Class DD

  • All Implemented Interfaces:
    java.io.Serializable, java.lang.Cloneable, java.lang.Comparable

    public final class DD
    extends java.lang.Object
    implements java.io.Serializable, java.lang.Comparable, java.lang.Cloneable
    Implements extended-precision floating-point numbers which maintain 106 bits (approximately 30 decimal digits) of precision.

    A DoubleDouble uses a representation containing two double-precision values. A number x is represented as a pair of doubles, x.hi and x.lo, such that the number represented by x is x.hi + x.lo, where

        |x.lo| <= 0.5*ulp(x.hi)
     
    and ulp(y) means "unit in the last place of y". The basic arithmetic operations are implemented using convenient properties of IEEE-754 floating-point arithmetic.

    The range of values which can be represented is the same as in IEEE-754. The precision of the representable numbers is twice as great as IEEE-754 double precision.

    The correctness of the arithmetic algorithms relies on operations being performed with standard IEEE-754 double precision and rounding. This is the Java standard arithmetic model, but for performance reasons Java implementations are not constrained to using this standard by default. Some processors (notably the Intel Pentium architecture) perform floating point operations in (non-IEEE-754-standard) extended-precision. A JVM implementation may choose to use the non-standard extended-precision as its default arithmetic mode. To prevent this from happening, this code uses the Java strictfp modifier, which forces all operations to take place in the standard IEEE-754 rounding model.

    The API provides both a set of value-oriented operations and a set of mutating operations. Value-oriented operations treat DoubleDouble values as immutable; operations on them return new objects carrying the result of the operation. This provides a simple and safe semantics for writing DoubleDouble expressions. However, there is a performance penalty for the object allocations required. The mutable interface updates object values in-place. It provides optimum memory performance, but requires care to ensure that aliasing errors are not created and constant values are not changed.

    For example, the following code example constructs three DD instances: two to hold the input values and one to hold the result of the addition.

         DD a = new DD(2.0);
         DD b = new DD(3.0);
         DD c = a.add(b);
     
    In contrast, the following approach uses only one object:
         DD a = new DD(2.0);
         a.selfAdd(3.0);
     

    This implementation uses algorithms originally designed variously by Knuth, Kahan, Dekker, and Linnainmaa. Douglas Priest developed the first C implementation of these techniques. Other more recent C++ implementation are due to Keith M. Briggs and David Bailey et al.

    References

    • Priest, D., Algorithms for Arbitrary Precision Floating Point Arithmetic, in P. Kornerup and D. Matula, Eds., Proc. 10th Symposium on Computer Arithmetic, IEEE Computer Society Press, Los Alamitos, Calif., 1991.
    • Yozo Hida, Xiaoye S. Li and David H. Bailey, Quad-Double Arithmetic: Algorithms, Implementation, and Application, manuscript, Oct 2000; Lawrence Berkeley National Laboratory Report BNL-46996.
    • David Bailey, High Precision Software Directory; http://crd.lbl.gov/~dhbailey/mpdist/index.html
    Author:
    Martin Davis
    See Also:
    Serialized Form
    • Field Summary

      Fields 
      Modifier and Type Field Description
      static DD E
      The value nearest to the constant e (the natural logarithm base).
      static double EPS
      The smallest representable relative difference between two {link @ DoubleDouble} values
      static DD NaN
      A value representing the result of an operation which does not return a valid number.
      static DD PI
      The value nearest to the constant Pi.
      static DD PI_2
      The value nearest to the constant Pi / 2.
      static DD TWO_PI
      The value nearest to the constant 2 * Pi.
    • Constructor Summary

      Constructors 
      Constructor Description
      DD()
      Creates a new DoubleDouble with value 0.0.
      DD​(double x)
      Creates a new DoubleDouble with value x.
      DD​(double hi, double lo)
      Creates a new DoubleDouble with value (hi, lo).
      DD​(java.lang.String str)
      Creates a new DoubleDouble with value equal to the argument.
      DD​(DD dd)
      Creates a new DoubleDouble with value equal to the argument.
    • Method Summary

      All Methods Static Methods Instance Methods Concrete Methods 
      Modifier and Type Method Description
      DD abs()
      Returns the absolute value of this value.
      DD add​(double y)
      Returns a new DoubleDouble whose value is (this + y).
      DD add​(DD y)
      Returns a new DoubleDouble whose value is (this + y).
      DD ceil()
      Returns the smallest (closest to negative infinity) value that is not less than the argument and is equal to a mathematical integer.
      java.lang.Object clone()
      Creates and returns a copy of this value.
      int compareTo​(java.lang.Object o)
      Compares two DoubleDouble objects numerically.
      static DD copy​(DD dd)
      Creates a new DoubleDouble with the value of the argument.
      static DD determinant​(double x1, double y1, double x2, double y2)
      Computes the determinant of the 2x2 matrix with the given entries.
      static DD determinant​(DD x1, DD y1, DD x2, DD y2)
      Computes the determinant of the 2x2 matrix with the given entries.
      DD divide​(double y)
      Computes a new DoubleDouble whose value is (this / y).
      DD divide​(DD y)
      Computes a new DoubleDouble whose value is (this / y).
      double doubleValue()
      Converts this value to the nearest double-precision number.
      java.lang.String dump()
      Dumps the components of this number to a string.
      boolean equals​(DD y)
      Tests whether this value is equal to another DoubleDouble value.
      DD floor()
      Returns the largest (closest to positive infinity) value that is not greater than the argument and is equal to a mathematical integer.
      boolean ge​(DD y)
      Tests whether this value is greater than or equals to another DoubleDouble value.
      boolean gt​(DD y)
      Tests whether this value is greater than another DoubleDouble value.
      int intValue()
      Converts this value to the nearest integer.
      boolean isNaN()
      Tests whether this value is NaN.
      boolean isNegative()
      Tests whether this value is less than 0.
      boolean isPositive()
      Tests whether this value is greater than 0.
      boolean isZero()
      Tests whether this value is equal to 0.
      boolean le​(DD y)
      Tests whether this value is less than or equal to another DoubleDouble value.
      boolean lt​(DD y)
      Tests whether this value is less than another DoubleDouble value.
      DD max​(DD x)
      Computes the maximum of this and another DD number.
      DD min​(DD x)
      Computes the minimum of this and another DD number.
      DD multiply​(double y)
      Returns a new DoubleDouble whose value is (this * y).
      DD multiply​(DD y)
      Returns a new DoubleDouble whose value is (this * y).
      DD negate()
      Returns a new DoubleDouble whose value is -this.
      static DD parse​(java.lang.String str)
      Converts a string representation of a real number into a DoubleDouble value.
      DD pow​(int exp)
      Computes the value of this number raised to an integral power.
      DD reciprocal()
      Returns a DoubleDouble whose value is 1 / this.
      DD rint()
      Rounds this value to the nearest integer.
      DD selfAdd​(double y)
      Adds the argument to the value of this.
      DD selfAdd​(DD y)
      Adds the argument to the value of this.
      DD selfDivide​(double y)
      Divides this object by the argument, returning this.
      DD selfDivide​(DD y)
      Divides this object by the argument, returning this.
      DD selfMultiply​(double y)
      Multiplies this object by the argument, returning this.
      DD selfMultiply​(DD y)
      Multiplies this object by the argument, returning this.
      DD selfSqr()
      Squares this object.
      DD selfSubtract​(double y)
      Subtracts the argument from the value of this.
      DD selfSubtract​(DD y)
      Subtracts the argument from the value of this.
      DD setValue​(double value)
      Set the value for the DD object.
      DD setValue​(DD value)
      Set the value for the DD object.
      int signum()
      Returns an integer indicating the sign of this value.
      DD sqr()
      Computes the square of this value.
      static DD sqr​(double x)
      Computes the square of this value.
      DD sqrt()
      Computes the positive square root of this value.
      static DD sqrt​(double x)  
      DD subtract​(double y)
      Computes a new DoubleDouble object whose value is (this - y).
      DD subtract​(DD y)
      Computes a new DoubleDouble object whose value is (this - y).
      java.lang.String toSciNotation()
      Returns the string representation of this value in scientific notation.
      java.lang.String toStandardNotation()
      Returns the string representation of this value in standard notation.
      java.lang.String toString()
      Returns a string representation of this number, in either standard or scientific notation.
      DD trunc()
      Returns the integer which is largest in absolute value and not further from zero than this value.
      static DD valueOf​(double x)
      Converts the double argument to a DoubleDouble number.
      static DD valueOf​(java.lang.String str)
      Converts the string argument to a DoubleDouble number.
      • Methods inherited from class java.lang.Object

        equals, getClass, hashCode, notify, notifyAll, wait, wait, wait
    • Field Detail

      • PI

        public static final DD PI
        The value nearest to the constant Pi.
      • TWO_PI

        public static final DD TWO_PI
        The value nearest to the constant 2 * Pi.
      • PI_2

        public static final DD PI_2
        The value nearest to the constant Pi / 2.
      • E

        public static final DD E
        The value nearest to the constant e (the natural logarithm base).
      • NaN

        public static final DD NaN
        A value representing the result of an operation which does not return a valid number.
      • EPS

        public static final double EPS
        The smallest representable relative difference between two {link @ DoubleDouble} values
        See Also:
        Constant Field Values
    • Constructor Detail

      • DD

        public DD()
        Creates a new DoubleDouble with value 0.0.
      • DD

        public DD​(double x)
        Creates a new DoubleDouble with value x.
        Parameters:
        x - the value to initialize
      • DD

        public DD​(double hi,
                  double lo)
        Creates a new DoubleDouble with value (hi, lo).
        Parameters:
        hi - the high-order component
        lo - the high-order component
      • DD

        public DD​(DD dd)
        Creates a new DoubleDouble with value equal to the argument.
        Parameters:
        dd - the value to initialize
      • DD

        public DD​(java.lang.String str)
           throws java.lang.NumberFormatException
        Creates a new DoubleDouble with value equal to the argument.
        Parameters:
        str - the value to initialize by
        Throws:
        java.lang.NumberFormatException - if str is not a valid representation of a number
    • Method Detail

      • valueOf

        public static DD valueOf​(java.lang.String str)
                          throws java.lang.NumberFormatException
        Converts the string argument to a DoubleDouble number.
        Parameters:
        str - a string containing a representation of a numeric value
        Returns:
        the extended precision version of the value
        Throws:
        java.lang.NumberFormatException - if s is not a valid representation of a number
      • valueOf

        public static DD valueOf​(double x)
        Converts the double argument to a DoubleDouble number.
        Parameters:
        x - a numeric value
        Returns:
        the extended precision version of the value
      • copy

        public static DD copy​(DD dd)
        Creates a new DoubleDouble with the value of the argument.
        Parameters:
        dd - the DoubleDouble value to copy
        Returns:
        a copy of the input value
      • clone

        public java.lang.Object clone()
        Creates and returns a copy of this value.
        Returns:
        a copy of this value
      • setValue

        public DD setValue​(DD value)
        Set the value for the DD object. This method supports the mutating operations concept described in the class documentation (see above).
        Parameters:
        value - a DD instance supplying an extended-precision value.
        Returns:
        a self-reference to the DD instance.
      • setValue

        public DD setValue​(double value)
        Set the value for the DD object. This method supports the mutating operations concept described in the class documentation (see above).
        Parameters:
        value - a floating point value to be stored in the instance.
        Returns:
        a self-reference to the DD instance.
      • add

        public final DD add​(DD y)
        Returns a new DoubleDouble whose value is (this + y).
        Parameters:
        y - the addend
        Returns:
        (this + y)
      • add

        public final DD add​(double y)
        Returns a new DoubleDouble whose value is (this + y).
        Parameters:
        y - the addend
        Returns:
        (this + y)
      • selfAdd

        public final DD selfAdd​(DD y)
        Adds the argument to the value of this. To prevent altering constants, this method must only be used on values known to be newly created.
        Parameters:
        y - the addend
        Returns:
        this object, increased by y
      • selfAdd

        public final DD selfAdd​(double y)
        Adds the argument to the value of this. To prevent altering constants, this method must only be used on values known to be newly created.
        Parameters:
        y - the addend
        Returns:
        this object, increased by y
      • subtract

        public final DD subtract​(DD y)
        Computes a new DoubleDouble object whose value is (this - y).
        Parameters:
        y - the subtrahend
        Returns:
        (this - y)
      • subtract

        public final DD subtract​(double y)
        Computes a new DoubleDouble object whose value is (this - y).
        Parameters:
        y - the subtrahend
        Returns:
        (this - y)
      • selfSubtract

        public final DD selfSubtract​(DD y)
        Subtracts the argument from the value of this. To prevent altering constants, this method must only be used on values known to be newly created.
        Parameters:
        y - the addend
        Returns:
        this object, decreased by y
      • selfSubtract

        public final DD selfSubtract​(double y)
        Subtracts the argument from the value of this. To prevent altering constants, this method must only be used on values known to be newly created.
        Parameters:
        y - the addend
        Returns:
        this object, decreased by y
      • negate

        public final DD negate()
        Returns a new DoubleDouble whose value is -this.
        Returns:
        -this
      • multiply

        public final DD multiply​(DD y)
        Returns a new DoubleDouble whose value is (this * y).
        Parameters:
        y - the multiplicand
        Returns:
        (this * y)
      • multiply

        public final DD multiply​(double y)
        Returns a new DoubleDouble whose value is (this * y).
        Parameters:
        y - the multiplicand
        Returns:
        (this * y)
      • selfMultiply

        public final DD selfMultiply​(DD y)
        Multiplies this object by the argument, returning this. To prevent altering constants, this method must only be used on values known to be newly created.
        Parameters:
        y - the value to multiply by
        Returns:
        this object, multiplied by y
      • selfMultiply

        public final DD selfMultiply​(double y)
        Multiplies this object by the argument, returning this. To prevent altering constants, this method must only be used on values known to be newly created.
        Parameters:
        y - the value to multiply by
        Returns:
        this object, multiplied by y
      • divide

        public final DD divide​(DD y)
        Computes a new DoubleDouble whose value is (this / y).
        Parameters:
        y - the divisor
        Returns:
        a new object with the value (this / y)
      • divide

        public final DD divide​(double y)
        Computes a new DoubleDouble whose value is (this / y).
        Parameters:
        y - the divisor
        Returns:
        a new object with the value (this / y)
      • selfDivide

        public final DD selfDivide​(DD y)
        Divides this object by the argument, returning this. To prevent altering constants, this method must only be used on values known to be newly created.
        Parameters:
        y - the value to divide by
        Returns:
        this object, divided by y
      • selfDivide

        public final DD selfDivide​(double y)
        Divides this object by the argument, returning this. To prevent altering constants, this method must only be used on values known to be newly created.
        Parameters:
        y - the value to divide by
        Returns:
        this object, divided by y
      • reciprocal

        public final DD reciprocal()
        Returns a DoubleDouble whose value is 1 / this.
        Returns:
        the reciprocal of this value
      • floor

        public DD floor()
        Returns the largest (closest to positive infinity) value that is not greater than the argument and is equal to a mathematical integer. Special cases:
        • If this value is NaN, returns NaN.
        Returns:
        the largest (closest to positive infinity) value that is not greater than the argument and is equal to a mathematical integer.
      • ceil

        public DD ceil()
        Returns the smallest (closest to negative infinity) value that is not less than the argument and is equal to a mathematical integer. Special cases:
        • If this value is NaN, returns NaN.
        Returns:
        the smallest (closest to negative infinity) value that is not less than the argument and is equal to a mathematical integer.
      • signum

        public int signum()
        Returns an integer indicating the sign of this value.
        • if this value is > 0, returns 1
        • if this value is < 0, returns -1
        • if this value is = 0, returns 0
        • if this value is NaN, returns 0
        Returns:
        an integer indicating the sign of this value
      • rint

        public DD rint()
        Rounds this value to the nearest integer. The value is rounded to an integer by adding 1/2 and taking the floor of the result. Special cases:
        • If this value is NaN, returns NaN.
        Returns:
        this value rounded to the nearest integer
      • trunc

        public DD trunc()
        Returns the integer which is largest in absolute value and not further from zero than this value. Special cases:
        • If this value is NaN, returns NaN.
        Returns:
        the integer which is largest in absolute value and not further from zero than this value
      • abs

        public DD abs()
        Returns the absolute value of this value. Special cases:
        • If this value is NaN, it is returned.
        Returns:
        the absolute value of this value
      • sqr

        public DD sqr()
        Computes the square of this value.
        Returns:
        the square of this value.
      • selfSqr

        public DD selfSqr()
        Squares this object. To prevent altering constants, this method must only be used on values known to be newly created.
        Returns:
        the square of this value.
      • sqr

        public static DD sqr​(double x)
        Computes the square of this value.
        Returns:
        the square of this value.
      • sqrt

        public DD sqrt()
        Computes the positive square root of this value. If the number is NaN or negative, NaN is returned.
        Returns:
        the positive square root of this number. If the argument is NaN or less than zero, the result is NaN.
      • sqrt

        public static DD sqrt​(double x)
      • pow

        public DD pow​(int exp)
        Computes the value of this number raised to an integral power. Follows semantics of Java Math.pow as closely as possible.
        Parameters:
        exp - the integer exponent
        Returns:
        x raised to the integral power exp
      • determinant

        public static DD determinant​(double x1,
                                     double y1,
                                     double x2,
                                     double y2)
        Computes the determinant of the 2x2 matrix with the given entries.
        Parameters:
        x1 - a double value
        y1 - a double value
        x2 - a double value
        y2 - a double value
        Returns:
        the determinant of the values
      • determinant

        public static DD determinant​(DD x1,
                                     DD y1,
                                     DD x2,
                                     DD y2)
        Computes the determinant of the 2x2 matrix with the given entries.
        Parameters:
        x1 - a matrix entry
        y1 - a matrix entry
        x2 - a matrix entry
        y2 - a matrix entry
        Returns:
        the determinant of the matrix of values
      • min

        public DD min​(DD x)
        Computes the minimum of this and another DD number.
        Parameters:
        x - a DD number
        Returns:
        the minimum of the two numbers
      • max

        public DD max​(DD x)
        Computes the maximum of this and another DD number.
        Parameters:
        x - a DD number
        Returns:
        the maximum of the two numbers
      • doubleValue

        public double doubleValue()
        Converts this value to the nearest double-precision number.
        Returns:
        the nearest double-precision number to this value
      • intValue

        public int intValue()
        Converts this value to the nearest integer.
        Returns:
        the nearest integer to this value
      • isZero

        public boolean isZero()
        Tests whether this value is equal to 0.
        Returns:
        true if this value is equal to 0
      • isNegative

        public boolean isNegative()
        Tests whether this value is less than 0.
        Returns:
        true if this value is less than 0
      • isPositive

        public boolean isPositive()
        Tests whether this value is greater than 0.
        Returns:
        true if this value is greater than 0
      • isNaN

        public boolean isNaN()
        Tests whether this value is NaN.
        Returns:
        true if this value is NaN
      • equals

        public boolean equals​(DD y)
        Tests whether this value is equal to another DoubleDouble value.
        Parameters:
        y - a DoubleDouble value
        Returns:
        true if this value = y
      • gt

        public boolean gt​(DD y)
        Tests whether this value is greater than another DoubleDouble value.
        Parameters:
        y - a DoubleDouble value
        Returns:
        true if this value > y
      • ge

        public boolean ge​(DD y)
        Tests whether this value is greater than or equals to another DoubleDouble value.
        Parameters:
        y - a DoubleDouble value
        Returns:
        true if this value >= y
      • lt

        public boolean lt​(DD y)
        Tests whether this value is less than another DoubleDouble value.
        Parameters:
        y - a DoubleDouble value
        Returns:
        true if this value < y
      • le

        public boolean le​(DD y)
        Tests whether this value is less than or equal to another DoubleDouble value.
        Parameters:
        y - a DoubleDouble value
        Returns:
        true if this value <= y
      • compareTo

        public int compareTo​(java.lang.Object o)
        Compares two DoubleDouble objects numerically.
        Specified by:
        compareTo in interface java.lang.Comparable
        Returns:
        -1,0 or 1 depending on whether this value is less than, equal to or greater than the value of o
      • dump

        public java.lang.String dump()
        Dumps the components of this number to a string.
        Returns:
        a string showing the components of the number
      • toString

        public java.lang.String toString()
        Returns a string representation of this number, in either standard or scientific notation. If the magnitude of the number is in the range [ 10-3, 108 ] standard notation will be used. Otherwise, scientific notation will be used.
        Overrides:
        toString in class java.lang.Object
        Returns:
        a string representation of this number
      • toStandardNotation

        public java.lang.String toStandardNotation()
        Returns the string representation of this value in standard notation.
        Returns:
        the string representation in standard notation
      • toSciNotation

        public java.lang.String toSciNotation()
        Returns the string representation of this value in scientific notation.
        Returns:
        the string representation in scientific notation
      • parse

        public static DD parse​(java.lang.String str)
                        throws java.lang.NumberFormatException
        Converts a string representation of a real number into a DoubleDouble value. The format accepted is similar to the standard Java real number syntax. It is defined by the following regular expression:
         [+|-] {digit} [ . {digit} ] [ ( e | E ) [+|-] {digit}+
         
        Parameters:
        str - the string to parse
        Returns:
        the value of the parsed number
        Throws:
        java.lang.NumberFormatException - if str is not a valid representation of a number