Package org.apache.commons.numbers.core

Class DD

java.lang.Object

java.lang.Number

org.apache.commons.numbers.core.DD

All Implemented Interfaces:

Serializable, Addition<DD>, Multiplication<DD>, NativeOperators<DD>

public final class DD
extends Number
implements NativeOperators<DD>, 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 (x_hi, x_lo), 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
static final DD	ONE	A double-double number representing one.
static final DD	ZERO	A double-double number representing zero.

Method Summary

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.
DD	add(double y)	Returns a DD whose value is (this + y).
DD	add(DD y)	Returns a DD whose value is (this + y).
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.
DD	divide(double y)	Returns a DD whose value is (this / y).
DD	divide(DD y)	Returns a DD whose value is (this / y).
double	doubleValue()	Get the value as a double.
boolean	equals(Object other)	Test for equality with another object.
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.
DD	frexp(int[] exp)	Convert this number x to fractional f and integral 2^exp components.
static DD	from(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).
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).
int	intValue()	Get the value as an int.
boolean	isFinite()	Returns true if the evaluated sum of the parts is finite.
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.
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).
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.
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.
DD	subtract(double y)	Returns a DD whose value is (this - y).
DD	subtract(DD y)	Returns a DD whose value is (this - y).
String	toString()	Returns a string representation of the double-double number.
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 Details

ONE

public static final DD ONE

A double-double number representing one.

ZERO

public static final DD ZERO

A double-double number representing zero.

Method Details

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

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(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 x_hi for the number (x_hi, x_lo).

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 x_lo for the number (x_hi, x_lo).

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 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 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 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 Number

Returns:

the value converted to an int

See Also:

• bigDecimalValue()

bigDecimalValue

public 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:

NumberFormatException - if any part of the number is infinite or NaN

See Also:

• BigDecimal

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()

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.

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

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.

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)

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

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.

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

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

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[])

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

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()

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[])

equals

public boolean equals(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 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 Object

Returns:

A hash code value for this object.

See Also:

• Arrays.hashCode(double[])

toString

public 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 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()

multiply

public DD multiply(int n)

Repeated addition.

This computes the same result as multiply((double) y).

Specified by:

multiply in interface NativeOperators<DD>

Parameters:

n - Number of times to add this to itself.

Returns:

n * this.

See Also:

• multiply(double)