Class DfpMath

• public class DfpMath
extends Object
Mathematical routines for use with Dfp. The constants are defined in DfpField
• Method Summary

All Methods
Modifier and Type Method Description
static Dfp acos​(Dfp a)
computes the arc-cosine of the argument.
static Dfp asin​(Dfp a)
computes the arc-sine of the argument.
static Dfp atan​(Dfp a)
computes the arc tangent of the argument Uses the typical taylor series but may reduce arguments using the following identity tan(x+y) = (tan(x) + tan(y)) / (1 - tan(x)*tan(y)) since tan(PI/8) = sqrt(2)-1, atan(x) = atan( (x - sqrt(2) + 1) / (1+x*sqrt(2) - x) + PI/8.0
protected static Dfp atanInternal​(Dfp a)
computes the arc-tangent of the argument.
static Dfp cos​(Dfp a)
computes the cosine of the argument.
protected static Dfp cosInternal​(Dfp[] a)
Computes cos(a) Used when 0 < a < pi/4.
static Dfp exp​(Dfp a)
Computes e to the given power.
protected static Dfp expInternal​(Dfp a)
Computes e to the given power.
static Dfp log​(Dfp a)
Returns the natural logarithm of a.
Computes the natural log of a number between 0 and 2.
static Dfp pow​(Dfp base, int a)
Raises base to the power a by successive squaring.
static Dfp pow​(Dfp x, Dfp y)
Computes x to the y power.
static Dfp sin​(Dfp a)
computes the sine of the argument.
protected static Dfp sinInternal​(Dfp[] a)
Computes sin(a) Used when 0 < a < pi/4.
protected static Dfp[] split​(Dfp a)
Splits a Dfp into 2 Dfp's such that their sum is equal to the input Dfp.
protected static Dfp[] split​(DfpField field, String a)
Breaks a string representation up into two dfp's.
protected static Dfp[] splitDiv​(Dfp[] a, Dfp[] b)
Divide two numbers that are split in to two pieces that are meant to be added together.
protected static Dfp[] splitMult​(Dfp[] a, Dfp[] b)
Multiply two numbers that are split in to two pieces that are meant to be added together.
protected static Dfp splitPow​(Dfp[] base, int a)
Raise a split base to the a power.
static Dfp tan​(Dfp a)
computes the tangent of the argument.
• Method Detail

• split

protected static Dfp[] split​(DfpField field,
String a)
Breaks a string representation up into two dfp's.

The two dfp are such that the sum of them is equivalent to the input string, but has higher precision than using a single dfp. This is useful for improving accuracy of exponentiation and critical multiplies.

Parameters:
field - field to which the Dfp must belong
a - string representation to split
Returns:
an array of two Dfp which sum is a
• split

protected static Dfp[] split​(Dfp a)
Splits a Dfp into 2 Dfp's such that their sum is equal to the input Dfp.
Parameters:
a - number to split
Returns:
two elements array containing the split number
• splitMult

protected static Dfp[] splitMult​(Dfp[] a,
Dfp[] b)
Multiply two numbers that are split in to two pieces that are meant to be added together. Use binomial multiplication so ab = a0 b0 + a0 b1 + a1 b0 + a1 b1 Store the first term in result0, the rest in result1
Parameters:
a - first factor of the multiplication, in split form
b - second factor of the multiplication, in split form
Returns:
a × b, in split form
• splitDiv

protected static Dfp[] splitDiv​(Dfp[] a,
Dfp[] b)
Divide two numbers that are split in to two pieces that are meant to be added together. Inverse of split multiply above: (a+b) / (c+d) = (a/c) + ( (bc-ad)/(c**2+cd) )
Parameters:
a - dividend, in split form
b - divisor, in split form
Returns:
a / b, in split form
• splitPow

protected static Dfp splitPow​(Dfp[] base,
int a)
Raise a split base to the a power.
Parameters:
base - number to raise
a - power
Returns:
basea
• pow

public static Dfp pow​(Dfp base,
int a)
Raises base to the power a by successive squaring.
Parameters:
base - number to raise
a - power
Returns:
basea
• exp

public static Dfp exp​(Dfp a)
Computes e to the given power. a is broken into two parts, such that a = n+m where n is an integer. We use pow() to compute en and a Taylor series to compute em. We return e*n × em
Parameters:
a - power at which e should be raised
Returns:
ea
• expInternal

protected static Dfp expInternal​(Dfp a)
Computes e to the given power. Where -1 < a < 1. Use the classic Taylor series. 1 + x**2/2! + x**3/3! + x**4/4! ...
Parameters:
a - power at which e should be raised
Returns:
ea
• log

public static Dfp log​(Dfp a)
Returns the natural logarithm of a. a is first split into three parts such that a = (10000^h)(2^j)k. ln(a) is computed by ln(a) = ln(5)*h + ln(2)*(h+j) + ln(k) k is in the range 2/3 < k < 4/3 and is passed on to a series expansion.
Parameters:
a - number from which logarithm is requested
Returns:
log(a)

Computes the natural log of a number between 0 and 2. Let f(x) = ln(x), We know that f'(x) = 1/x, thus from Taylor's theorum we have: ----- n+1 n f(x) = \ (-1) (x - 1) / ---------------- for 1 <= n <= infinity ----- n or 2 3 4 (x-1) (x-1) (x-1) ln(x) = (x-1) - ----- + ------ - ------ + ... 2 3 4 alternatively, 2 3 4 x x x ln(x+1) = x - - + - - - + ... 2 3 4 This series can be used to compute ln(x), but it converges too slowly. If we substitute -x for x above, we get 2 3 4 x x x ln(1-x) = -x - - - - - - + ... 2 3 4 Note that all terms are now negative. Because the even powered ones absorbed the sign. Now, subtract the series above from the previous one to get ln(x+1) - ln(1-x). Note the even terms cancel out leaving only the odd ones 3 5 7 2x 2x 2x ln(x+1) - ln(x-1) = 2x + --- + --- + ---- + ... 3 5 7 By the property of logarithms that ln(a) - ln(b) = ln (a/b) we have: 3 5 7 x+1 / x x x \ ln ----- = 2 * | x + ---- + ---- + ---- + ... | x-1 \ 3 5 7 / But now we want to find ln(a), so we need to find the value of x such that a = (x+1)/(x-1). This is easily solved to find that x = (a-1)/(a+1).
Parameters:
a - number from which logarithm is requested, in split form
Returns:
log(a)
• pow

public static Dfp pow​(Dfp x,
Dfp y)
Computes x to the y power.

Uses the following method:

1. Set u = rint(y), v = y-u
2. Compute a = v * ln(x)
3. Compute b = rint( a/ln(2) )
4. Compute c = a - b*ln(2)
5. xy = xu * 2b * ec
if |y| > 1e8, then we compute by exp(y*ln(x))

Special Cases

• if y is 0.0 or -0.0 then result is 1.0
• if y is 1.0 then result is x
• if y is NaN then result is NaN
• if x is NaN and y is not zero then result is NaN
• if |x| > 1.0 and y is +Infinity then result is +Infinity
• if |x| < 1.0 and y is -Infinity then result is +Infinity
• if |x| > 1.0 and y is -Infinity then result is +0
• if |x| < 1.0 and y is +Infinity then result is +0
• if |x| = 1.0 and y is +/-Infinity then result is NaN
• if x = +0 and y > 0 then result is +0
• if x = +Inf and y < 0 then result is +0
• if x = +0 and y < 0 then result is +Inf
• if x = +Inf and y > 0 then result is +Inf
• if x = -0 and y > 0, finite, not odd integer then result is +0
• if x = -0 and y < 0, finite, and odd integer then result is -Inf
• if x = -Inf and y > 0, finite, and odd integer then result is -Inf
• if x = -0 and y < 0, not finite odd integer then result is +Inf
• if x = -Inf and y > 0, not finite odd integer then result is +Inf
• if x < 0 and y > 0, finite, and odd integer then result is -(|x|y)
• if x < 0 and y > 0, finite, and not integer then result is NaN
Parameters:
x - base to be raised
y - power to which base should be raised
Returns:
xy
• sinInternal

protected static Dfp sinInternal​(Dfp[] a)
Computes sin(a) Used when 0 < a < pi/4. Uses the classic Taylor series. x - x**3/3! + x**5/5! ...
Parameters:
a - number from which sine is desired, in split form
Returns:
sin(a)
• cosInternal

protected static Dfp cosInternal​(Dfp[] a)
Computes cos(a) Used when 0 < a < pi/4. Uses the classic Taylor series for cosine. 1 - x**2/2! + x**4/4! ...
Parameters:
a - number from which cosine is desired, in split form
Returns:
cos(a)
• sin

public static Dfp sin​(Dfp a)
computes the sine of the argument.
Parameters:
a - number from which sine is desired
Returns:
sin(a)
• cos

public static Dfp cos​(Dfp a)
computes the cosine of the argument.
Parameters:
a - number from which cosine is desired
Returns:
cos(a)
• tan

public static Dfp tan​(Dfp a)
computes the tangent of the argument.
Parameters:
a - number from which tangent is desired
Returns:
tan(a)
• atanInternal

protected static Dfp atanInternal​(Dfp a)
computes the arc-tangent of the argument.
Parameters:
a - number from which arc-tangent is desired
Returns:
atan(a)
• atan

public static Dfp atan​(Dfp a)
computes the arc tangent of the argument Uses the typical taylor series but may reduce arguments using the following identity tan(x+y) = (tan(x) + tan(y)) / (1 - tan(x)*tan(y)) since tan(PI/8) = sqrt(2)-1, atan(x) = atan( (x - sqrt(2) + 1) / (1+x*sqrt(2) - x) + PI/8.0
Parameters:
a - number from which arc-tangent is desired
Returns:
atan(a)
• asin

public static Dfp asin​(Dfp a)
computes the arc-sine of the argument.
Parameters:
a - number from which arc-sine is desired
Returns:
asin(a)
• acos

public static Dfp acos​(Dfp a)
computes the arc-cosine of the argument.
Parameters:
a - number from which arc-cosine is desired
Returns:
acos(a)