StrictMath.java

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 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
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 * This code is free software; you can redistribute it and/or modify it
 * under the terms of the GNU General Public License version 2 only, as
 * published by the Free Software Foundation.  Oracle designates this
 * particular file as subject to the "Classpath" exception as provided
 * by Oracle in the LICENSE file that accompanied this code.
 *
 * This code is distributed in the hope that it will be useful, but WITHOUT
 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
 * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
 * version 2 for more details (a copy is included in the LICENSE file that
 * accompanied this code).
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 * 2 along with this work; if not, write to the Free Software Foundation,
 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
 *
 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
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package java.lang;

import jdk.internal.HotSpotIntrinsicCandidate;
import jdk.internal.math.DoubleConsts;

import java.util.Random;

/**
 * The class {@code StrictMath} contains methods for performing basic
 * numeric operations such as the elementary exponential, logarithm,
 * square root, and trigonometric functions.
 *
 * <p>To help ensure portability of Java programs, the definitions of
 * some of the numeric functions in this package require that they
 * produce the same results as certain published algorithms. These
 * algorithms are available from the well-known network library
 * {@code netlib} as the package "Freely Distributable Math
 * Library," <a
 * href="ftp://ftp.netlib.org/fdlibm.tar">{@code fdlibm}</a>. These
 * algorithms, which are written in the C programming language, are
 * then to be understood as executed with all floating-point
 * operations following the rules of Java floating-point arithmetic.
 *
 * <p>The Java math library is defined with respect to
 * {@code fdlibm} version 5.3. Where {@code fdlibm} provides
 * more than one definition for a function (such as
 * {@code acos}), use the "IEEE 754 core function" version
 * (residing in a file whose name begins with the letter
 * {@code e}).  The methods which require {@code fdlibm}
 * semantics are {@code sin}, {@code cos}, {@code tan},
 * {@code asin}, {@code acos}, {@code atan},
 * {@code exp}, {@code log}, {@code log10},
 * {@code cbrt}, {@code atan2}, {@code pow},
 * {@code sinh}, {@code cosh}, {@code tanh},
 * {@code hypot}, {@code expm1}, and {@code log1p}.
 *
 * <p>
 * The platform uses signed two's complement integer arithmetic with
 * int and long primitive types.  The developer should choose
 * the primitive type to ensure that arithmetic operations consistently
 * produce correct results, which in some cases means the operations
 * will not overflow the range of values of the computation.
 * The best practice is to choose the primitive type and algorithm to avoid
 * overflow. In cases where the size is {@code int} or {@code long} and
 * overflow errors need to be detected, the methods {@code addExact},
 * {@code subtractExact}, {@code multiplyExact}, and {@code toIntExact}
 * throw an {@code ArithmeticException} when the results overflow.
 * For other arithmetic operations such as divide, absolute value,
 * increment by one, decrement by one, and negation overflow occurs only with
 * a specific minimum or maximum value and should be checked against
 * the minimum or maximum as appropriate.
 *
 * @author unascribed
 * @author Joseph D. Darcy
 * @since 1.3
 */
// 数学运算类，相当于一个严格版本的Math类，要求同样的运算在不同平台上结果一致
public final class StrictMath {
    
    /**
     * The {@code double} value that is closer than any other to
     * <i>e</i>, the base of the natural logarithms.
     */
    public static final double E = 2.7182818284590452354;
    
    /**
     * The {@code double} value that is closer than any other to
     * <i>pi</i>, the ratio of the circumference of a circle to its
     * diameter.
     */
    public static final double PI = 3.14159265358979323846;
    
    /**
     * Constant by which to multiply an angular value in degrees to obtain an angular value in radians.
     */
    private static final double DEGREES_TO_RADIANS = 0.017453292519943295;
    
    /**
     * Constant by which to multiply an angular value in radians to obtain an angular value in degrees.
     */
    private static final double RADIANS_TO_DEGREES = 57.29577951308232;
    
    
    
    /**
     * Don't let anyone instantiate this class.
     */
    private StrictMath() {
    }
    
    
    
    /*▼ 基本运算 ████████████████████████████████████████████████████████████████████████████████┓ */
    
    /**
     * Returns the sum of its arguments,
     * throwing an exception if the result overflows an {@code int}.
     *
     * @param x the first value
     * @param y the second value
     *
     * @return the result
     *
     * @throws ArithmeticException if the result overflows an int
     * @see Math#addExact(int, int)
     * @since 1.8
     */
    // 加法
    public static int addExact(int x, int y) {
        return Math.addExact(x, y);
    }
    
    /**
     * Returns the sum of its arguments,
     * throwing an exception if the result overflows a {@code long}.
     *
     * @param x the first value
     * @param y the second value
     *
     * @return the result
     *
     * @throws ArithmeticException if the result overflows a long
     * @see Math#addExact(long, long)
     * @since 1.8
     */
    // 加法
    public static long addExact(long x, long y) {
        return Math.addExact(x, y);
    }
    
    /**
     * Returns the difference of the arguments,
     * throwing an exception if the result overflows an {@code int}.
     *
     * @param x the first value
     * @param y the second value to subtract from the first
     *
     * @return the result
     *
     * @throws ArithmeticException if the result overflows an int
     * @see Math#subtractExact(int, int)
     * @since 1.8
     */
    // 减法
    public static int subtractExact(int x, int y) {
        return Math.subtractExact(x, y);
    }
    
    /**
     * Returns the difference of the arguments,
     * throwing an exception if the result overflows a {@code long}.
     *
     * @param x the first value
     * @param y the second value to subtract from the first
     *
     * @return the result
     *
     * @throws ArithmeticException if the result overflows a long
     * @see Math#subtractExact(long, long)
     * @since 1.8
     */
    // 减法
    public static long subtractExact(long x, long y) {
        return Math.subtractExact(x, y);
    }
    
    /**
     * Returns the product of the arguments,
     * throwing an exception if the result overflows an {@code int}.
     *
     * @param x the first value
     * @param y the second value
     *
     * @return the result
     *
     * @throws ArithmeticException if the result overflows an int
     * @see Math#multiplyExact(int, int)
     * @since 1.8
     */
    // 乘法
    public static int multiplyExact(int x, int y) {
        return Math.multiplyExact(x, y);
    }
    
    /**
     * Returns the product of the arguments, throwing an exception if the result
     * overflows a {@code long}.
     *
     * @param x the first value
     * @param y the second value
     *
     * @return the result
     *
     * @throws ArithmeticException if the result overflows a long
     * @see Math#multiplyExact(long, int)
     * @since 9
     */
    // 乘法
    public static long multiplyExact(long x, int y) {
        return Math.multiplyExact(x, y);
    }
    
    /**
     * Returns the product of the arguments,
     * throwing an exception if the result overflows a {@code long}.
     *
     * @param x the first value
     * @param y the second value
     *
     * @return the result
     *
     * @throws ArithmeticException if the result overflows a long
     * @see Math#multiplyExact(long, long)
     * @since 1.8
     */
    // 乘法
    public static long multiplyExact(long x, long y) {
        return Math.multiplyExact(x, y);
    }
    
    /**
     * Returns the exact mathematical product of the arguments.
     *
     * @param x the first value
     * @param y the second value
     *
     * @return the result
     *
     * @see Math#multiplyFull(int, int)
     * @since 9
     */
    // 乘法
    public static long multiplyFull(int x, int y) {
        return Math.multiplyFull(x, y);
    }
    
    /**
     * Returns as a {@code long} the most significant 64 bits of the 128-bit
     * product of two 64-bit factors.
     *
     * @param x the first value
     * @param y the second value
     *
     * @return the result
     *
     * @see Math#multiplyHigh(long, long)
     * @since 9
     */
    // 乘法，返回两个long乘积的高64位
    public static long multiplyHigh(long x, long y) {
        return Math.multiplyHigh(x, y);
    }
    
    /**
     * Returns the largest (closest to positive infinity)
     * {@code int} value that is less than or equal to the algebraic quotient.
     * There is one special case, if the dividend is the
     * {@linkplain Integer#MIN_VALUE Integer.MIN_VALUE} and the divisor is {@code -1},
     * then integer overflow occurs and
     * the result is equal to the {@code Integer.MIN_VALUE}.
     * <p>
     * See {@link Math#floorDiv(int, int) Math.floorDiv} for examples and
     * a comparison to the integer division {@code /} operator.
     *
     * @param x the dividend
     * @param y the divisor
     *
     * @return the largest (closest to positive infinity)
     * {@code int} value that is less than or equal to the algebraic quotient.
     *
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see Math#floorDiv(int, int)
     * @see Math#floor(double)
     * @since 1.8
     */
    // 除法，如果两数符号不同，则向下取整
    public static int floorDiv(int x, int y) {
        return Math.floorDiv(x, y);
    }
    
    /**
     * Returns the largest (closest to positive infinity)
     * {@code long} value that is less than or equal to the algebraic quotient.
     * There is one special case, if the dividend is the
     * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
     * then integer overflow occurs and
     * the result is equal to {@code Long.MIN_VALUE}.
     * <p>
     * See {@link Math#floorDiv(int, int) Math.floorDiv} for examples and
     * a comparison to the integer division {@code /} operator.
     *
     * @param x the dividend
     * @param y the divisor
     *
     * @return the largest (closest to positive infinity)
     * {@code int} value that is less than or equal to the algebraic quotient.
     *
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see Math#floorDiv(long, int)
     * @see Math#floor(double)
     * @since 9
     */
    // 除法，如果两数符号不同，则向下取整
    public static long floorDiv(long x, int y) {
        return Math.floorDiv(x, y);
    }
    
    /**
     * Returns the largest (closest to positive infinity)
     * {@code long} value that is less than or equal to the algebraic quotient.
     * There is one special case, if the dividend is the
     * {@linkplain Long#MIN_VALUE Long.MIN_VALUE} and the divisor is {@code -1},
     * then integer overflow occurs and
     * the result is equal to the {@code Long.MIN_VALUE}.
     * <p>
     * See {@link Math#floorDiv(int, int) Math.floorDiv} for examples and
     * a comparison to the integer division {@code /} operator.
     *
     * @param x the dividend
     * @param y the divisor
     *
     * @return the largest (closest to positive infinity)
     * {@code long} value that is less than or equal to the algebraic quotient.
     *
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see Math#floorDiv(long, long)
     * @see Math#floor(double)
     * @since 1.8
     */
    // 除法，如果两数符号不同，则向下取整
    public static long floorDiv(long x, long y) {
        return Math.floorDiv(x, y);
    }
    
    /**
     * Returns the floor modulus of the {@code int} arguments.
     * <p>
     * The floor modulus is {@code x - (floorDiv(x, y) * y)},
     * has the same sign as the divisor {@code y}, and
     * is in the range of {@code -abs(y) < r < +abs(y)}.
     * <p>
     * The relationship between {@code floorDiv} and {@code floorMod} is such that:
     * <ul>
     * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
     * </ul>
     * <p>
     * See {@link Math#floorMod(int, int) Math.floorMod} for examples and
     * a comparison to the {@code %} operator.
     *
     * @param x the dividend
     * @param y the divisor
     *
     * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
     *
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see Math#floorMod(int, int)
     * @see StrictMath#floorDiv(int, int)
     * @since 1.8
     */
    // 取余，相当于(x % y + y) % y
    public static int floorMod(int x, int y) {
        return Math.floorMod(x, y);
    }
    
    /**
     * Returns the floor modulus of the {@code long} and {@code int} arguments.
     * <p>
     * The floor modulus is {@code x - (floorDiv(x, y) * y)},
     * has the same sign as the divisor {@code y}, and
     * is in the range of {@code -abs(y) < r < +abs(y)}.
     *
     * <p>
     * The relationship between {@code floorDiv} and {@code floorMod} is such that:
     * <ul>
     * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
     * </ul>
     * <p>
     * See {@link Math#floorMod(int, int) Math.floorMod} for examples and
     * a comparison to the {@code %} operator.
     *
     * @param x the dividend
     * @param y the divisor
     *
     * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
     *
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see Math#floorMod(long, int)
     * @see StrictMath#floorDiv(long, int)
     * @since 9
     */
    // 取余，相当于(x % y + y) % y
    public static int floorMod(long x, int y) {
        return Math.floorMod(x, y);
    }
    
    /**
     * Returns the floor modulus of the {@code long} arguments.
     * <p>
     * The floor modulus is {@code x - (floorDiv(x, y) * y)},
     * has the same sign as the divisor {@code y}, and
     * is in the range of {@code -abs(y) < r < +abs(y)}.
     * <p>
     * The relationship between {@code floorDiv} and {@code floorMod} is such that:
     * <ul>
     * <li>{@code floorDiv(x, y) * y + floorMod(x, y) == x}
     * </ul>
     * <p>
     * See {@link Math#floorMod(int, int) Math.floorMod} for examples and
     * a comparison to the {@code %} operator.
     *
     * @param x the dividend
     * @param y the divisor
     *
     * @return the floor modulus {@code x - (floorDiv(x, y) * y)}
     *
     * @throws ArithmeticException if the divisor {@code y} is zero
     * @see Math#floorMod(long, long)
     * @see StrictMath#floorDiv(long, long)
     * @since 1.8
     */
    // 取余，相当于(x % y + y) % y
    public static long floorMod(long x, long y) {
        return Math.floorMod(x, y);
    }
    
    /**
     * Computes the remainder operation on two arguments as prescribed
     * by the IEEE 754 standard.
     * The remainder value is mathematically equal to
     * <code>f1&nbsp;-&nbsp;f2</code>&nbsp;&times;&nbsp;<i>n</i>,
     * where <i>n</i> is the mathematical integer closest to the exact
     * mathematical value of the quotient {@code f1/f2}, and if two
     * mathematical integers are equally close to {@code f1/f2},
     * then <i>n</i> is the integer that is even. If the remainder is
     * zero, its sign is the same as the sign of the first argument.
     * Special cases:
     * <ul><li>If either argument is NaN, or the first argument is infinite,
     * or the second argument is positive zero or negative zero, then the
     * result is NaN.
     * <li>If the first argument is finite and the second argument is
     * infinite, then the result is the same as the first argument.</ul>
     *
     * @param f1 the dividend.
     * @param f2 the divisor.
     *
     * @return the remainder when {@code f1} is divided by
     * {@code f2}.
     */
    // 取余，返回精确的余数
    public static native double IEEEremainder(double f1, double f2);
    
    /**
     * Returns the fused multiply add of the three arguments; that is,
     * returns the exact product of the first two arguments summed
     * with the third argument and then rounded once to the nearest
     * {@code double}.
     *
     * The rounding is done using the {@linkplain
     * java.math.RoundingMode#HALF_EVEN round to nearest even
     * rounding mode}.
     *
     * In contrast, if {@code a * b + c} is evaluated as a regular
     * floating-point expression, two rounding errors are involved,
     * the first for the multiply operation, the second for the
     * addition operation.
     *
     * <p>Special cases:
     * <ul>
     * <li> If any argument is NaN, the result is NaN.
     *
     * <li> If one of the first two arguments is infinite and the
     * other is zero, the result is NaN.
     *
     * <li> If the exact product of the first two arguments is infinite
     * (in other words, at least one of the arguments is infinite and
     * the other is neither zero nor NaN) and the third argument is an
     * infinity of the opposite sign, the result is NaN.
     *
     * </ul>
     *
     * <p>Note that {@code fusedMac(a, 1.0, c)} returns the same
     * result as ({@code a + c}).  However,
     * {@code fusedMac(a, b, +0.0)} does <em>not</em> always return the
     * same result as ({@code a * b}) since
     * {@code fusedMac(-0.0, +0.0, +0.0)} is {@code +0.0} while
     * ({@code -0.0 * +0.0}) is {@code -0.0}; {@code fusedMac(a, b, -0.0)} is
     * equivalent to ({@code a * b}) however.
     *
     * @param a a value
     * @param b a value
     * @param c a value
     *
     * @return (< i > a < / i > & nbsp ; & times ; & nbsp ; < i > b < / i > & nbsp ; + & nbsp ; < i > c < / i >)
     * computed, as if with unlimited range and precision, and rounded
     * once to the nearest {@code double} value
     *
     * @apiNote This method corresponds to the fusedMultiplyAdd
     * operation defined in IEEE 754-2008.
     * @since 9
     */
    // 计算a*b+c
    public static double fma(double a, double b, double c) {
        return Math.fma(a, b, c);
    }
    
    /**
     * Returns the fused multiply add of the three arguments; that is,
     * returns the exact product of the first two arguments summed
     * with the third argument and then rounded once to the nearest
     * {@code float}.
     *
     * The rounding is done using the {@linkplain
     * java.math.RoundingMode#HALF_EVEN round to nearest even
     * rounding mode}.
     *
     * In contrast, if {@code a * b + c} is evaluated as a regular
     * floating-point expression, two rounding errors are involved,
     * the first for the multiply operation, the second for the
     * addition operation.
     *
     * <p>Special cases:
     * <ul>
     * <li> If any argument is NaN, the result is NaN.
     *
     * <li> If one of the first two arguments is infinite and the
     * other is zero, the result is NaN.
     *
     * <li> If the exact product of the first two arguments is infinite
     * (in other words, at least one of the arguments is infinite and
     * the other is neither zero nor NaN) and the third argument is an
     * infinity of the opposite sign, the result is NaN.
     *
     * </ul>
     *
     * <p>Note that {@code fma(a, 1.0f, c)} returns the same
     * result as ({@code a + c}).  However,
     * {@code fma(a, b, +0.0f)} does <em>not</em> always return the
     * same result as ({@code a * b}) since
     * {@code fma(-0.0f, +0.0f, +0.0f)} is {@code +0.0f} while
     * ({@code -0.0f * +0.0f}) is {@code -0.0f}; {@code fma(a, b, -0.0f)} is
     * equivalent to ({@code a * b}) however.
     *
     * @param a a value
     * @param b a value
     * @param c a value
     *
     * @return (< i > a < / i > & nbsp ; & times ; & nbsp ; < i > b < / i > & nbsp ; + & nbsp ; < i > c < / i >)
     * computed, as if with unlimited range and precision, and rounded
     * once to the nearest {@code float} value
     *
     * @apiNote This method corresponds to the fusedMultiplyAdd
     * operation defined in IEEE 754-2008.
     * @since 9
     */
    // 计算a*b+c
    public static float fma(float a, float b, float c) {
        return Math.fma(a, b, c);
    }
    
    /*▲ 基本运算 ████████████████████████████████████████████████████████████████████████████████┛ */
    
    
    
    /*▼ 乘方/开方 ████████████████████████████████████████████████████████████████████████████████┓ */
    
    /**
     * Returns the value of the first argument raised to the power of the
     * second argument. Special cases:
     *
     * <ul><li>If the second argument is positive or negative zero, then the
     * result is 1.0.
     * <li>If the second argument is 1.0, then the result is the same as the
     * first argument.
     * <li>If the second argument is NaN, then the result is NaN.
     * <li>If the first argument is NaN and the second argument is nonzero,
     * then the result is NaN.
     *
     * <li>If
     * <ul>
     * <li>the absolute value of the first argument is greater than 1
     * and the second argument is positive infinity, or
     * <li>the absolute value of the first argument is less than 1 and
     * the second argument is negative infinity,
     * </ul>
     * then the result is positive infinity.
     *
     * <li>If
     * <ul>
     * <li>the absolute value of the first argument is greater than 1 and
     * the second argument is negative infinity, or
     * <li>the absolute value of the
     * first argument is less than 1 and the second argument is positive
     * infinity,
     * </ul>
     * then the result is positive zero.
     *
     * <li>If the absolute value of the first argument equals 1 and the
     * second argument is infinite, then the result is NaN.
     *
     * <li>If
     * <ul>
     * <li>the first argument is positive zero and the second argument
     * is greater than zero, or
     * <li>the first argument is positive infinity and the second
     * argument is less than zero,
     * </ul>
     * then the result is positive zero.
     *
     * <li>If
     * <ul>
     * <li>the first argument is positive zero and the second argument
     * is less than zero, or
     * <li>the first argument is positive infinity and the second
     * argument is greater than zero,
     * </ul>
     * then the result is positive infinity.
     *
     * <li>If
     * <ul>
     * <li>the first argument is negative zero and the second argument
     * is greater than zero but not a finite odd integer, or
     * <li>the first argument is negative infinity and the second
     * argument is less than zero but not a finite odd integer,
     * </ul>
     * then the result is positive zero.
     *
     * <li>If
     * <ul>
     * <li>the first argument is negative zero and the second argument
     * is a positive finite odd integer, or
     * <li>the first argument is negative infinity and the second
     * argument is a negative finite odd integer,
     * </ul>
     * then the result is negative zero.
     *
     * <li>If
     * <ul>
     * <li>the first argument is negative zero and the second argument
     * is less than zero but not a finite odd integer, or
     * <li>the first argument is negative infinity and the second
     * argument is greater than zero but not a finite odd integer,
     * </ul>
     * then the result is positive infinity.
     *
     * <li>If
     * <ul>
     * <li>the first argument is negative zero and the second argument
     * is a negative finite odd integer, or
     * <li>the first argument is negative infinity and the second
     * argument is a positive finite odd integer,
     * </ul>
     * then the result is negative infinity.
     *
     * <li>If the first argument is finite and less than zero
     * <ul>
     * <li> if the second argument is a finite even integer, the
     * result is equal to the result of raising the absolute value of
     * the first argument to the power of the second argument
     *
     * <li>if the second argument is a finite odd integer, the result
     * is equal to the negative of the result of raising the absolute
     * value of the first argument to the power of the second
     * argument
     *
     * <li>if the second argument is finite and not an integer, then
     * the result is NaN.
     * </ul>
     *
     * <li>If both arguments are integers, then the result is exactly equal
     * to the mathematical result of raising the first argument to the power
     * of the second argument if that result can in fact be represented
     * exactly as a {@code double} value.</ul>
     *
     * <p>(In the foregoing descriptions, a floating-point value is
     * considered to be an integer if and only if it is finite and a
     * fixed point of the method {@link #ceil ceil} or,
     * equivalently, a fixed point of the method {@link #floor
     * floor}. A value is a fixed point of a one-argument
     * method if and only if the result of applying the method to the
     * value is equal to the value.)
     *
     * @param a base.
     * @param b the exponent.
     *
     * @return the value {@code a}<sup>{@code b}</sup>.
     */
    // a的b次方
    public static double pow(double a, double b) {
        return FdLibm.Pow.compute(a, b);
    }
    
    /**
     * Returns the correctly rounded positive square root of a
     * {@code double} value.
     * Special cases:
     * <ul><li>If the argument is NaN or less than zero, then the result
     * is NaN.
     * <li>If the argument is positive infinity, then the result is positive
     * infinity.
     * <li>If the argument is positive zero or negative zero, then the
     * result is the same as the argument.</ul>
     * Otherwise, the result is the {@code double} value closest to
     * the true mathematical square root of the argument value.
     *
     * @param a a value.
     *
     * @return the positive square root of {@code a}.
     */
    // 平方根
    @HotSpotIntrinsicCandidate
    public static native double sqrt(double a);
    
    /**
     * Returns the cube root of a {@code double} value.  For
     * positive finite {@code x}, {@code cbrt(-x) ==
     * -cbrt(x)}; that is, the cube root of a negative value is
     * the negative of the cube root of that value's magnitude.
     * Special cases:
     *
     * <ul>
     *
     * <li>If the argument is NaN, then the result is NaN.
     *
     * <li>If the argument is infinite, then the result is an infinity
     * with the same sign as the argument.
     *
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.
     *
     * </ul>
     *
     * @param a a value.
     *
     * @return the cube root of {@code a}.
     *
     * @since 1.5
     */
    // 立方根
    public static double cbrt(double a) {
        return FdLibm.Cbrt.compute(a);
    }
    
    /*▲ 乘方/开方 ████████████████████████████████████████████████████████████████████████████████┛ */
    
    
    
    /*▼ 指数/对数 ████████████████████████████████████████████████████████████████████████████████┓ */
    
    /**
     * Returns Euler's number <i>e</i> raised to the power of a
     * {@code double} value. Special cases:
     * <ul><li>If the argument is NaN, the result is NaN.
     * <li>If the argument is positive infinity, then the result is
     * positive infinity.
     * <li>If the argument is negative infinity, then the result is
     * positive zero.</ul>
     *
     * @param a the exponent to raise <i>e</i> to.
     *
     * @return the value <i>e</i><sup>{@code a}</sup>,
     * where <i>e</i> is the base of the natural logarithms.
     */
    // e的a次方
    public static double exp(double a) {
        return FdLibm.Exp.compute(a);
    }
    
    /**
     * Returns the natural logarithm (base <i>e</i>) of a {@code double}
     * value. Special cases:
     * <ul><li>If the argument is NaN or less than zero, then the result
     * is NaN.
     * <li>If the argument is positive infinity, then the result is
     * positive infinity.
     * <li>If the argument is positive zero or negative zero, then the
     * result is negative infinity.</ul>
     *
     * @param a a value
     *
     * @return the value ln&nbsp;{@code a}, the natural logarithm of
     * {@code a}.
     */
    // 以e为底a的对数，自然对数
    public static native double log(double a);
    
    /**
     * Returns the base 10 logarithm of a {@code double} value.
     * Special cases:
     *
     * <ul><li>If the argument is NaN or less than zero, then the result
     * is NaN.
     * <li>If the argument is positive infinity, then the result is
     * positive infinity.
     * <li>If the argument is positive zero or negative zero, then the
     * result is negative infinity.
     * <li> If the argument is equal to 10<sup><i>n</i></sup> for
     * integer <i>n</i>, then the result is <i>n</i>.
     * </ul>
     *
     * @param a a value
     *
     * @return the base 10 logarithm of  {@code a}.
     *
     * @since 1.5
     */
    // 以10为底a的对数
    public static native double log10(double a);
    
    /**
     * Returns <i>e</i><sup>x</sup>&nbsp;-1.  Note that for values of
     * <i>x</i> near 0, the exact sum of
     * {@code expm1(x)}&nbsp;+&nbsp;1 is much closer to the true
     * result of <i>e</i><sup>x</sup> than {@code exp(x)}.
     *
     * <p>Special cases:
     * <ul>
     * <li>If the argument is NaN, the result is NaN.
     *
     * <li>If the argument is positive infinity, then the result is
     * positive infinity.
     *
     * <li>If the argument is negative infinity, then the result is
     * -1.0.
     *
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.
     *
     * </ul>
     *
     * @param x the exponent to raise <i>e</i> to in the computation of
     *          <i>e</i><sup>{@code x}</sup>&nbsp;-1.
     *
     * @return the value <i>e</i><sup>{@code x}</sup>&nbsp;-&nbsp;1.
     *
     * @since 1.5
     */
    // e的x次方减1
    public static native double expm1(double x);
    
    /**
     * Returns the natural logarithm of the sum of the argument and 1.
     * Note that for small values {@code x}, the result of
     * {@code log1p(x)} is much closer to the true result of ln(1
     * + {@code x}) than the floating-point evaluation of
     * {@code log(1.0+x)}.
     *
     * <p>Special cases:
     * <ul>
     *
     * <li>If the argument is NaN or less than -1, then the result is
     * NaN.
     *
     * <li>If the argument is positive infinity, then the result is
     * positive infinity.
     *
     * <li>If the argument is negative one, then the result is
     * negative infinity.
     *
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.
     *
     * </ul>
     *
     * @param x a value
     *
     * @return the value ln({@code x}&nbsp;+&nbsp;1), the natural
     * log of {@code x}&nbsp;+&nbsp;1
     *
     * @since 1.5
     */
    // 以e为底x+1的对数
    public static native double log1p(double x);
    
    
    /**
     * Returns {@code d} &times;
     * 2<sup>{@code scaleFactor}</sup> rounded as if performed
     * by a single correctly rounded floating-point multiply to a
     * member of the double value set.  See the Java
     * Language Specification for a discussion of floating-point
     * value sets.  If the exponent of the result is between {@link
     * Double#MIN_EXPONENT} and {@link Double#MAX_EXPONENT}, the
     * answer is calculated exactly.  If the exponent of the result
     * would be larger than {@code Double.MAX_EXPONENT}, an
     * infinity is returned.  Note that if the result is subnormal,
     * precision may be lost; that is, when {@code scalb(x, n)}
     * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
     * <i>x</i>.  When the result is non-NaN, the result has the same
     * sign as {@code d}.
     *
     * <p>Special cases:
     * <ul>
     * <li> If the first argument is NaN, NaN is returned.
     * <li> If the first argument is infinite, then an infinity of the
     * same sign is returned.
     * <li> If the first argument is zero, then a zero of the same
     * sign is returned.
     * </ul>
     *
     * @param d           number to be scaled by a power of two.
     * @param scaleFactor power of 2 used to scale {@code d}
     *
     * @return {@code d} &times; 2<sup>{@code scaleFactor}</sup>
     *
     * @since 1.6
     */
    // d乘以2的scaleFactor次方
    public static double scalb(double d, int scaleFactor) {
        return Math.scalb(d, scaleFactor);
    }
    
    /**
     * Returns {@code f} &times;
     * 2<sup>{@code scaleFactor}</sup> rounded as if performed
     * by a single correctly rounded floating-point multiply to a
     * member of the float value set.  See the Java
     * Language Specification for a discussion of floating-point
     * value sets.  If the exponent of the result is between {@link
     * Float#MIN_EXPONENT} and {@link Float#MAX_EXPONENT}, the
     * answer is calculated exactly.  If the exponent of the result
     * would be larger than {@code Float.MAX_EXPONENT}, an
     * infinity is returned.  Note that if the result is subnormal,
     * precision may be lost; that is, when {@code scalb(x, n)}
     * is subnormal, {@code scalb(scalb(x, n), -n)} may not equal
     * <i>x</i>.  When the result is non-NaN, the result has the same
     * sign as {@code f}.
     *
     * <p>Special cases:
     * <ul>
     * <li> If the first argument is NaN, NaN is returned.
     * <li> If the first argument is infinite, then an infinity of the
     * same sign is returned.
     * <li> If the first argument is zero, then a zero of the same
     * sign is returned.
     * </ul>
     *
     * @param f           number to be scaled by a power of two.
     * @param scaleFactor power of 2 used to scale {@code f}
     *
     * @return {@code f} &times; 2<sup>{@code scaleFactor}</sup>
     *
     * @since 1.6
     */
    // f乘以2的scaleFactor次方
    public static float scalb(float f, int scaleFactor) {
        return Math.scalb(f, scaleFactor);
    }
    
    /*▲ 指数/对数 ████████████████████████████████████████████████████████████████████████████████┛ */
    
    
    
    /*▼ 平面坐标 ████████████████████████████████████████████████████████████████████████████████┓ */
    
    /**
     * Converts an angle measured in degrees to an approximately
     * equivalent angle measured in radians.  The conversion from
     * degrees to radians is generally inexact.
     *
     * @param angdeg an angle, in degrees
     *
     * @return the measurement of the angle {@code angdeg}
     * in radians.
     */
    // 角度 --> 弧度
    public static strictfp double toRadians(double angdeg) {
        // Do not delegate to Math.toRadians(angdeg) because this method has the strictfp modifier.
        return angdeg * DEGREES_TO_RADIANS;
    }
    
    /**
     * Converts an angle measured in radians to an approximately
     * equivalent angle measured in degrees.  The conversion from
     * radians to degrees is generally inexact; users should
     * <i>not</i> expect {@code cos(toRadians(90.0))} to exactly
     * equal {@code 0.0}.
     *
     * @param angrad an angle, in radians
     *
     * @return the measurement of the angle {@code angrad}
     * in degrees.
     */
    // 弧度 --> 角度
    public static strictfp double toDegrees(double angrad) {
        // Do not delegate to Math.toDegrees(angrad) because this method has the strictfp modifier.
        return angrad * RADIANS_TO_DEGREES;
    }
    
    /**
     * Returns sqrt(<i>x</i><sup>2</sup>&nbsp;+<i>y</i><sup>2</sup>)
     * without intermediate overflow or underflow.
     *
     * <p>Special cases:
     * <ul>
     *
     * <li> If either argument is infinite, then the result
     * is positive infinity.
     *
     * <li> If either argument is NaN and neither argument is infinite,
     * then the result is NaN.
     *
     * </ul>
     *
     * @param x a value
     * @param y a value
     *
     * @return sqrt(< i > x < / i > < sup > 2 < / sup > & nbsp ; + < i > y < / i > < sup > 2 < / sup >)
     * without intermediate overflow or underflow
     *
     * @since 1.5
     */
    // 根号下x与y的平方和，可用于计算直角三角形的斜边长
    public static double hypot(double x, double y) {
        return FdLibm.Hypot.compute(x, y);
    }
    
    /**
     * Returns the angle <i>theta</i> from the conversion of rectangular
     * coordinates ({@code x},&nbsp;{@code y}) to polar
     * coordinates (r,&nbsp;<i>theta</i>).
     * This method computes the phase <i>theta</i> by computing an arc tangent
     * of {@code y/x} in the range of -<i>pi</i> to <i>pi</i>. Special
     * cases:
     * <ul><li>If either argument is NaN, then the result is NaN.
     * <li>If the first argument is positive zero and the second argument
     * is positive, or the first argument is positive and finite and the
     * second argument is positive infinity, then the result is positive
     * zero.
     * <li>If the first argument is negative zero and the second argument
     * is positive, or the first argument is negative and finite and the
     * second argument is positive infinity, then the result is negative zero.
     * <li>If the first argument is positive zero and the second argument
     * is negative, or the first argument is positive and finite and the
     * second argument is negative infinity, then the result is the
     * {@code double} value closest to <i>pi</i>.
     * <li>If the first argument is negative zero and the second argument
     * is negative, or the first argument is negative and finite and the
     * second argument is negative infinity, then the result is the
     * {@code double} value closest to -<i>pi</i>.
     * <li>If the first argument is positive and the second argument is
     * positive zero or negative zero, or the first argument is positive
     * infinity and the second argument is finite, then the result is the
     * {@code double} value closest to <i>pi</i>/2.
     * <li>If the first argument is negative and the second argument is
     * positive zero or negative zero, or the first argument is negative
     * infinity and the second argument is finite, then the result is the
     * {@code double} value closest to -<i>pi</i>/2.
     * <li>If both arguments are positive infinity, then the result is the
     * {@code double} value closest to <i>pi</i>/4.
     * <li>If the first argument is positive infinity and the second argument
     * is negative infinity, then the result is the {@code double}
     * value closest to 3*<i>pi</i>/4.
     * <li>If the first argument is negative infinity and the second argument
     * is positive infinity, then the result is the {@code double} value
     * closest to -<i>pi</i>/4.
     * <li>If both arguments are negative infinity, then the result is the
     * {@code double} value closest to -3*<i>pi</i>/4.</ul>
     *
     * @param y the ordinate coordinate
     * @param x the abscissa coordinate
     *
     * @return the <i>theta</i> component of the point
     * (<i>r</i>,&nbsp;<i>theta</i>)
     * in polar coordinates that corresponds to the point
     * (<i>x</i>,&nbsp;<i>y</i>) in Cartesian coordinates.
     */
    // 计算y/x的反正切值。可用在将矩形坐标(x, y)转换成极坐标(r, θ)时，计算逆时针旋转的角度θ。
    public static native double atan2(double y, double x);
    
    /*▲ 平面坐标 ████████████████████████████████████████████████████████████████████████████████┛ */
    
    
    
    /*▼ 三角函数 ████████████████████████████████████████████████████████████████████████████████┓ */
    
    /**
     * Returns the trigonometric sine of an angle. Special cases:
     * <ul><li>If the argument is NaN or an infinity, then the
     * result is NaN.
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.</ul>
     *
     * @param a an angle, in radians.
     *
     * @return the sine of the argument.
     */
    // 正弦，参数是弧度
    public static native double sin(double a);
    
    /**
     * Returns the trigonometric cosine of an angle. Special cases:
     * <ul><li>If the argument is NaN or an infinity, then the
     * result is NaN.</ul>
     *
     * @param a an angle, in radians.
     *
     * @return the cosine of the argument.
     */
    // 余弦，参数是弧度
    public static native double cos(double a);
    
    /**
     * Returns the trigonometric tangent of an angle. Special cases:
     * <ul><li>If the argument is NaN or an infinity, then the result
     * is NaN.
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.</ul>
     *
     * @param a an angle, in radians.
     *
     * @return the tangent of the argument.
     */
    // 正切，参数是弧度
    public static native double tan(double a);
    
    /**
     * Returns the arc sine of a value; the returned angle is in the
     * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:
     * <ul><li>If the argument is NaN or its absolute value is greater
     * than 1, then the result is NaN.
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.</ul>
     *
     * @param a the value whose arc sine is to be returned.
     *
     * @return the arc sine of the argument.
     */
    // 反正弦，返回值是弧度
    public static native double asin(double a);
    
    /**
     * Returns the arc cosine of a value; the returned angle is in the
     * range 0.0 through <i>pi</i>.  Special case:
     * <ul><li>If the argument is NaN or its absolute value is greater
     * than 1, then the result is NaN.</ul>
     *
     * @param a the value whose arc cosine is to be returned.
     *
     * @return the arc cosine of the argument.
     */
    // 反余弦，返回值是弧度
    public static native double acos(double a);
    
    /**
     * Returns the arc tangent of a value; the returned angle is in the
     * range -<i>pi</i>/2 through <i>pi</i>/2.  Special cases:
     * <ul><li>If the argument is NaN, then the result is NaN.
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.</ul>
     *
     * @param a the value whose arc tangent is to be returned.
     *
     * @return the arc tangent of the argument.
     */
    // 反正切，返回值是弧度
    public static native double atan(double a);
    
    /*▲ 三角函数 ████████████████████████████████████████████████████████████████████████████████┛ */
    
    
    
    /*▼ 双曲函数 ████████████████████████████████████████████████████████████████████████████████┓ */
    
    /**
     * Returns the hyperbolic sine of a {@code double} value.
     * The hyperbolic sine of <i>x</i> is defined to be
     * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/2
     * where <i>e</i> is {@linkplain Math#E Euler's number}.
     *
     * <p>Special cases:
     * <ul>
     *
     * <li>If the argument is NaN, then the result is NaN.
     *
     * <li>If the argument is infinite, then the result is an infinity
     * with the same sign as the argument.
     *
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.
     *
     * </ul>
     *
     * @param x The number whose hyperbolic sine is to be returned.
     *
     * @return The hyperbolic sine of {@code x}.
     *
     * @since 1.5
     */
    // 双曲正弦函数
    public static native double sinh(double x);
    
    /**
     * Returns the hyperbolic cosine of a {@code double} value.
     * The hyperbolic cosine of <i>x</i> is defined to be
     * (<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>)/2
     * where <i>e</i> is {@linkplain Math#E Euler's number}.
     *
     * <p>Special cases:
     * <ul>
     *
     * <li>If the argument is NaN, then the result is NaN.
     *
     * <li>If the argument is infinite, then the result is positive
     * infinity.
     *
     * <li>If the argument is zero, then the result is {@code 1.0}.
     *
     * </ul>
     *
     * @param x The number whose hyperbolic cosine is to be returned.
     *
     * @return The hyperbolic cosine of {@code x}.
     *
     * @since 1.5
     */
    // 双曲余弦函数
    public static native double cosh(double x);
    
    /**
     * Returns the hyperbolic tangent of a {@code double} value.
     * The hyperbolic tangent of <i>x</i> is defined to be
     * (<i>e<sup>x</sup>&nbsp;-&nbsp;e<sup>-x</sup></i>)/(<i>e<sup>x</sup>&nbsp;+&nbsp;e<sup>-x</sup></i>),
     * in other words, {@linkplain Math#sinh
     * sinh(<i>x</i>)}/{@linkplain Math#cosh cosh(<i>x</i>)}.  Note
     * that the absolute value of the exact tanh is always less than
     * 1.
     *
     * <p>Special cases:
     * <ul>
     *
     * <li>If the argument is NaN, then the result is NaN.
     *
     * <li>If the argument is zero, then the result is a zero with the
     * same sign as the argument.
     *
     * <li>If the argument is positive infinity, then the result is
     * {@code +1.0}.
     *
     * <li>If the argument is negative infinity, then the result is
     * {@code -1.0}.
     *
     * </ul>
     *
     * @param x The number whose hyperbolic tangent is to be returned.
     *
     * @return The hyperbolic tangent of {@code x}.
     *
     * @since 1.5
     */
    // 双曲正切函数
    public static native double tanh(double x);
    
    /*▲ 双曲函数 ████████████████████████████████████████████████████████████████████████████████┛ */
    
    
    
    /*▼ 舍入操作 ████████████████████████████████████████████████████████████████████████████████┓ */
    
    /**
     * Returns the smallest (closest to negative infinity)
     * {@code double} value that is greater than or equal to the
     * argument and is equal to a mathematical integer. Special cases:
     * <ul><li>If the argument value is already equal to a
     * mathematical integer, then the result is the same as the
     * argument.  <li>If the argument is NaN or an infinity or
     * positive zero or negative zero, then the result is the same as
     * the argument.  <li>If the argument value is less than zero but
     * greater than -1.0, then the result is negative zero.</ul> Note
     * that the value of {@code StrictMath.ceil(x)} is exactly the
     * value of {@code -StrictMath.floor(-x)}.
     *
     * @param a a value.
     *
     * @return the smallest (closest to negative infinity)
     * floating-point value that is greater than or equal to
     * the argument and is equal to a mathematical integer.
     */
    // 向上取整，大于a的最小整数
    public static double ceil(double a) {
        return floorOrCeil(a, -0.0, 1.0, 1.0);
    }
    
    /**
     * Returns the largest (closest to positive infinity)
     * {@code double} value that is less than or equal to the
     * argument and is equal to a mathematical integer. Special cases:
     * <ul><li>If the argument value is already equal to a
     * mathematical integer, then the result is the same as the
     * argument.  <li>If the argument is NaN or an infinity or
     * positive zero or negative zero, then the result is the same as
     * the argument.</ul>
     *
     * @param a a value.
     *
     * @return the largest (closest to positive infinity)
     * floating-point value that less than or equal to the argument
     * and is equal to a mathematical integer.
     */
    // 向下取整，小于a的最大整数
    public static double floor(double a) {
        return floorOrCeil(a, -1.0, 0.0, -1.0);
    }
    
    /**
     * Returns the closest {@code int} to the argument, with ties
     * rounding to positive infinity.
     *
     * <p>Special cases:
     * <ul><li>If the argument is NaN, the result is 0.
     * <li>If the argument is negative infinity or any value less than or
     * equal to the value of {@code Integer.MIN_VALUE}, the result is
     * equal to the value of {@code Integer.MIN_VALUE}.
     * <li>If the argument is positive infinity or any value greater than or
     * equal to the value of {@code Integer.MAX_VALUE}, the result is
     * equal to the value of {@code Integer.MAX_VALUE}.</ul>
     *
     * @param a a floating-point value to be rounded to an integer.
     *
     * @return the value of the argument rounded to the nearest
     * {@code int} value.
     *
     * @see java.lang.Integer#MAX_VALUE
     * @see java.lang.Integer#MIN_VALUE
     */
    /*
     * 返回离a最近的整数，如果a位于两个整数之间，向大数舍入
     *
     * 举例：
     * round(1.4)==1，round(1.5)==2
     * round(-1.5)==-1，round(-1.6)==-2
     */
    public static int round(float a) {
        return Math.round(a);
    }
    
    /**
     * Returns the closest {@code long} to the argument, with ties
     * rounding to positive infinity.
     *
     * <p>Special cases:
     * <ul><li>If the argument is NaN, the result is 0.
     * <li>If the argument is negative infinity or any value less than or
     * equal to the value of {@code Long.MIN_VALUE}, the result is
     * equal to the value of {@code Long.MIN_VALUE}.
     * <li>If the argument is positive infinity or any value greater than or
     * equal to the value of {@code Long.MAX_VALUE}, the result is
     * equal to the value of {@code Long.MAX_VALUE}.</ul>
     *
     * @param a a floating-point value to be rounded to a
     *          {@code long}.
     *
     * @return the value of the argument rounded to the nearest
     * {@code long} value.
     *
     * @see java.lang.Long#MAX_VALUE
     * @see java.lang.Long#MIN_VALUE
     */
    // 返回离a最近的整数，如果a位于两个整数之间，向大数舍入
    public static long round(double a) {
        return Math.round(a);
    }
    
    /**
     * Returns the {@code double} value that is closest in value
     * to the argument and is equal to a mathematical integer. If two
     * {@code double} values that are mathematical integers are
     * equally close to the value of the argument, the result is the
     * integer value that is even. Special cases:
     * <ul><li>If the argument value is already equal to a mathematical
     * integer, then the result is the same as the argument.
     * <li>If the argument is NaN or an infinity or positive zero or negative
     * zero, then the result is the same as the argument.</ul>
     *
     * @param a a value.
     *
     * @return the closest floating-point value to {@code a} that is
     * equal to a mathematical integer.
     *
     * @author Joseph D. Darcy
     */
    /*
     * 返回离a最近的整数，如果a位于两个整数之间，向偶数舍入
     *
     * 举例：
     * rint(2.4) = 2.0，rint(2.6) = 3.0
     * rint(2.5) = 2.0，rint(3.5) = 4.0
     * rint(-2.4) = -2.0，rint(-2.6) = -3.0
     * rint(-2.5) = -2.0，rint(-3.5) = -4.0
     */
    public static double rint(double a) {
        /*
         * If the absolute value of a is not less than 2^52, it
         * is either a finite integer (the double format does not have
         * enough significand bits for a number that large to have any
         * fractional portion), an infinity, or a NaN.  In any of
         * these cases, rint of the argument is the argument.
         *
         * Otherwise, the sum (twoToThe52 + a ) will properly round
         * away any fractional portion of a since ulp(twoToThe52) ==
         * 1.0; subtracting out twoToThe52 from this sum will then be
         * exact and leave the rounded integer portion of a.
         *
         * This method does *not* need to be declared strictfp to get
         * fully reproducible results.  Whether or not a method is
         * declared strictfp can only make a difference in the
         * returned result if some operation would overflow or
         * underflow with strictfp semantics.  The operation
         * (twoToThe52 + a ) cannot overflow since large values of a
         * are screened out; the add cannot underflow since twoToThe52
         * is too large.  The subtraction ((twoToThe52 + a ) -
         * twoToThe52) will be exact as discussed above and thus
         * cannot overflow or meaningfully underflow.  Finally, the
         * last multiply in the return statement is by plus or minus
         * 1.0, which is exact too.
         */
        double twoToThe52 = (double) (1L << 52); // 2^52
        double sign = Math.copySign(1.0, a); // preserve sign info
        a = Math.abs(a);
        
        if(a<twoToThe52) { // E_min <= ilogb(a) <= 51
            a = ((twoToThe52 + a) - twoToThe52);
        }
        
        return sign * a; // restore original sign
    }
    
    /**
     * Internal method to share logic between floor and ceil.
     *
     * @param a                the value to be floored or ceiled
     * @param negativeBoundary result for values in (-1, 0)
     * @param positiveBoundary result for values in (0, 1)
     * @param sign             value to add when the argument is non-integral
     */
    // 内部使用，实现floor和Ceil
    private static double floorOrCeil(double a, double negativeBoundary, double positiveBoundary, double sign) {
        int exponent = Math.getExponent(a);
        
        if(exponent<0) {
            /*
             * Absolute value of argument is less than 1.
             * floorOrceil(-0.0) => -0.0
             * floorOrceil(+0.0) => +0.0
             */
            return ((a == 0.0) ? a : ((a<0.0) ? negativeBoundary : positiveBoundary));
        } else if(exponent >= 52) {
            /*
             * Infinity, NaN, or a value so large it must be integral.
             */
            return a;
        }
        
        // Else the argument is either an integral value already XOR it has to be rounded to one.
        assert exponent >= 0 && exponent<=51;
        
        long doppel = Double.doubleToRawLongBits(a);
        long mask = DoubleConsts.SIGNIF_BIT_MASK >> exponent;
        
        if((mask & doppel) == 0L)
            return a; // integral value
        else {
            double result = Double.longBitsToDouble(doppel & (~mask));
            if(sign * a>0.0)
                result = result + sign;
            return result;
        }
    }
    
    /*▲ 舍入操作 ████████████████████████████████████████████████████████████████████████████████┛ */
    
    
    
    /*▼ 绝对值 ████████████████████████████████████████████████████████████████████████████████┓ */
    
    /**
     * Returns the absolute value of an {@code int} value.
     * If the argument is not negative, the argument is returned.
     * If the argument is negative, the negation of the argument is returned.
     *
     * <p>Note that if the argument is equal to the value of
     * {@link Integer#MIN_VALUE}, the most negative representable
     * {@code int} value, the result is that same value, which is
     * negative.
     *
     * @param a the  argument whose absolute value is to be determined.
     *
     * @return the absolute value of the argument.
     */
    // 绝对值
    public static int abs(int a) {
        return Math.abs(a);
    }
    
    /**
     * Returns the absolute value of a {@code long} value.
     * If the argument is not negative, the argument is returned.
     * If the argument is negative, the negation of the argument is returned.
     *
     * <p>Note that if the argument is equal to the value of
     * {@link Long#MIN_VALUE}, the most negative representable
     * {@code long} value, the result is that same value, which
     * is negative.
     *
     * @param a the  argument whose absolute value is to be determined.
     *
     * @return the absolute value of the argument.
     */
    // 绝对值
    public static long abs(long a) {
        return Math.abs(a);
    }
    
    /**
     * Returns the absolute value of a {@code float} value.
     * If the argument is not negative, the argument is returned.
     * If the argument is negative, the negation of the argument is returned.
     * Special cases:
     * <ul><li>If the argument is positive zero or negative zero, the
     * result is positive zero.
     * <li>If the argument is infinite, the result is positive infinity.
     * <li>If the argument is NaN, the result is NaN.</ul>
     *
     * @param a the argument whose absolute value is to be determined
     *
     * @return the absolute value of the argument.
     *
     * @apiNote As implied by the above, one valid implementation of
     * this method is given by the expression below which computes a
     * {@code float} with the same exponent and significand as the
     * argument but with a guaranteed zero sign bit indicating a
     * positive value: <br>
     * {@code Float.intBitsToFloat(0x7fffffff & Float.floatToRawIntBits(a))}
     */
    // 绝对值
    public static float abs(float a) {
        return Math.abs(a);
    }
    
    /**
     * Returns the absolute value of a {@code double} value.
     * If the argument is not negative, the argument is returned.
     * If the argument is negative, the negation of the argument is returned.
     * Special cases:
     * <ul><li>If the argument is positive zero or negative zero, the result
     * is positive zero.
     * <li>If the argument is infinite, the result is positive infinity.
     * <li>If the argument is NaN, the result is NaN.</ul>
     *
     * @param a the argument whose absolute value is to be determined
     *
     * @return the absolute value of the argument.
     *
     * @apiNote As implied by the above, one valid implementation of
     * this method is given by the expression below which computes a
     * {@code double} with the same exponent and significand as the
     * argument but with a guaranteed zero sign bit indicating a
     * positive value: <br>
     * {@code Double.longBitsToDouble((Double.doubleToRawLongBits(a)<<1)>>>1)}
     */
    // 绝对值
    public static double abs(double a) {
        return Math.abs(a);
    }
    
    /*▲ 绝对值 ████████████████████████████████████████████████████████████████████████████████┛ */
    
    
    
    /*▼ 最大值/最小值 ████████████████████████████████████████████████████████████████████████████████┓ */
    
    /**
     * Returns the greater of two {@code int} values. That is, the
     * result is the argument closer to the value of
     * {@link Integer#MAX_VALUE}. If the arguments have the same value,
     * the result is that same value.
     *
     * @param a an argument.
     * @param b another argument.
     *
     * @return the larger of {@code a} and {@code b}.
     */
    // 最大值
    @HotSpotIntrinsicCandidate
    public static int max(int a, int b) {
        return Math.max(a, b);
    }
    
    /**
     * Returns the greater of two {@code long} values. That is, the
     * result is the argument closer to the value of
     * {@link Long#MAX_VALUE}. If the arguments have the same value,
     * the result is that same value.
     *
     * @param a an argument.
     * @param b another argument.
     *
     * @return the larger of {@code a} and {@code b}.
     */
    // 最大值
    public static long max(long a, long b) {
        return Math.max(a, b);
    }
    
    /**
     * Returns the greater of two {@code float} values.  That is,
     * the result is the argument closer to positive infinity. If the
     * arguments have the same value, the result is that same
     * value. If either value is NaN, then the result is NaN.  Unlike
     * the numerical comparison operators, this method considers
     * negative zero to be strictly smaller than positive zero. If one
     * argument is positive zero and the other negative zero, the
     * result is positive zero.
     *
     * @param a an argument.
     * @param b another argument.
     *
     * @return the larger of {@code a} and {@code b}.
     */
    // 最大值
    public static float max(float a, float b) {
        return Math.max(a, b);
    }
    
    /**
     * Returns the greater of two {@code double} values.  That
     * is, the result is the argument closer to positive infinity. If
     * the arguments have the same value, the result is that same
     * value. If either value is NaN, then the result is NaN.  Unlike
     * the numerical comparison operators, this method considers
     * negative zero to be strictly smaller than positive zero. If one
     * argument is positive zero and the other negative zero, the
     * result is positive zero.
     *
     * @param a an argument.
     * @param b another argument.
     *
     * @return the larger of {@code a} and {@code b}.
     */
    // 最大值
    public static double max(double a, double b) {
        return Math.max(a, b);
    }
    
    /**
     * Returns the smaller of two {@code int} values. That is,
     * the result the argument closer to the value of
     * {@link Integer#MIN_VALUE}.  If the arguments have the same
     * value, the result is that same value.
     *
     * @param a an argument.
     * @param b another argument.
     *
     * @return the smaller of {@code a} and {@code b}.
     */
    // 最小值
    @HotSpotIntrinsicCandidate
    public static int min(int a, int b) {
        return Math.min(a, b);
    }
    
    /**
     * Returns the smaller of two {@code long} values. That is,
     * the result is the argument closer to the value of
     * {@link Long#MIN_VALUE}. If the arguments have the same
     * value, the result is that same value.
     *
     * @param a an argument.
     * @param b another argument.
     *
     * @return the smaller of {@code a} and {@code b}.
     */
    // 最小值
    public static long min(long a, long b) {
        return Math.min(a, b);
    }
    
    /**
     * Returns the smaller of two {@code float} values.  That is,
     * the result is the value closer to negative infinity. If the
     * arguments have the same value, the result is that same
     * value. If either value is NaN, then the result is NaN.  Unlike
     * the numerical comparison operators, this method considers
     * negative zero to be strictly smaller than positive zero.  If
     * one argument is positive zero and the other is negative zero,
     * the result is negative zero.
     *
     * @param a an argument.
     * @param b another argument.
     *
     * @return the smaller of {@code a} and {@code b.}
     */
    // 最小值
    public static float min(float a, float b) {
        return Math.min(a, b);
    }
    
    /**
     * Returns the smaller of two {@code double} values.  That
     * is, the result is the value closer to negative infinity. If the
     * arguments have the same value, the result is that same
     * value. If either value is NaN, then the result is NaN.  Unlike
     * the numerical comparison operators, this method considers
     * negative zero to be strictly smaller than positive zero. If one
     * argument is positive zero and the other is negative zero, the
     * result is negative zero.
     *
     * @param a an argument.
     * @param b another argument.
     *
     * @return the smaller of {@code a} and {@code b}.
     */
    // 最小值
    public static double min(double a, double b) {
        return Math.min(a, b);
    }
    
    /*▲ 最大值/最小值 ████████████████████████████████████████████████████████████████████████████████┛ */
    
    
    
    /*▼ 符号函数 ████████████████████████████████████████████████████████████████████████████████┓ */
    
    /**
     * Returns the signum function of the argument; zero if the argument
     * is zero, 1.0 if the argument is greater than zero, -1.0 if the
     * argument is less than zero.
     *
     * <p>Special Cases:
     * <ul>
     * <li> If the argument is NaN, then the result is NaN.
     * <li> If the argument is positive zero or negative zero, then the
     * result is the same as the argument.
     * </ul>
     *
     * @param d the floating-point value whose signum is to be returned
     *
     * @return the signum function of the argument
     *
     * @author Joseph D. Darcy
     * @since 1.5
     */
    // d>0返回1，d<0返回-1，d==0返回0
    public static double signum(double d) {
        return Math.signum(d);
    }
    
    /**
     * Returns the signum function of the argument; zero if the argument
     * is zero, 1.0f if the argument is greater than zero, -1.0f if the
     * argument is less than zero.
     *
     * <p>Special Cases:
     * <ul>
     * <li> If the argument is NaN, then the result is NaN.
     * <li> If the argument is positive zero or negative zero, then the
     * result is the same as the argument.
     * </ul>
     *
     * @param f the floating-point value whose signum is to be returned
     *
     * @return the signum function of the argument
     *
     * @author Joseph D. Darcy
     * @since 1.5
     */
    // f>0返回1，f<0返回-1，f==0返回0
    public static float signum(float f) {
        return Math.signum(f);
    }
    
    /**
     * Returns the first floating-point argument with the sign of the
     * second floating-point argument.  For this method, a NaN
     * {@code sign} argument is always treated as if it were
     * positive.
     *
     * @param magnitude the parameter providing the magnitude of the result
     * @param sign      the parameter providing the sign of the result
     *
     * @return a value with the magnitude of {@code magnitude}
     * and the sign of {@code sign}.
     *
     * @since 1.6
     */
    // sign>=0返回magnitude，sign<0返回-magnitude
    public static double copySign(double magnitude, double sign) {
        return Math.copySign(magnitude, (Double.isNaN(sign) ? 1.0d : sign));
    }
    
    /**
     * Returns the first floating-point argument with the sign of the
     * second floating-point argument.  For this method, a NaN
     * {@code sign} argument is always treated as if it were
     * positive.
     *
     * @param magnitude the parameter providing the magnitude of the result
     * @param sign      the parameter providing the sign of the result
     *
     * @return a value with the magnitude of {@code magnitude}
     * and the sign of {@code sign}.
     *
     * @since 1.6
     */
    // sign>=0返回magnitude，sign<0返回-magnitude
    public static float copySign(float magnitude, float sign) {
        return Math.copySign(magnitude, (Float.isNaN(sign) ? 1.0f : sign));
    }
    
    /*▲ 符号函数 ████████████████████████████████████████████████████████████████████████████████┛ */
    
    
    
    /*▼ 浮点数操作 ████████████████████████████████████████████████████████████████████████████████┓ */
    
    /**
     * Returns the unbiased exponent used in the representation of a
     * {@code float}.  Special cases:
     *
     * <ul>
     * <li>If the argument is NaN or infinite, then the result is
     * {@link Float#MAX_EXPONENT} + 1.
     * <li>If the argument is zero or subnormal, then the result is
     * {@link Float#MIN_EXPONENT} -1.
     * </ul>
     *
     * @param f a {@code float} value
     *
     * @return the unbiased exponent of the argument
     *
     * @since 1.6
     */
    // 返回e-127，其中e是浮点数指数位的无符号值
    public static int getExponent(float f) {
        return Math.getExponent(f);
    }
    
    /**
     * Returns the unbiased exponent used in the representation of a
     * {@code double}.  Special cases:
     *
     * <ul>
     * <li>If the argument is NaN or infinite, then the result is
     * {@link Double#MAX_EXPONENT} + 1.
     * <li>If the argument is zero or subnormal, then the result is
     * {@link Double#MIN_EXPONENT} -1.
     * </ul>
     *
     * @param d a {@code double} value
     *
     * @return the unbiased exponent of the argument
     *
     * @since 1.6
     */
    // 返回e-1023，其中e是浮点数指数位的无符号值
    public static int getExponent(double d) {
        return Math.getExponent(d);
    }
    
    /**
     * Returns the size of an ulp of the argument.  An ulp, unit in
     * the last place, of a {@code float} value is the positive
     * distance between this floating-point value and the {@code
     * float} value next larger in magnitude.  Note that for non-NaN
     * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
     *
     * <p>Special Cases:
     * <ul>
     * <li> If the argument is NaN, then the result is NaN.
     * <li> If the argument is positive or negative infinity, then the
     * result is positive infinity.
     * <li> If the argument is positive or negative zero, then the result is
     * {@code Float.MIN_VALUE}.
     * <li> If the argument is &plusmn;{@code Float.MAX_VALUE}, then
     * the result is equal to 2<sup>104</sup>.
     * </ul>
     *
     * @param f the floating-point value whose ulp is to be returned
     *
     * @return the size of an ulp of the argument
     *
     * @author Joseph D. Darcy
     * @since 1.5
     */
    // 返回1/n，n代表区间[d-1, d)之间的浮点数个数
    public static float ulp(float f) {
        return Math.ulp(f);
    }
    
    /**
     * Returns the size of an ulp of the argument.  An ulp, unit in
     * the last place, of a {@code double} value is the positive
     * distance between this floating-point value and the {@code
     * double} value next larger in magnitude.  Note that for non-NaN
     * <i>x</i>, <code>ulp(-<i>x</i>) == ulp(<i>x</i>)</code>.
     *
     * <p>Special Cases:
     * <ul>
     * <li> If the argument is NaN, then the result is NaN.
     * <li> If the argument is positive or negative infinity, then the
     * result is positive infinity.
     * <li> If the argument is positive or negative zero, then the result is
     * {@code Double.MIN_VALUE}.
     * <li> If the argument is &plusmn;{@code Double.MAX_VALUE}, then
     * the result is equal to 2<sup>971</sup>.
     * </ul>
     *
     * @param d the floating-point value whose ulp is to be returned
     *
     * @return the size of an ulp of the argument
     *
     * @author Joseph D. Darcy
     * @since 1.5
     */
    // 返回1/n，n代表区间[f-1, f)之间的浮点数个数
    public static double ulp(double d) {
        return Math.ulp(d);
    }
    
    /**
     * Returns the floating-point value adjacent to {@code f} in
     * the direction of positive infinity.  This method is
     * semantically equivalent to {@code nextAfter(f,
     * Float.POSITIVE_INFINITY)}; however, a {@code nextUp}
     * implementation may run faster than its equivalent
     * {@code nextAfter} call.
     *
     * <p>Special Cases:
     * <ul>
     * <li> If the argument is NaN, the result is NaN.
     *
     * <li> If the argument is positive infinity, the result is
     * positive infinity.
     *
     * <li> If the argument is zero, the result is
     * {@link Float#MIN_VALUE}
     *
     * </ul>
     *
     * @param f starting floating-point value
     *
     * @return The adjacent floating-point value closer to positive
     * infinity.
     *
     * @since 1.6
     */
    // 返回比当前浮点数大的最小浮点数（浮点数是非均匀且不连续的）
    public static float nextUp(float f) {
        return Math.nextUp(f);
    }
    
    /**
     * Returns the floating-point value adjacent to {@code d} in
     * the direction of positive infinity.  This method is
     * semantically equivalent to {@code nextAfter(d,
     * Double.POSITIVE_INFINITY)}; however, a {@code nextUp}
     * implementation may run faster than its equivalent
     * {@code nextAfter} call.
     *
     * <p>Special Cases:
     * <ul>
     * <li> If the argument is NaN, the result is NaN.
     *
     * <li> If the argument is positive infinity, the result is
     * positive infinity.
     *
     * <li> If the argument is zero, the result is
     * {@link Double#MIN_VALUE}
     *
     * </ul>
     *
     * @param d starting floating-point value
     *
     * @return The adjacent floating-point value closer to positive
     * infinity.
     *
     * @since 1.6
     */
    // 返回比当前浮点数大的最小浮点数（浮点数是非均匀且不连续的）
    public static double nextUp(double d) {
        return Math.nextUp(d);
    }
    
    /**
     * Returns the floating-point value adjacent to {@code f} in
     * the direction of negative infinity.  This method is
     * semantically equivalent to {@code nextAfter(f,
     * Float.NEGATIVE_INFINITY)}; however, a
     * {@code nextDown} implementation may run faster than its
     * equivalent {@code nextAfter} call.
     *
     * <p>Special Cases:
     * <ul>
     * <li> If the argument is NaN, the result is NaN.
     *
     * <li> If the argument is negative infinity, the result is
     * negative infinity.
     *
     * <li> If the argument is zero, the result is
     * {@code -Float.MIN_VALUE}
     *
     * </ul>
     *
     * @param f starting floating-point value
     *
     * @return The adjacent floating-point value closer to negative
     * infinity.
     *
     * @since 1.8
     */
    // 返回比当前浮点数小的最大浮点数（浮点数是非均匀且不连续的）
    public static float nextDown(float f) {
        return Math.nextDown(f);
    }
    
    /**
     * Returns the floating-point value adjacent to {@code d} in
     * the direction of negative infinity.  This method is
     * semantically equivalent to {@code nextAfter(d,
     * Double.NEGATIVE_INFINITY)}; however, a
     * {@code nextDown} implementation may run faster than its
     * equivalent {@code nextAfter} call.
     *
     * <p>Special Cases:
     * <ul>
     * <li> If the argument is NaN, the result is NaN.
     *
     * <li> If the argument is negative infinity, the result is
     * negative infinity.
     *
     * <li> If the argument is zero, the result is
     * {@code -Double.MIN_VALUE}
     *
     * </ul>
     *
     * @param d starting floating-point value
     *
     * @return The adjacent floating-point value closer to negative
     * infinity.
     *
     * @since 1.8
     */
    // 返回比当前浮点数小的最大浮点数（浮点数是非均匀且不连续的）
    public static double nextDown(double d) {
        return Math.nextDown(d);
    }
    
    /**
     * Returns the floating-point number adjacent to the first
     * argument in the direction of the second argument.  If both
     * arguments compare as equal a value equivalent to the second argument
     * is returned.
     *
     * <p>Special cases:
     * <ul>
     * <li> If either argument is a NaN, then NaN is returned.
     *
     * <li> If both arguments are signed zeros, a value equivalent
     * to {@code direction} is returned.
     *
     * <li> If {@code start} is
     * &plusmn;{@link Float#MIN_VALUE} and {@code direction}
     * has a value such that the result should have a smaller
     * magnitude, then a zero with the same sign as {@code start}
     * is returned.
     *
     * <li> If {@code start} is infinite and
     * {@code direction} has a value such that the result should
     * have a smaller magnitude, {@link Float#MAX_VALUE} with the
     * same sign as {@code start} is returned.
     *
     * <li> If {@code start} is equal to &plusmn;
     * {@link Float#MAX_VALUE} and {@code direction} has a
     * value such that the result should have a larger magnitude, an
     * infinity with same sign as {@code start} is returned.
     * </ul>
     *
     * @param start     starting floating-point value
     * @param direction value indicating which of
     *                  {@code start}'s neighbors or {@code start} should
     *                  be returned
     *
     * @return The floating-point number adjacent to {@code start} in the
     * direction of {@code direction}.
     *
     * @since 1.6
     */
    /*
     * 如果start==direction，返回direction
     * 如果start<direction，返回比start大的最小浮点数（浮点数是非均匀且不连续的）
     * 如果start>direction，返回比start小的最大浮点数（浮点数是非均匀且不连续的）
     */
    public static float nextAfter(float start, double direction) {
        return Math.nextAfter(start, direction);
    }
    
    /**
     * Returns the floating-point number adjacent to the first
     * argument in the direction of the second argument.  If both
     * arguments compare as equal the second argument is returned.
     *
     * <p>Special cases:
     * <ul>
     * <li> If either argument is a NaN, then NaN is returned.
     *
     * <li> If both arguments are signed zeros, {@code direction}
     * is returned unchanged (as implied by the requirement of
     * returning the second argument if the arguments compare as
     * equal).
     *
     * <li> If {@code start} is
     * &plusmn;{@link Double#MIN_VALUE} and {@code direction}
     * has a value such that the result should have a smaller
     * magnitude, then a zero with the same sign as {@code start}
     * is returned.
     *
     * <li> If {@code start} is infinite and
     * {@code direction} has a value such that the result should
     * have a smaller magnitude, {@link Double#MAX_VALUE} with the
     * same sign as {@code start} is returned.
     *
     * <li> If {@code start} is equal to &plusmn;
     * {@link Double#MAX_VALUE} and {@code direction} has a
     * value such that the result should have a larger magnitude, an
     * infinity with same sign as {@code start} is returned.
     * </ul>
     *
     * @param start     starting floating-point value
     * @param direction value indicating which of
     *                  {@code start}'s neighbors or {@code start} should
     *                  be returned
     *
     * @return The floating-point number adjacent to {@code start} in the
     * direction of {@code direction}.
     *
     * @since 1.6
     */
    /*
     * 如果start==direction，返回direction
     * 如果start<direction，返回比start大的最小浮点数（浮点数是非均匀且不连续的）
     * 如果start>direction，返回比start小的最大浮点数（浮点数是非均匀且不连续的）
     */
    public static double nextAfter(double start, double direction) {
        return Math.nextAfter(start, direction);
    }
    
    /*▲ 浮点数操作 ████████████████████████████████████████████████████████████████████████████████┛ */
    
    
    
    /*▼ 杂项 ████████████████████████████████████████████████████████████████████████████████┓ */
    
    /**
     * Returns a {@code double} value with a positive sign, greater
     * than or equal to {@code 0.0} and less than {@code 1.0}.
     * Returned values are chosen pseudorandomly with (approximately)
     * uniform distribution from that range.
     *
     * <p>When this method is first called, it creates a single new
     * pseudorandom-number generator, exactly as if by the expression
     *
     * <blockquote>{@code new java.util.Random()}</blockquote>
     *
     * This new pseudorandom-number generator is used thereafter for
     * all calls to this method and is used nowhere else.
     *
     * <p>This method is properly synchronized to allow correct use by
     * more than one thread. However, if many threads need to generate
     * pseudorandom numbers at a great rate, it may reduce contention
     * for each thread to have its own pseudorandom-number generator.
     *
     * @return a pseudorandom {@code double} greater than or equal
     * to {@code 0.0} and less than {@code 1.0}.
     *
     * @see Random#nextDouble()
     */
    // 随机均匀地生成一个double值，值的范围是[0, 1)，正数
    public static double random() {
        return RandomNumberGeneratorHolder.randomNumberGenerator.nextDouble();
    }
    
    /**
     * Returns the value of the {@code long} argument;
     * throwing an exception if the value overflows an {@code int}.
     *
     * @param value the long value
     *
     * @return the argument as an int
     *
     * @throws ArithmeticException if the {@code argument} overflows an int
     * @see Math#toIntExact(long)
     * @since 1.8
     */
    // 将long转为int，要保证不溢出
    public static int toIntExact(long value) {
        return Math.toIntExact(value);
    }
    
    /*▲ 杂项 ████████████████████████████████████████████████████████████████████████████████┛ */
    
    
    
    // 持有一个Random属性
    private static final class RandomNumberGeneratorHolder {
        static final Random randomNumberGenerator = new Random();
    }
}