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:

  base^a

• 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:

  base^a

• 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 e^m. We return e*ⁿ × e^m

  Parameters:

  a - power at which e should be raised

  Returns:

  e^a

• 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:

  e^a

• 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)

• logInternal

  protected static Dfp[] logInternal(Dfp[] 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. x^y = x^u * 2^b * e^c

  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:

  x^y

• 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)