Package org.apache.commons.numbers.core

Class DD

  • All Implemented Interfaces:
    java.io.Serializable, Addition<DD>, Multiplication<DD>, NativeOperators<DD>
    public final class DD
extends java.lang.Number
implements NativeOperators<DD>, java.io.Serializable
    Computes double-double floating-point operations.

    A double-double is an unevaluated sum of two IEEE double precision numbers capable of representing at least 106 bits of significand. A normalized double-double number (x, xx) satisfies the condition that the parts are non-overlapping in magnitude such that: 

     |x| > |xx|
 x == x + xx

    This implementation assumes a normalized representation during operations on a DD number and computes results as a normalized representation. Any double-double number can be normalized by summation of the parts (see ofSum). Note that the number (x, xx) may also be referred to using the labels high and low to indicate the magnitude of the parts as (xhi, xlo), or using a numerical suffix for the parts as (x0, x1). The numerical suffix is typically used when the number has an arbitrary number of parts.

    The double-double class is immutable.

    Construction

    Factory methods to create a DD that are exact use the prefix of. Methods that create the closest possible representation use the prefix from. These methods may suffer a possible loss of precision during conversion.

    Primitive values of type double, int and long are converted exactly to a DD.

    The DD class can also be created as the result of an arithmetic operation on a pair of double operands. The resulting DD has the IEEE754 double result of the operation in the first part, and the second part contains the round-off lost from the operation due to rounding. Construction using add (+), subtract (-) and multiply (*) operators are exact. Construction using division (/) may be inexact if the quotient is not representable.

    Note that it is more efficient to create a DD from a double operation than to create two DD values and combine them with the same operation. The result will be the same for add, subtract and multiply but may lose precision for divide. 

    
 // Inefficient
 DD a = DD.of(1.23).add(DD.of(4.56));
 // Optimal
 DD b = DD.ofSum(1.23, 4.56);

 // Inefficient and may lose precision
 DD c = DD.of(1.23).divide(DD.of(4.56));
 // Optimal
 DD d = DD.fromQuotient(1.23, 4.56);

    It is not possible to directly specify the two parts of the number. The two parts must be added using ofSum. If the two parts already represent a number (x, xx) such that x == x + xx then the magnitudes of the parts will be unchanged; any signed zeros may be subject to a sign change.

    Primitive operands

    Operations are provided using a DD operand or a double operand. Implicit type conversion allows methods with a double operand to be used with other primitives such as int or long. Note that casting of a long to a double may result in loss of precision. To maintain the full precision of a long first convert the value to a DD using of(long) and use the same arithmetic operation using the DD operand.

    Accuracy

    Add and multiply operations using two double values operands are computed to an exact DD result (see ofSum and ofProduct). Operations involving a DD and another operand, either double or DD, are not exact.

    This class is not intended to perform exact arithmetic. Arbitrary precision arithmetic is available using BigDecimal. Single operations will compute the DD result within a tolerance of the 106-bit exact result. This far exceeds the accuracy of double arithmetic. The reduced accuracy is a compromise to deliver increased performance. The class is intended to reduce error in equivalent double arithmetic operations where the double valued result is required to high accuracy. Although it is possible to reduce error to 2-106 for all operations, the additional computation would impact performance and would require multiple chained operations to potentially observe a different result when the final DD is converted to a double.

    Canonical representation

    The double-double number is the sum of its parts. The canonical representation of the number is the explicit value of the parts. The toString() method is provided to convert to a String representation of the parts formatted as a tuple.

    The class implements equals(Object) and hashCode() and allows usage as a key in a Set or Map. Equality requires binary equivalence of the parts. Note that representations of zero using different combinations of +/- 0.0 are not considered equal. Also note that many non-normalized double-double numbers can represent the same number. Double-double numbers can be normalized before operations that involve equals(Object) by adding the parts; this is exact for a finite sum and provides equality support for non-zero numbers. Alternatively exact numerical equality and comparisons are supported by conversion to a BigDecimal representation. Note that BigDecimal does not support non-finite values.

    Overflow, underflow and non-finite support

    A double-double number is limited to the same finite range as a double (4.9E-324 to 1.7976931348623157E308). This class is intended for use when the ultimate result is finite and intermediate values do not approach infinity or zero.

    This implementation does not support IEEE standards for handling infinite and NaN when used in arithmetic operations. Computations may split a 64-bit double into two parts and/or use subtraction of intermediate terms to compute round-off parts. These operations may generate infinite values due to overflow which then propagate through further operations to NaN, for example computing the round-off using Inf - Inf = NaN.

    Operations that involve splitting a double (multiply, divide) are safe when the base 2 exponent is below 996. This puts an upper limit of approximately +/-6.7e299 on any values to be split; in practice the arguments to multiply and divide operations are further constrained by the expected finite value of the product or quotient.

    Likewise the smallest value that can be represented is Double.MIN_VALUE. The full 106-bit accuracy will be lost when intermediates are within 253 of Double.MIN_NORMAL.

    The DD result can be verified by checking it is a finite evaluated sum. Computations expecting to approach over or underflow must use scaling of intermediate terms (see frexp and scalb) and appropriate management of the current base 2 scale.

    References:

    1. Dekker, T.J. (1971) A floating-point technique for extending the available precision Numerische Mathematik, 18:224–242.
    2. Shewchuk, J.R. (1997) Arbitrary Precision Floating-Point Arithmetic.
    3. Hide, Y, Li, X.S. and Bailey, D.H. (2008) Library for Double-Double and Quad-Double Arithmetic.
    Since:
    1.2
    See Also:
    Serialized Form

    • Field Summary

      Fields 
      Modifier and Type Field Description
      private static int CMP_UNSIGNED_1022
      The value 1022 converted for use if using Integer.compareUnsigned(int, int).
      private static int CMP_UNSIGNED_2046
      The value 2046 converted for use if using Integer.compareUnsigned(int, int).
      private static int CMP_UNSIGNED_MINUS_1
      The value -1 converted for use if using Integer.compareUnsigned(int, int).
      private static int EXP_MASK
      The mask to extract the raw 11-bit exponent.
      private static int EXPONENT_OFFSET
      Exponent offset in IEEE754 representation.
      private static char FORMAT_END
      String representation.
      private static char FORMAT_SEP
      String representation.
      private static char FORMAT_START
      String representation.
      private static double HALF
      0.5.
      private static long HIGH32_MASK
      Mask to extract the high 32-bits from a long.
      private static long MANTISSA_MASK
      Mask to extract the 52-bit mantissa from a long representation of a double.
      private static double MULTIPLIER
      The multiplier used to split the double value into high and low parts.
      static DD ONE
      A double-double number representing one.
      private static double SAFE_MULTIPLY
      The limit for safe multiplication of x*y, assuming values above 1.
      private static long serialVersionUID
      Serializable version identifier.
      private static int TO_STRING_SIZE
      The size of the buffer for toString().
      private static double TWO_POW_512
      2^512.
      private static double TWO_POW_53
      2^53.
      private static double TWO_POW_M512
      2^-512.
      private static long UNSIGN_MASK
      Mask to remove the sign bit from a long.
      private double x
      The high part of the double-double number.
      private double xx
      The low part of the double-double number.
      static DD ZERO
      A double-double number representing zero.

    • Constructor Summary

      Constructors 
      Modifier Constructor Description
      private DD​(double x, double xx)
      Create a double-double number (x, xx).

    • Method Summary

      All Methods Static Methods Instance Methods Concrete Methods 
      Modifier and Type Method Description
      DD abs()
      Returns a DD whose value is the absolute value of the number (x, xx) This method assumes that the low part xx is the smaller magnitude.
      (package private) static DD accurateAdd​(double x, double xx, double y)
      Compute the sum of (x, xx) and y.
      (package private) static DD accurateAdd​(double x, double xx, double y, double yy)
      Compute the sum of (x, xx) and (y, yy).
      DD add​(double y)
      Returns a DD whose value is (this + y).
      (package private) static DD add​(double x, double xx, double y, double yy)
      Compute the sum of (x, xx) and (y, yy).
      DD add​(DD y)
      Returns a DD whose value is (this + y).
      java.math.BigDecimal bigDecimalValue()
      Get the value as a BigDecimal.
      DD ceil()
      Returns the smallest (closest to negative infinity) DD value that is greater than or equal to this number (x, xx) and is equal to a mathematical integer.
      private static DD computePow​(double x, double xx, int n)
      Compute the number x (non-zero finite) raised to the power n.
      private static DD computePowScaled​(long b, double x, double xx, int n, long[] exp)
      Compute the number x (non-zero finite) raised to the power n.
      DD divide​(double y)
      Returns a DD whose value is (this / y).
      private static DD divide​(double x, double xx, double y)
      Compute the division of (x, xx) by y.
      private static DD divide​(double x, double xx, double y, double yy)
      Compute the division of (x, xx) by (y, yy).
      DD divide​(DD y)
      Returns a DD whose value is (this / y).
      double doubleValue()
      Get the value as a double.
      private static boolean equals​(double x, double y)
      Returns true if the values are numerically equal.
      boolean equals​(java.lang.Object other)
      Test for equality with another object.
      (package private) static DD fastTwoDiff​(double a, double b)
      Compute the difference of two numbers a and b using Dekker's two-sum algorithm.
      private static double fastTwoDiffLow​(double a, double b, double x)
      Compute the round-off of the difference of two numbers a and b using Dekker's two-sum algorithm.
      (package private) static DD fastTwoSum​(double a, double b)
      Compute the sum of two numbers a and b using Dekker's two-sum algorithm.
      (package private) static double fastTwoSumLow​(double a, double b, double x)
      Compute the round-off of the sum of two numbers a and b using Dekker's two-sum algorithm.
      float floatValue()
      Get the value as a float.
      DD floor()
      Returns the largest (closest to positive infinity) DD value that is less than or equal to this number (x, xx) and is equal to a mathematical integer.
      private static DD floorOrCeil​(double x, double xx, java.util.function.DoubleUnaryOperator op)
      Implementation of the floor and ceiling functions.
      DD frexp​(int[] exp)
      Convert this number x to fractional f and integral 2^exp components.
      static DD from​(java.math.BigDecimal x)
      Creates the double-double number (z, zz) using the double representation of the argument x; the low part is the double representation of the round-off error.
      static DD fromQuotient​(double x, double y)
      Returns a DD whose value is (x / y).
      private static int getScale​(double a)
      Returns a scale suitable for use with Math.scalb(double, int) to normalise the number to the interval [1, 2).
      int hashCode()
      Gets a hash code for the double-double number.
      double hi()
      Gets the first part x of the double-double number (x, xx).
      (package private) static double highPart​(double value)
      Implement Dekker's method to split a value into two parts.
      int intValue()
      Get the value as an int.
      boolean isFinite()
      Returns true if the evaluated sum of the parts is finite.
      (package private) static boolean isNotNormal​(double a)
      Checks if the number is not normal.
      boolean isOne()
      Check if this is a neutral element of multiplication, i.e.
      boolean isZero()
      Check if this is a neutral element of addition, i.e.
      double lo()
      Gets the second part xx of the double-double number (x, xx).
      long longValue()
      Get the value as a long.
      DD multiply​(double y)
      Returns a DD whose value is this * y.
      private static DD multiply​(double x, double xx, double y)
      Compute the multiplication product of (x, xx) and y.
      private static DD multiply​(double x, double xx, double y, double yy)
      Compute the multiplication product of (x, xx) and (y, yy).
      DD multiply​(int n)
      Repeated addition.
      DD multiply​(DD y)
      Returns a DD whose value is this * y.
      DD negate()
      Returns a DD whose value is the negation of both parts of double-double number.
      static DD of​(double x)
      Creates the double-double number as the value (x, 0).
      (package private) static DD of​(double x, double xx)
      Creates the double-double number as the value (x, xx).
      static DD of​(int x)
      Creates the double-double number as the value (x, 0).
      static DD of​(long x)
      Creates the double-double number with the high part equal to (double) x and the low part equal to any remaining bits.
      static DD ofDifference​(double x, double y)
      Returns a DD whose value is (x - y).
      static DD ofProduct​(double x, double y)
      Returns a DD whose value is (x * y).
      static DD ofSquare​(double x)
      Returns a DD whose value is (x * x).
      static DD ofSum​(double x, double y)
      Returns a DD whose value is (x + y).
      DD one()
      Identity element.
      DD pow​(int n)
      Compute this number (x, xx) raised to the power n.
      DD pow​(int n, long[] exp)
      Compute this number x raised to the power n.
      DD reciprocal()
      Compute the reciprocal of this.
      private static DD reciprocal​(double y, double yy)
      Compute the inverse of (y, yy).
      DD scalb​(int exp)
      Multiply this number (x, xx) by an integral power of two.
      DD sqrt()
      Compute the square root of this number (x, xx).
      DD square()
      Returns a DD whose value is this * this.
      private static DD square​(double x, double xx)
      Compute the square of (x, xx).
      DD subtract​(double y)
      Returns a DD whose value is (this - y).
      DD subtract​(DD y)
      Returns a DD whose value is (this - y).
      java.lang.String toString()
      Returns a string representation of the double-double number.
      (package private) static DD twoDiff​(double a, double b)
      Compute the difference of two numbers a and b using Knuth's two-sum algorithm.
      private static double twoDiffLow​(double a, double b, double x)
      Compute the round-off of the difference of two numbers a and b using Knuth two-sum algorithm.
      (package private) static double twoPow​(int n)
      Create a normalized double with the value 2^n.
      (package private) static DD twoProd​(double x, double y)
      Compute the double-double number (z,zz) for the exact product of x and y.
      (package private) static double twoProductLow​(double x, double y, double xy)
      Compute the low part of the double length number (z,zz) for the exact product of x and y using Dekker's mult12 algorithm.
      (package private) static double twoProductLow​(double hx, double lx, double hy, double ly, double xy)
      Compute the low part of the double length number (z,zz) for the exact product of x and y using Dekker's mult12 algorithm.
      (package private) static DD twoSquare​(double x)
      Compute the double-double number (z,zz) for the exact square of x.
      (package private) static double twoSquareLow​(double x, double x2)
      Compute the low part of the double length number (z,zz) for the exact square of x using Dekker's mult12 algorithm.
      (package private) static double twoSquareLow​(double hx, double lx, double x2)
      Compute the low part of the double length number (z,zz) for the exact square of x using Dekker's mult12 algorithm.
      (package private) static DD twoSum​(double a, double b)
      Compute the sum of two numbers a and b using Knuth's two-sum algorithm.
      (package private) static double twoSumLow​(double a, double b, double x)
      Compute the round-off of the sum of two numbers a and b using Knuth two-sum algorithm.
      DD zero()
      Identity element.

      • Methods inherited from class java.lang.Number

        byteValue, shortValue

      • Methods inherited from class java.lang.Object

        clone, finalize, getClass, notify, notifyAll, wait, wait, wait

    • Field Detail

      • ONE

        public static final DD ONE
        A double-double number representing one.

      • ZERO

        public static final DD ZERO
        A double-double number representing zero.

      • MULTIPLIER

        private static final double MULTIPLIER
        The multiplier used to split the double value into high and low parts. From Dekker (1971): "The constant should be chosen equal to 2^(p - p/2) + 1, where p is the number of binary digits in the mantissa". Here p is 53 and the multiplier is 2^27 + 1.
        See Also:
        Constant Field Values

      • EXP_MASK

        private static final int EXP_MASK
        The mask to extract the raw 11-bit exponent. The value must be shifted 52-bits to remove the mantissa bits.
        See Also:
        Constant Field Values

      • CMP_UNSIGNED_2046

        private static final int CMP_UNSIGNED_2046
        The value 2046 converted for use if using Integer.compareUnsigned(int, int). This requires adding Integer.MIN_VALUE to 2046.
        See Also:
        Constant Field Values

      • CMP_UNSIGNED_MINUS_1

        private static final int CMP_UNSIGNED_MINUS_1
        The value -1 converted for use if using Integer.compareUnsigned(int, int). This requires adding Integer.MIN_VALUE to -1.
        See Also:
        Constant Field Values

      • CMP_UNSIGNED_1022

        private static final int CMP_UNSIGNED_1022
        The value 1022 converted for use if using Integer.compareUnsigned(int, int). This requires adding Integer.MIN_VALUE to 1022.
        See Also:
        Constant Field Values

      • TWO_POW_53

        private static final double TWO_POW_53
        2^53. Any double with a magnitude above this is an even integer.
        See Also:
        Constant Field Values

      • HIGH32_MASK

        private static final long HIGH32_MASK
        Mask to extract the high 32-bits from a long.
        See Also:
        Constant Field Values

      • UNSIGN_MASK

        private static final long UNSIGN_MASK
        Mask to remove the sign bit from a long.
        See Also:
        Constant Field Values

      • MANTISSA_MASK

        private static final long MANTISSA_MASK
        Mask to extract the 52-bit mantissa from a long representation of a double.
        See Also:
        Constant Field Values

      • EXPONENT_OFFSET

        private static final int EXPONENT_OFFSET
        Exponent offset in IEEE754 representation.
        See Also:
        Constant Field Values

      • SAFE_MULTIPLY

        private static final double SAFE_MULTIPLY
        The limit for safe multiplication of x*y, assuming values above 1. Used to maintain positive values during the power computation.
        See Also:
        Constant Field Values

      • TO_STRING_SIZE

        private static final int TO_STRING_SIZE
        The size of the buffer for toString().

        The longest double will require a sign, a maximum of 17 digits, the decimal place and the exponent, e.g. for max value this is 24 chars: -1.7976931348623157e+308. Set the buffer size to twice this and round up to a power of 2 thus allowing for formatting characters. The size is 64.

        See Also:
        Constant Field Values

      • serialVersionUID

        private static final long serialVersionUID
        Serializable version identifier.
        See Also:
        Constant Field Values

      • x

        private final double x
        The high part of the double-double number.

      • xx

        private final double xx
        The low part of the double-double number.

    • Constructor Detail

      • DD

        private DD​(double x,
           double xx)
        Create a double-double number (x, xx).
        Parameters:
        x - High part.
        xx - Low part.

    • Method Detail

      • of

        public static DD of​(double x)
        Creates the double-double number as the value (x, 0).
        Parameters:
        x - Value.
        Returns:
        the double-double

      • of

        static DD of​(double x,
             double xx)
        Creates the double-double number as the value (x, xx).

        Warning

        The arguments are used directly. No checks are made that they represent a normalized double-double number: x == x + xx.

        This method is exposed for testing.

        Parameters:
        x - High part.
        xx - Low part.
        Returns:
        the double-double
        See Also:
        twoSum(double, double)

      • of

        public static DD of​(int x)
        Creates the double-double number as the value (x, 0).

        Note this method exists to avoid using of(long) for integer arguments; the long variation is slower as it preserves all 64-bits of information.

        Parameters:
        x - Value.
        Returns:
        the double-double
        See Also:
        of(long)

      • of

        public static DD of​(long x)
        Creates the double-double number with the high part equal to (double) x and the low part equal to any remaining bits.

        Note this method preserves all 64-bits of precision. Faster construction can be achieved using up to 53-bits of precision using of((double) x).

        Parameters:
        x - Value.
        Returns:
        the double-double
        See Also:
        of(double)

      • from

        public static DD from​(java.math.BigDecimal x)
        Creates the double-double number (z, zz) using the double representation of the argument x; the low part is the double representation of the round-off error. 
         double z = x.doubleValue();
 double zz = x.subtract(new BigDecimal(z)).doubleValue();

        If the value cannot be represented as a finite value the result will have an infinite high part and the low part is undefined.

        Note: This conversion can lose information about the precision of the BigDecimal value. The result is the closest double-double representation to the value.

        Parameters:
        x - Value.
        Returns:
        the double-double

      • ofSum

        public static DD ofSum​(double x,
                       double y)
        Returns a DD whose value is (x + y). The values are not required to be ordered by magnitude, i.e. the result is commutative: x + y == y + x.

        This method ignores special handling of non-normal numbers and overflow within the extended precision computation. This creates the following special cases:

        • If x + y is infinite then the low part is NaN.
        • If x or y is infinite or NaN then the low part is NaN.
        • If x + y is sub-normal or zero then the low part is +/-0.0.

        An invalid result can be identified using isFinite().

        The result is the exact double-double representation of the sum.

        Parameters:
        x - Addend.
        y - Addend.
        Returns:
        the sum x + y.
        See Also:
        ofDifference(double, double)

      • ofDifference

        public static DD ofDifference​(double x,
                              double y)
        Returns a DD whose value is (x - y). The values are not required to be ordered by magnitude, i.e. the result matches a negation and addition: x - y == -y + x.

        Computes the same results as ofSum(a, -b). See that method for details of special cases.

        An invalid result can be identified using isFinite().

        The result is the exact double-double representation of the difference.

        Parameters:
        x - Minuend.
        y - Subtrahend.
        Returns:
        x - y.
        See Also:
        ofSum(double, double)

      • ofProduct

        public static DD ofProduct​(double x,
                           double y)
        Returns a DD whose value is (x * y).

        This method ignores special handling of non-normal numbers and intermediate overflow within the extended precision computation. This creates the following special cases:

        • If either |x| or |y| multiplied by 1 + 2^27 is infinite (intermediate overflow) then the low part is NaN.
        • If x * y is infinite then the low part is NaN.
        • If x or y is infinite or NaN then the low part is NaN.
        • If x * y is sub-normal or zero then the low part is +/-0.0.

        An invalid result can be identified using isFinite().

        Note: Ignoring special cases is a design choice for performance. The method is therefore not a drop-in replacement for roundOff = Math.fma(x, y, -x * y).

        The result is the exact double-double representation of the product.

        Parameters:
        x - Factor.
        y - Factor.
        Returns:
        the product x * y.

      • ofSquare

        public static DD ofSquare​(double x)
        Returns a DD whose value is (x * x).

        This method is an optimisation of multiply(x, x). See that method for details of special cases.

        An invalid result can be identified using isFinite().

        The result is the exact double-double representation of the square.

        Parameters:
        x - Factor.
        Returns:
        the square x * x.
        See Also:
        ofProduct(double, double)

      • fromQuotient

        public static DD fromQuotient​(double x,
                              double y)
        Returns a DD whose value is (x / y). If y = 0 the result is undefined.

        This method ignores special handling of non-normal numbers and intermediate overflow within the extended precision computation. This creates the following special cases:

        • If either |x / y| or |y| multiplied by 1 + 2^27 is infinite (intermediate overflow) then the low part is NaN.
        • If x / y is infinite then the low part is NaN.
        • If x or y is infinite or NaN then the low part is NaN.
        • If x / y is sub-normal or zero, excluding the previous cases, then the low part is +/-0.0.

        An invalid result can be identified using isFinite().

        The result is the closest double-double representation to the quotient.

        Parameters:
        x - Dividend.
        y - Divisor.
        Returns:
        the quotient x / y.

      • hi

        public double hi()
        Gets the first part x of the double-double number (x, xx). In a normalized double-double number this part will have the greatest magnitude.

        This is equivalent to returning the high-part xhi for the number (xhi, xlo).

        Returns:
        the first part

      • lo

        public double lo()
        Gets the second part xx of the double-double number (x, xx). In a normalized double-double number this part will have the smallest magnitude.

        This is equivalent to returning the low part xlo for the number (xhi, xlo).

        Returns:
        the second part

      • isFinite

        public boolean isFinite()
        Returns true if the evaluated sum of the parts is finite.

        This method is provided as a utility to check the result of a DD computation. Note that for performance the DD class does not follow IEEE754 arithmetic for infinite and NaN, and does not protect from overflow of intermediate values in multiply and divide operations. If this method returns false following DD arithmetic then the computation is not supported to extended precision.

        Note: Any number that returns true may be converted to the exact BigDecimal value.

        Returns:
        true if this instance represents a finite double value.
        See Also:
        Double.isFinite(double), bigDecimalValue()

      • doubleValue

        public double doubleValue()
        Get the value as a double. This is the evaluated sum of the parts.

        Note that even when the return value is finite, this conversion can lose information about the precision of the DD value.

        Conversion of a finite DD can also be performed using the BigDecimal representation.

        Specified by:
        doubleValue in class java.lang.Number
        Returns:
        the value converted to a double
        See Also:
        bigDecimalValue()

      • floatValue

        public float floatValue()
        Get the value as a float. This is the narrowing primitive conversion of the doubleValue(). This conversion can lose range, resulting in a float zero from a nonzero double and a float infinity from a finite double. A double NaN is converted to a float NaN and a double infinity is converted to the same-signed float infinity.

        Note that even when the return value is finite, this conversion can lose information about the precision of the DD value.

        Conversion of a finite DD can also be performed using the BigDecimal representation.

        Specified by:
        floatValue in class java.lang.Number
        Returns:
        the value converted to a float
        See Also:
        bigDecimalValue()

      • intValue

        public int intValue()
        Get the value as an int. This conversion discards the fractional part of the number and effectively rounds the value to the closest whole number in the direction of zero. This is the equivalent of a cast of a floating-point number to an integer, for example (int) -2.75 => -2.

        Note that this conversion can lose information about the precision of the DD value.

        Special cases:

        • If the DD value is infinite the result is Integer.MAX_VALUE.
        • If the DD value is -infinite the result is Integer.MIN_VALUE.
        • If the DD value is NaN the result is 0.

        Conversion of a finite DD can also be performed using the BigDecimal representation. Note that BigDecimal conversion rounds to the BigInteger whole number representation and returns the low-order 32-bits. Numbers too large for an int may change sign. This method ensures the sign is correct by directly rounding to an int and returning the respective upper or lower limit for numbers too large for an int.

        Specified by:
        intValue in class java.lang.Number
        Returns:
        the value converted to an int
        See Also:
        bigDecimalValue()

      • longValue

        public long longValue()
        Get the value as a long. This conversion discards the fractional part of the number and effectively rounds the value to the closest whole number in the direction of zero. This is the equivalent of a cast of a floating-point number to an integer, for example (long) -2.75 => -2.

        Note that this conversion can lose information about the precision of the DD value.

        Special cases:

        • If the DD value is infinite the result is Long.MAX_VALUE.
        • If the DD value is -infinite the result is Long.MIN_VALUE.
        • If the DD value is NaN the result is 0.

        Conversion of a finite DD can also be performed using the BigDecimal representation. Note that BigDecimal conversion rounds to the BigInteger whole number representation and returns the low-order 64-bits. Numbers too large for a long may change sign. This method ensures the sign is correct by directly rounding to a long and returning the respective upper or lower limit for numbers too large for a long.

        Specified by:
        longValue in class java.lang.Number
        Returns:
        the value converted to an int
        See Also:
        bigDecimalValue()

      • bigDecimalValue

        public java.math.BigDecimal bigDecimalValue()
        Get the value as a BigDecimal. This is the evaluated sum of the parts; the conversion is exact.

        The conversion will raise a NumberFormatException if the number is non-finite.

        Returns:
        the double-double as a BigDecimal.
        Throws:
        java.lang.NumberFormatException - if any part of the number is infinite or NaN
        See Also:
        BigDecimal

      • fastTwoSum

        static DD fastTwoSum​(double a,
                     double b)
        Compute the sum of two numbers a and b using Dekker's two-sum algorithm. The values are required to be ordered by magnitude: |a| >= |b|.

        If a is zero and b is non-zero the returned value is (b, 0).

        Parameters:
        a - First part of sum.
        b - Second part of sum.
        Returns:
        the sum
        See Also:
        fastTwoDiff(double, double), Shewchuk (1997) Theorum 6

      • fastTwoSumLow

        static double fastTwoSumLow​(double a,
                            double b,
                            double x)
        Compute the round-off of the sum of two numbers a and b using Dekker's two-sum algorithm. The values are required to be ordered by magnitude: |a| >= |b|.

        If a is zero and b is non-zero the returned value is zero.

        Parameters:
        a - First part of sum.
        b - Second part of sum.
        x - Sum.
        Returns:
        the sum round-off
        See Also:
        fastTwoSum(double, double)

      • fastTwoDiff

        static DD fastTwoDiff​(double a,
                      double b)
        Compute the difference of two numbers a and b using Dekker's two-sum algorithm. The values are required to be ordered by magnitude: |a| >= |b|.

        Computes the same results as fastTwoSum(a, -b).

        Parameters:
        a - Minuend.
        b - Subtrahend.
        Returns:
        the difference
        See Also:
        fastTwoSum(double, double), Shewchuk (1997) Theorum 6

      • fastTwoDiffLow

        private static double fastTwoDiffLow​(double a,
                                     double b,
                                     double x)
        Compute the round-off of the difference of two numbers a and b using Dekker's two-sum algorithm. The values are required to be ordered by magnitude: |a| >= |b|.
        Parameters:
        a - Minuend.
        b - Subtrahend.
        x - Difference.
        Returns:
        the difference round-off
        See Also:
        fastTwoDiff(double, double)

      • twoSum

        static DD twoSum​(double a,
                 double b)
        Compute the sum of two numbers a and b using Knuth's two-sum algorithm. The values are not required to be ordered by magnitude, i.e. the result is commutative s = a + b == b + a.
        Parameters:
        a - First part of sum.
        b - Second part of sum.
        Returns:
        the sum
        See Also:
        twoDiff(double, double), Shewchuk (1997) Theorum 7

      • twoSumLow

        static double twoSumLow​(double a,
                        double b,
                        double x)
        Compute the round-off of the sum of two numbers a and b using Knuth two-sum algorithm. The values are not required to be ordered by magnitude, i.e. the result is commutative s = a + b == b + a.
        Parameters:
        a - First part of sum.
        b - Second part of sum.
        x - Sum.
        Returns:
        the sum round-off
        See Also:
        twoSum(double, double)

      • twoDiff

        static DD twoDiff​(double a,
                  double b)
        Compute the difference of two numbers a and b using Knuth's two-sum algorithm. The values are not required to be ordered by magnitude.

        Computes the same results as twoSum(a, -b).

        Parameters:
        a - Minuend.
        b - Subtrahend.
        Returns:
        the difference
        See Also:
        twoSum(double, double)

      • twoDiffLow

        private static double twoDiffLow​(double a,
                                 double b,
                                 double x)
        Compute the round-off of the difference of two numbers a and b using Knuth two-sum algorithm. The values are not required to be ordered by magnitude,
        Parameters:
        a - Minuend.
        b - Subtrahend.
        x - Difference.
        Returns:
        the difference round-off
        See Also:
        twoDiff(double, double)

      • twoProd

        static DD twoProd​(double x,
                  double y)
        Compute the double-double number (z,zz) for the exact product of x and y.

        The high part of the number is equal to the product z = x * y. The low part is set to the round-off of the double product.

        This method ignores special handling of non-normal numbers and intermediate overflow within the extended precision computation. This creates the following special cases:

        • If x * y is sub-normal or zero then the low part is +/-0.0.
        • If x * y is infinite then the low part is NaN.
        • If x or y is infinite or NaN then the low part is NaN.
        • If either |x| or |y| multiplied by 1 + 2^27 is infinite (intermediate overflow) then the low part is NaN.

        Note: Ignoring special cases is a design choice for performance. The method is therefore not a drop-in replacement for round_off = Math.fma(x, y, -x * y).

        Parameters:
        x - First factor.
        y - Second factor.
        Returns:
        the product

      • twoProductLow

        static double twoProductLow​(double x,
                            double y,
                            double xy)
        Compute the low part of the double length number (z,zz) for the exact product of x and y using Dekker's mult12 algorithm. The standard precision product x*y must be provided. The numbers x and y are split into high and low parts using Dekker's algorithm.

        Warning: This method does not perform scaling in Dekker's split and large finite numbers can create NaN results.

        Parameters:
        x - First factor.
        y - Second factor.
        xy - Product of the factors (x * y).
        Returns:
        the low part of the product double length number
        See Also:
        highPart(double)

      • twoProductLow

        static double twoProductLow​(double hx,
                            double lx,
                            double hy,
                            double ly,
                            double xy)
        Compute the low part of the double length number (z,zz) for the exact product of x and y using Dekker's mult12 algorithm. The standard precision product x*y, and the high and low parts of the factors must be provided.
        Parameters:
        hx - High-part of first factor.
        lx - Low-part of first factor.
        hy - High-part of second factor.
        ly - Low-part of second factor.
        xy - Product of the factors (x * y).
        Returns:
        the low part of the product double length number

      • twoSquare

        static DD twoSquare​(double x)
        Compute the double-double number (z,zz) for the exact square of x.

        The high part of the number is equal to the square z = x * x. The low part is set to the round-off of the double square.

        This method is an optimisation of twoProd(x, x). See that method for details of special cases.

        Parameters:
        x - Factor.
        Returns:
        the square
        See Also:
        twoProd(double, double)

      • twoSquareLow

        static double twoSquareLow​(double x,
                           double x2)
        Compute the low part of the double length number (z,zz) for the exact square of x using Dekker's mult12 algorithm. The standard precision square x*x must be provided. The number x is split into high and low parts using Dekker's algorithm.

        Warning: This method does not perform scaling in Dekker's split and large finite numbers can create NaN results.

        Parameters:
        x - Factor.
        x2 - Square of the factor (x * x).
        Returns:
        the low part of the square double length number
        See Also:
        highPart(double), twoProductLow(double, double, double)

      • twoSquareLow

        static double twoSquareLow​(double hx,
                           double lx,
                           double x2)
        Compute the low part of the double length number (z,zz) for the exact square of x using Dekker's mult12 algorithm. The standard precision square x*x, and the high and low parts of the factors must be provided.
        Parameters:
        hx - High-part of factor.
        lx - Low-part of factor.
        x2 - Square of the factor (x * x).
        Returns:
        the low part of the square double length number

      • highPart

        static double highPart​(double value)
        Implement Dekker's method to split a value into two parts. Multiplying by (2^s + 1) creates a big value from which to derive the two split parts. 
         c = (2^s + 1) * a
 a_big = c - a
 a_hi = c - a_big
 a_lo = a - a_hi
 a = a_hi + a_lo

        The multiplicand allows a p-bit value to be split into (p-s)-bit value a_hi and a non-overlapping (s-1)-bit value a_lo. Combined they have (p-1) bits of significand but the sign bit of a_lo contains a bit of information. The constant is chosen so that s is ceil(p/2) where the precision p for a double is 53-bits (1-bit of the mantissa is assumed to be 1 for a non sub-normal number) and s is 27.

        This conversion does not use scaling and the result of overflow is NaN. Overflow may occur when the exponent of the input value is above 996.

        Splitting a NaN or infinite value will return NaN.

        Parameters:
        value - Value.
        Returns:
        the high part of the value.
        See Also:
        Math.getExponent(double)

      • negate

        public DD negate()
        Returns a DD whose value is the negation of both parts of double-double number.
        Specified by:
        negate in interface Addition<DD>
        Returns:
        the negation

      • abs

        public DD abs()
        Returns a DD whose value is the absolute value of the number (x, xx) This method assumes that the low part xx is the smaller magnitude.

        Cases:

        • If the x value is negative the result is (-x, -xx).
        • If the x value is +/- 0.0 the result is (0.0, 0.0); this will remove sign information from the round-off component assumed to be zero.
        • Otherwise the result is this.
        Returns:
        the absolute value
        See Also:
        negate(), ZERO

      • floor

        public DD floor()
        Returns the largest (closest to positive infinity) DD value that is less than or equal to this number (x, xx) and is equal to a mathematical integer.

        This method may change the representation of zero and non-finite values; the result is equivalent to Math.floor(x) and the xx part is ignored.

        Cases:

        • If x is NaN, then the result is (NaN, 0).
        • If x is infinite, then the result is (x, 0).
        • If x is +/-0.0, then the result is (x, 0).
        • If x != Math.floor(x), then the result is (Math.floor(x), 0).
        • Otherwise the result is the DD value equal to the sum Math.floor(x) + Math.floor(xx).

        The result may generate a high part smaller (closer to negative infinity) than Math.floor(x) if x is a representable integer and the xx value is negative.

        Returns:
        the largest (closest to positive infinity) value that is less than or equal to this and is equal to a mathematical integer
        See Also:
        Math.floor(double), isFinite()

      • ceil

        public DD ceil()
        Returns the smallest (closest to negative infinity) DD value that is greater than or equal to this number (x, xx) and is equal to a mathematical integer.

        This method may change the representation of zero and non-finite values; the result is equivalent to Math.ceil(x) and the xx part is ignored.

        Cases:

        • If x is NaN, then the result is (NaN, 0).
        • If x is infinite, then the result is (x, 0).
        • If x is +/-0.0, then the result is (x, 0).
        • If x != Math.ceil(x), then the result is (Math.ceil(x), 0).
        • Otherwise the result is the DD value equal to the sum Math.ceil(x) + Math.ceil(xx).

        The result may generate a high part larger (closer to positive infinity) than Math.ceil(x) if x is a representable integer and the xx value is positive.

        Returns:
        the smallest (closest to negative infinity) value that is greater than or equal to this and is equal to a mathematical integer
        See Also:
        Math.ceil(double), isFinite()

      • floorOrCeil

        private static DD floorOrCeil​(double x,
                              double xx,
                              java.util.function.DoubleUnaryOperator op)
        Implementation of the floor and ceiling functions.

        Cases:

        • If x is non-finite or zero, then the result is (x, 0).
        • If x is rounded by the operator to a new value y, then the result is (y, 0).
        • Otherwise the result is the DD value equal to the sum op(x) + op(xx).
        Parameters:
        x - High part of x.
        xx - Low part of x.
        op - Floor or ceiling operator.
        Returns:
        the result

      • add

        public DD add​(double y)
        Returns a DD whose value is (this + y).

        This computes the same result as add(DD.of(y)).

        The computed result is within 2 eps of the exact result where eps is 2-106.

        Parameters:
        y - Value to be added to this number.
        Returns:
        this + y.
        See Also:
        add(DD)

      • add

        public DD add​(DD y)
        Returns a DD whose value is (this + y).

        The computed result is within 4 eps of the exact result where eps is 2-106.

        Specified by:
        add in interface Addition<DD>
        Parameters:
        y - Value to be added to this number.
        Returns:
        this + y.

      • add

        static DD add​(double x,
              double xx,
              double y,
              double yy)
        Compute the sum of (x, xx) and (y, yy).

        The computed result is within 4 eps of the exact result where eps is 2-106.

        Parameters:
        x - High part of x.
        xx - Low part of x.
        y - High part of y.
        yy - Low part of y.
        Returns:
        the sum
        See Also:
        accurateAdd(double, double, double, double)

      • accurateAdd

        static DD accurateAdd​(double x,
                      double xx,
                      double y)
        Compute the sum of (x, xx) and y.

        This computes the same result as accurateAdd(x, xx, y, 0).

        Note: This is an internal helper method used when accuracy is required. The computed result is within 1 eps of the exact result where eps is 2-106. The performance is approximately 1.5-fold slower than add(double).

        Parameters:
        x - High part of x.
        xx - Low part of x.
        y - y.
        Returns:
        the sum

      • accurateAdd

        static DD accurateAdd​(double x,
                      double xx,
                      double y,
                      double yy)
        Compute the sum of (x, xx) and (y, yy).

        The high-part of the result is within 1 ulp of the true sum e. The low-part of the result is within 1 ulp of the result of the high-part subtracted from the true sum e - hi.

        Note: This is an internal helper method used when accuracy is required. The computed result is within 1 eps of the exact result where eps is 2-106. The performance is approximately 2-fold slower than add(DD).

        Parameters:
        x - High part of x.
        xx - Low part of x.
        y - High part of y.
        yy - Low part of y.
        Returns:
        the sum

      • subtract

        public DD subtract​(double y)
        Returns a DD whose value is (this - y).

        This computes the same result as add(-y).

        The computed result is within 2 eps of the exact result where eps is 2-106.

        Parameters:
        y - Value to be subtracted from this number.
        Returns:
        this - y.
        See Also:
        subtract(DD)

      • subtract

        public DD subtract​(DD y)
        Returns a DD whose value is (this - y).

        This computes the same result as add(y.negate()).

        The computed result is within 4 eps of the exact result where eps is 2-106.

        Specified by:
        subtract in interface NativeOperators<DD>
        Parameters:
        y - Value to be subtracted from this number.
        Returns:
        this - y.

      • multiply

        public DD multiply​(double y)
        Returns a DD whose value is this * y.

        This computes the same result as multiply(DD.of(y)).

        The computed result is within 4 eps of the exact result where eps is 2-106.

        Parameters:
        y - Factor.
        Returns:
        this * y.
        See Also:
        multiply(DD)

      • multiply

        private static DD multiply​(double x,
                           double xx,
                           double y)
        Compute the multiplication product of (x, xx) and y.

        This computes the same result as multiply(x, xx, y, 0).

        The computed result is within 4 eps of the exact result where eps is 2-106.

        Parameters:
        x - High part of x.
        xx - Low part of x.
        y - High part of y.
        Returns:
        the product
        See Also:
        multiply(double, double, double, double)

      • multiply

        public DD multiply​(DD y)
        Returns a DD whose value is this * y.

        The computed result is within 4 eps of the exact result where eps is 2-106.

        Specified by:
        multiply in interface Multiplication<DD>
        Parameters:
        y - Factor.
        Returns:
        this * y.

      • multiply

        private static DD multiply​(double x,
                           double xx,
                           double y,
                           double yy)
        Compute the multiplication product of (x, xx) and (y, yy).

        The computed result is within 4 eps of the exact result where eps is 2-106.

        Parameters:
        x - High part of x.
        xx - Low part of x.
        y - High part of y.
        yy - Low part of y.
        Returns:
        the product

      • square

        public DD square()
        Returns a DD whose value is this * this.

        This method is an optimisation of multiply(this).

        The computed result is within 4 eps of the exact result where eps is 2-106.

        Returns:
        this2
        See Also:
        multiply(DD)

      • square

        private static DD square​(double x,
                         double xx)
        Compute the square of (x, xx).
        Parameters:
        x - High part of x.
        xx - Low part of x.
        Returns:
        the square

      • divide

        public DD divide​(double y)
        Returns a DD whose value is (this / y). If y = 0 the result is undefined.

        The computed result is within 1 eps of the exact result where eps is 2-106.

        Parameters:
        y - Divisor.
        Returns:
        this / y.

      • divide

        private static DD divide​(double x,
                         double xx,
                         double y)
        Compute the division of (x, xx) by y. If y = 0 the result is undefined.

        The computed result is within 1 eps of the exact result where eps is 2-106.

        Parameters:
        x - High part of x.
        xx - Low part of x.
        y - High part of y.
        Returns:
        the quotient

      • divide

        public DD divide​(DD y)
        Returns a DD whose value is (this / y). If y = 0 the result is undefined.

        The computed result is within 4 eps of the exact result where eps is 2-106.

        Specified by:
        divide in interface NativeOperators<DD>
        Parameters:
        y - Divisor.
        Returns:
        this / y.

      • divide

        private static DD divide​(double x,
                         double xx,
                         double y,
                         double yy)
        Compute the division of (x, xx) by (y, yy). If y = 0 the result is undefined.

        The computed result is within 4 eps of the exact result where eps is 2-106.

        Parameters:
        x - High part of x.
        xx - Low part of x.
        y - High part of y.
        yy - Low part of y.
        Returns:
        the quotient

      • reciprocal

        public DD reciprocal()
        Compute the reciprocal of this. If this value is zero the result is undefined.

        The computed result is within 4 eps of the exact result where eps is 2-106.

        Specified by:
        reciprocal in interface Multiplication<DD>
        Returns:
        this-1

      • reciprocal

        private static DD reciprocal​(double y,
                             double yy)
        Compute the inverse of (y, yy). If y = 0 the result is undefined.

        The computed result is within 4 eps of the exact result where eps is 2-106.

        Parameters:
        y - High part of y.
        yy - Low part of y.
        Returns:
        the inverse

      • sqrt

        public DD sqrt()
        Compute the square root of this number (x, xx).

        Uses the result Math.sqrt(x) if that result is not a finite normalized double.

        Special cases:

        • If x is NaN or less than zero, then the result is (NaN, 0).
        • If x is positive infinity, then the result is (+infinity, 0).
        • If x is positive zero or negative zero, then the result is (x, 0).

        The computed result is within 4 eps of the exact result where eps is 2-106.

        Returns:
        sqrt(this)
        See Also:
        Math.sqrt(double), Double.MIN_NORMAL

      • isNotNormal

        static boolean isNotNormal​(double a)
        Checks if the number is not normal. This is functionally equivalent to: 
        
 final double abs = Math.abs(a);
 return (abs <= Double.MIN_NORMAL || !(abs <= Double.MAX_VALUE));
        Parameters:
        a - The value.
        Returns:
        true if the value is not normal

      • scalb

        public DD scalb​(int exp)
        Multiply this number (x, xx) by an integral power of two. 
         (y, yy) = (x, xx) * 2^exp

        The result is rounded as if performed by a single correctly rounded floating-point multiply. This performs the same result as: 

         y = Math.scalb(x, exp);
 yy = Math.scalb(xx, exp);

        The implementation computes using a single multiplication if exp is in [-1022, 1023]. Otherwise the parts (x, xx) are scaled by repeated multiplication by power-of-two factors. The result is exact unless the scaling generates sub-normal parts; in this case precision may be lost by a single rounding.

        Parameters:
        exp - Power of two scale factor.
        Returns:
        the result
        See Also:
        Math.scalb(double, int), frexp(int[])

      • twoPow

        static double twoPow​(int n)
        Create a normalized double with the value 2^n.

        Warning: Do not call with n = -1023. This will create zero.

        Parameters:
        n - Exponent (in the range [-1022, 1023]).
        Returns:
        the double

      • frexp

        public DD frexp​(int[] exp)
        Convert this number x to fractional f and integral 2^exp components. 
         x = f * 2^exp

        The combined fractional part (f, ff) is in the range [0.5, 1).

        Special cases:

        • If x is zero, then the normalized fraction is zero and the exponent is zero.
        • If x is NaN, then the normalized fraction is NaN and the exponent is unspecified.
        • If x is infinite, then the normalized fraction is infinite and the exponent is unspecified.
        • If high-part x is an exact power of 2 and the low-part xx has an opposite signed non-zero magnitude then fraction high-part f will be +/-1 such that the double-double number is in the range [0.5, 1).

        This is named using the equivalent function in the standard C math.h library.

        Parameters:
        exp - Power of two scale factor (integral exponent).
        Returns:
        Fraction part.
        See Also:
        Math.getExponent(double), scalb(int), C math.h frexp

      • getScale

        private static int getScale​(double a)
        Returns a scale suitable for use with Math.scalb(double, int) to normalise the number to the interval [1, 2).

        In contrast to Math.getExponent(double) this handles sub-normal numbers by computing the number of leading zeros in the mantissa and shifting the unbiased exponent. The result is that for all finite, non-zero, numbers, the magnitude of scalb(x, -getScale(x)) is always in the range [1, 2).

        This method is a functional equivalent of the c function ilogb(double).

        The result is to be used to scale a number using Math.scalb(double, int). Hence the special case of a zero argument is handled using the return value for NaN as zero cannot be scaled. This is different from Math.getExponent(double).

        Special cases:

        • If the argument is NaN or infinite, then the result is Double.MAX_EXPONENT + 1.
        • If the argument is zero, then the result is Double.MAX_EXPONENT + 1.
        Parameters:
        a - Value.
        Returns:
        The unbiased exponent of the value to be used for scaling, or 1024 for 0, NaN or Inf
        See Also:
        Math.getExponent(double), Math.scalb(double, int), ilogb

      • pow

        public DD pow​(int n)
        Compute this number (x, xx) raised to the power n.

        Special cases:

        • If x is not a finite normalized double, the low part xx is ignored and the result is Math.pow(x, n).
        • If n = 0 the result is (1, 0).
        • If n = 1 the result is (x, xx).
        • If n = -1 the result is the reciprocal.
        • If the computation overflows the result is undefined.

        Computation uses multiplication by factors generated by repeat squaring of the value. These multiplications have no special case handling for overflow; in the event of overflow the result is undefined. The pow(int, long[]) method can be used to generate a scaled fraction result for any finite DD number and exponent.

        The computed result is approximately 16 * (n - 1) * eps of the exact result where eps is 2-106.

        Specified by:
        pow in interface NativeOperators<DD>
        Parameters:
        n - Exponent.
        Returns:
        thisn
        See Also:
        Math.pow(double, double), pow(int, long[]), isFinite()

      • computePow

        private static DD computePow​(double x,
                             double xx,
                             int n)
        Compute the number x (non-zero finite) raised to the power n.

        The input power is treated as an unsigned integer. Thus the negative value Integer.MIN_VALUE is 2^31.

        Parameters:
        x - Fractional high part of x.
        xx - Fractional low part of x.
        n - Power (in [2, 2^31]).
        Returns:
        x^n.

      • pow

        public DD pow​(int n,
              long[] exp)
        Compute this number x raised to the power n.

        The value is returned as fractional f and integral 2^exp components. 

         (x+xx)^n = (f+ff) * 2^exp

        The combined fractional part (f, ff) is in the range [0.5, 1).

        Special cases:

        • If (x, xx) is zero the high part of the fractional part is computed using Math.pow(x, n) and the exponent is 0.
        • If n = 0 the fractional part is 0.5 and the exponent is 1.
        • If (x, xx) is an exact power of 2 the fractional part is 0.5 and the exponent is the power of 2 minus 1.
        • If the result high-part is an exact power of 2 and the low-part has an opposite signed non-zero magnitude then the fraction high-part f will be +/-1 such that the double-double number is in the range [0.5, 1).
        • If the argument is not finite then a fractional representation is not possible. In this case the fraction and the scale factor is undefined.

        The computed result is approximately 16 * (n - 1) * eps of the exact result where eps is 2-106.

        Parameters:
        n - Power.
        exp - Result power of two scale factor (integral exponent).
        Returns:
        Fraction part.
        See Also:
        frexp(int[])

      • computePowScaled

        private static DD computePowScaled​(long b,
                                   double x,
                                   double xx,
                                   int n,
                                   long[] exp)
        Compute the number x (non-zero finite) raised to the power n.

        The input power is treated as an unsigned integer. Thus the negative value Integer.MIN_VALUE is 2^31.

        Parameters:
        b - Integral component 2^exp of x.
        x - Fractional high part of x.
        xx - Fractional low part of x.
        n - Power (in [2, 2^31]).
        exp - Result power of two scale factor (integral exponent).
        Returns:
        Fraction part.

      • equals

        public boolean equals​(java.lang.Object other)
        Test for equality with another object. If the other object is a DD then a comparison is made of the parts; otherwise false is returned.

        If both parts of two double-double numbers are numerically equivalent the two DD objects are considered to be equal. For this purpose, two double values are considered to be the same if and only if the method call Double.doubleToLongBits(value + 0.0) returns the identical long when applied to each value. This provides numeric equality of different representations of zero as per -0.0 == 0.0, and equality of NaN values.

        Note that in most cases, for two instances of class DD, x and y, the value of x.equals(y) is true if and only if 

          x.hi() == y.hi() && x.lo() == y.lo()

        also has the value true. However, there are exceptions:

        • Instances that contain NaN values in the same part are considered to be equal for that part, even though Double.NaN == Double.NaN has the value false.
        • Instances that share a NaN value in one part but have different values in the other part are not considered equal.

        The behavior is the same as if the components of the two double-double numbers were passed to Arrays.equals(double[], double[]): 

          Arrays.equals(new double[]{x.hi() + 0.0, x.lo() + 0.0},
                new double[]{y.hi() + 0.0, y.lo() + 0.0});

        Note: Addition of 0.0 converts signed representations of zero values -0.0 and 0.0 to a canonical 0.0.

        Overrides:
        equals in class java.lang.Object
        Parameters:
        other - Object to test for equality with this instance.
        Returns:
        true if the objects are equal, false if object is null, not an instance of DD, or not equal to this instance.
        See Also:
        Double.doubleToLongBits(double), Arrays.equals(double[], double[])

      • hashCode

        public int hashCode()
        Gets a hash code for the double-double number.

        The behavior is the same as if the parts of the double-double number were passed to Arrays.hashCode(double[]): 

          Arrays.hashCode(new double[] {hi() + 0.0, lo() + 0.0})

        Note: Addition of 0.0 provides the same hash code for different signed representations of zero values -0.0 and 0.0.

        Overrides:
        hashCode in class java.lang.Object
        Returns:
        A hash code value for this object.
        See Also:
        Arrays.hashCode(double[])

      • equals

        private static boolean equals​(double x,
                              double y)
        Returns true if the values are numerically equal.

        Two double values are considered to be the same if and only if the method call Double.doubleToLongBits(value + 0.0) returns the identical long when applied to each value. This provides numeric equality of different representations of zero as per -0.0 == 0.0, and equality of NaN values.

        Parameters:
        x - Value
        y - Value
        Returns:
        true if the values are numerically equal

      • toString

        public java.lang.String toString()
        Returns a string representation of the double-double number.

        The string will represent the numeric values of the parts. The values are split by a separator and surrounded by parentheses.

        The format for a double-double number is "(x,xx)", with x and xx converted as if using Double.toString(double).

        Note: A numerical string representation of a finite double-double number can be generated by conversion to a BigDecimal before formatting.

        Overrides:
        toString in class java.lang.Object
        Returns:
        A string representation of the double-double number.
        See Also:
        Double.toString(double), bigDecimalValue()

      • zero

        public DD zero()
        Identity element.

        Note: Addition of this value with any element a may not create an element equal to a if the element contains sign zeros. In this case the magnitude of the result will be identical.

        Specified by:
        zero in interface Addition<DD>
        Returns:
        the field element such that for all a, zero().add(a).equals(a) is true.

      • isZero

        public boolean isZero()
        Check if this is a neutral element of addition, i.e. this.add(a) returns a or an element representing the same value as a.

        The default implementation calls equals(zero()). Implementations may want to employ more a efficient method. This may even be required if an implementation has multiple representations of zero and its equals method differentiates between them.

        Specified by:
        isZero in interface Addition<DD>
        Returns:
        true if this is a neutral element of addition.
        See Also:
        Addition.zero()

      • one

        public DD one()
        Identity element.

        Note: Multiplication of this value with any element a may not create an element equal to a if the element contains sign zeros. In this case the magnitude of the result will be identical.

        Specified by:
        one in interface Multiplication<DD>
        Returns:
        the field element such that for all a, one().multiply(a).equals(a) is true.

      • isOne

        public boolean isOne()
        Check if this is a neutral element of multiplication, i.e. this.multiply(a) returns a or an element representing the same value as a.

        The default implementation calls equals(one()). Implementations may want to employ more a efficient method. This may even be required if an implementation has multiple representations of one and its equals method differentiates between them.

        Specified by:
        isOne in interface Multiplication<DD>
        Returns:
        true if this is a neutral element of multiplication.
        See Also:
        Multiplication.one()