Package org.apache.commons.numbers.core

Class DDMath

java.lang.Object

org.apache.commons.numbers.core.DDMath

public final class DDMath
extends java.lang.Object

Computes extended precision floating-point operations.

This class supplements the arithmetic operations in the DD class providing greater accuracy at the cost of performance.

Since:

1.2

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
private static class	DDMath.MDD	Mutable double-double number used for working.

Field Summary

Fields

Modifier and Type	Field	Description
private static double	HALF	0.5.
private static double	SAFE_MULTIPLY	The limit for safe multiplication of x*y, assuming values above 1.

Constructor Summary

Constructors

Modifier	Constructor	Description
private	DDMath()	No instances.

Method Summary

Modifier and Type	Method	Description
private static double	add3(double a0, double a1, double a2, double b0, double b1, double b2, DDMath.MDD s12)	Compute the sum of (a0,a1,a2) and (b0,b1,b2)).
private static double	add3(double a0, double a1, double a2, double b, DDMath.MDD s12)	Compute the sum of (a0,a1,a2) and b.
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.
private static DD	inverse3(double y, double yy, double yyy)	Compute the inverse of (y, yy, yyy).
private static double	multiply3(double a0, double a1, double a2, double b, DDMath.MDD s12)	Compute the multiplication product of (a0,a1,a2) and b.
private static double	norm3(double s0, double s1, double s2, double s3, DDMath.MDD s12)	Normalize (s0, s1, s2, s3) to (s0, s1, s2).
static DD	pow(DD x, int n, long[] exp)	Compute the number x raised to the power n.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

HALF

private static final double HALF

0.5.

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

Constructor Detail

DDMath

private DDMath()

No instances.

Method Detail

pow

public static DD pow(DD x,
                     int n,
                     long[] exp)

Compute the 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 within 1 eps of the exact result where eps is 2^-106.

The performance is approximately 4-fold slower than DD.pow(int, long[]).

Parameters:

x - Number.

n - Power.

exp - Result power of two scale factor (integral exponent).

Returns:

Fraction part.

See Also:

DD.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.

Performs the computation using triple-double precision. If the input power is negative the result is computed using the absolute value of n and then inverted by dividing into 1.

Parameters:

b - Integral component 2^b 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.

norm3

private static double norm3(double s0,
                            double s1,
                            double s2,
                            double s3,
                            DDMath.MDD s12)

Normalize (s0, s1, s2, s3) to (s0, s1, s2).

Parameters:

s0 - High part of s.

s1 - Second part of s.

s2 - Third part of s.

s3 - Fourth part of s.

s12 - Output parts (s1, s2)

Returns:

s0

inverse3

private static DD inverse3(double y,
                           double yy,
                           double yyy)

Compute the inverse of (y, yy, yyy). If y = 0 the result is undefined.

This is special routine used in #pow(int, long[]) to invert the triple precision result.

Parameters:

y - First part of y.

yy - Second part of y.

yyy - Third part of y.

Returns:

the inverse

multiply3

private static double multiply3(double a0,
                                double a1,
                                double a2,
                                double b,
                                DDMath.MDD s12)

Compute the multiplication product of (a0,a1,a2) and b.

Parameters:

a0 - High part of a.

a1 - Second part of a.

a2 - Third part of a.

b - Factor.

s12 - Output parts (s1, s2)

Returns:

s0

add3

private static double add3(double a0,
                           double a1,
                           double a2,
                           double b,
                           DDMath.MDD s12)

Compute the sum of (a0,a1,a2) and b.

Parameters:

a0 - High part of a.

a1 - Second part of a.

a2 - Third part of a.

b - Addend.

s12 - Output parts (s1, s2)

Returns:

s0

add3

private static double add3(double a0,
                           double a1,
                           double a2,
                           double b0,
                           double b1,
                           double b2,
                           DDMath.MDD s12)

Compute the sum of (a0,a1,a2) and (b0,b1,b2)). It is assumed the absolute magnitudes of a and b are equal and the sign of a and b are opposite.

Parameters:

a0 - High part of a.

a1 - Second part of a.

a2 - Third part of a.

b0 - High part of b.

b1 - Second part of b.

b2 - Third part of b.

s12 - Output parts (s1, s2)

Returns:

s0