try with a shorter number less than v... + if (recovers(BigDecimal.valueOf(c / 10, -q - 1))) { + return false; + } + + // ... and with a shorter number greater than v + if (recovers(BigDecimal.valueOf(c / 10 + 1, -q - 1))) { + return false; + } + } + + // Try with the decimal predecessor... + BigDecimal dp = c == 10 ? + BigDecimal.valueOf(99, -q + 1) : + BigDecimal.valueOf(c - 1, -q); + if (recovers(dp)) { + BigDecimal bv = toBigDecimal(); + BigDecimal deltav = bv.subtract(BigDecimal.valueOf(c, -q)); + if (deltav.signum() >= 0) { + return true; + } + BigDecimal delta = dp.subtract(bv); + if (delta.signum() >= 0) { + return false; + } + int cmp = deltav.compareTo(delta); + return cmp > 0 || cmp == 0 && (c & 0x1) == 0; + } + + // ... and with the decimal successor + BigDecimal ds = BigDecimal.valueOf(c + 1, -q); + if (recovers(ds)) { + BigDecimal bv = toBigDecimal(); + BigDecimal deltav = bv.subtract(BigDecimal.valueOf(c, -q)); + if (deltav.signum() <= 0) { + return true; + } + BigDecimal delta = ds.subtract(bv); + if (delta.signum() <= 0) { + return false; + } + int cmp = deltav.compareTo(delta); + return cmp < 0 || cmp == 0 && (c & 0x1) == 0; + } + + return true; + } + + abstract BigDecimal toBigDecimal(); + + abstract boolean recovers(BigDecimal b); + + abstract boolean recovers(String s); + + abstract String hexBits(); + + abstract int minExp(); + + abstract int maxExp(); + + abstract int maxLen10(); + + abstract boolean isZero(); + + abstract boolean isInfinity(); + + abstract void negate(); + + abstract boolean isNegative(); + + abstract boolean isNaN(); + +}