module BigMath

Core BigMath methods for BigDecimal (log, exp) are defined here. Other methods (sin, cos, atan) are defined in ‘bigdecimal/math.rb’.

Provides mathematical functions.

Example:

require "bigdecimal/math"

include BigMath

a = BigDecimal((PI(100)/2).to_s)
puts sin(a,100) # => 0.99999999999999999999......e0

Public Class Methods

exp(decimal, numeric) → BigDecimal

Source

# File vendor/bundle/ruby/3.4.0/gems/bigdecimal-3.2.3/lib/bigdecimal.rb, line 295
def self.exp(x, prec)
  BigDecimal::Internal.validate_prec(prec, :exp)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :exp)
  return BigDecimal::Internal.nan_computation_result if x.nan?
  return x.positive? ? BigDecimal::Internal.infinity_computation_result : BigDecimal(0) if x.infinite?
  return BigDecimal(1) if x.zero?
  return BigDecimal(1).div(exp(-x, prec), prec) if x < 0

  # exp(x * 10**cnt) = exp(x)**(10**cnt)
  cnt = x > 1 ? x.exponent : 0
  prec2 = prec + BigDecimal.double_fig + cnt
  x = x._decimal_shift(-cnt)
  xn = BigDecimal(1)
  y = BigDecimal(1)

  # Taylor series for exp(x) around 0
  1.step do |i|
    n = prec2 + xn.exponent
    break if n <= 0 || xn.zero?
    x = x.mult(1, n)
    xn = xn.mult(x, n).div(i, n)
    y = y.add(xn, prec2)
  end

  # calculate exp(x * 10**cnt) from exp(x)
  # exp(x * 10**k) = exp(x * 10**(k - 1)) ** 10
  cnt.times do
    y2 = y.mult(y, prec2)
    y5 = y2.mult(y2, prec2).mult(y, prec2)
    y = y5.mult(y5, prec2)
  end

  y.mult(1, prec)
end

Computes the value of e (the base of natural logarithms) raised to the power of decimal, to the specified number of digits of precision.

If decimal is infinity, returns Infinity.

If decimal is NaN, returns NaN.

log(decimal, numeric) → BigDecimal

Source

# File vendor/bundle/ruby/3.4.0/gems/bigdecimal-3.2.3/lib/bigdecimal.rb, line 229
def self.log(x, prec)
  BigDecimal::Internal.validate_prec(prec, :log)
  raise Math::DomainError, 'Complex argument for BigMath.log' if Complex === x

  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :log)
  return BigDecimal::Internal.nan_computation_result if x.nan?
  raise Math::DomainError, 'Zero or negative argument for log' if x <= 0
  return BigDecimal::Internal.infinity_computation_result if x.infinite?
  return BigDecimal(0) if x == 1

  BigDecimal.save_limit do
    BigDecimal.limit(0)
    if x > 10 || x < 0.1
      log10 = log(BigDecimal(10), prec)
      exponent = x.exponent
      x = x._decimal_shift(-exponent)
      if x < 0.3
        x *= 10
        exponent -= 1
      end
      return log10 * exponent + log(x, prec)
    end

    x_minus_one_exponent = (x - 1).exponent
    prec += BigDecimal.double_fig

    # log(x) = log(sqrt(sqrt(sqrt(sqrt(x))))) * 2**sqrt_steps
    sqrt_steps = [Integer.sqrt(prec) + 3 * x_minus_one_exponent, 0].max

    lg2 = 0.3010299956639812
    prec2 = prec + [-x_minus_one_exponent, 0].max + (sqrt_steps * lg2).ceil

    sqrt_steps.times do
      x = x.sqrt(prec2)

      # Workaround for https://github.com/ruby/bigdecimal/issues/354
      x = x.mult(1, prec2 + BigDecimal.double_fig)
    end

    # Taylor series for log(x) around 1
    # log(x) = -log((1 + X) / (1 - X)) where X = (x - 1) / (x + 1)
    # log(x) = 2 * (X + X**3 / 3 + X**5 / 5 + X**7 / 7 + ...)
    x = (x - 1).div(x + 1, prec2)
    y = x
    x2 = x.mult(x, prec)
    1.step do |i|
      n = prec + x.exponent - y.exponent + x2.exponent
      break if n <= 0 || x.zero?
      x = x.mult(x2.round(n - x2.exponent), n)
      y = y.add(x.div(2 * i + 1, n), prec)
    end

    y.mult(2 ** (sqrt_steps + 1), prec)
  end
end

Computes the natural logarithm of decimal to the specified number of digits of precision, numeric.

If decimal is zero or negative, raises Math::DomainError.

If decimal is positive infinity, returns Infinity.

If decimal is NaN, returns NaN.

Public Instance Methods

E(numeric) → BigDecimal

Source

# File vendor/bundle/ruby/3.4.0/gems/bigdecimal-3.2.3/lib/bigdecimal/math.rb, line 230
def E(prec)
  raise ArgumentError, "Zero or negative precision for E" if prec <= 0
  BigMath.exp(1, prec)
end

Computes e (the base of natural logarithms) to the specified number of digits of precision, numeric.

BigMath.E(10).to_s
#=> "0.271828182845904523536028752390026306410273e1"

PI(numeric) → BigDecimal

Source

# File vendor/bundle/ruby/3.4.0/gems/bigdecimal-3.2.3/lib/bigdecimal/math.rb, line 185
def PI(prec)
  raise ArgumentError, "Zero or negative precision for PI" if prec <= 0
  n      = prec + BigDecimal.double_fig
  zero   = BigDecimal("0")
  one    = BigDecimal("1")
  two    = BigDecimal("2")

  m25    = BigDecimal("-0.04")
  m57121 = BigDecimal("-57121")

  pi     = zero

  d = one
  k = one
  t = BigDecimal("-80")
  while d.nonzero? && ((m = n - (pi.exponent - d.exponent).abs) > 0)
    m = BigDecimal.double_fig if m < BigDecimal.double_fig
    t   = t*m25
    d   = t.div(k,m)
    k   = k+two
    pi  = pi + d
  end

  d = one
  k = one
  t = BigDecimal("956")
  while d.nonzero? && ((m = n - (pi.exponent - d.exponent).abs) > 0)
    m = BigDecimal.double_fig if m < BigDecimal.double_fig
    t   = t.div(m57121,n)
    d   = t.div(k,m)
    pi  = pi + d
    k   = k+two
  end
  pi
end

Computes the value of pi to the specified number of digits of precision, numeric.

BigMath.PI(10).to_s
#=> "0.3141592653589793238462643388813853786957412e1"

atan(decimal, numeric) → BigDecimal

Source

# File vendor/bundle/ruby/3.4.0/gems/bigdecimal-3.2.3/lib/bigdecimal/math.rb, line 148
def atan(x, prec)
  raise ArgumentError, "Zero or negative precision for atan" if prec <= 0
  return BigDecimal("NaN") if x.nan?
  pi = PI(prec)
  x = -x if neg = x < 0
  return pi.div(neg ? -2 : 2, prec) if x.infinite?
  return pi / (neg ? -4 : 4) if x.round(prec) == 1
  n = prec + BigDecimal.double_fig
  x = BigDecimal("1").div(x, n) if inv = x > 1
  x = (-1 + sqrt(1 + x.mult(x, n), n)).div(x, n) if dbl = x > 0.5
  y = x
  d = y
  t = x
  r = BigDecimal("3")
  x2 = x.mult(x,n)
  while d.nonzero? && ((m = n - (y.exponent - d.exponent).abs) > 0)
    m = BigDecimal.double_fig if m < BigDecimal.double_fig
    t = -t.mult(x2,n)
    d = t.div(r,m)
    y += d
    r += 2
  end
  y *= 2 if dbl
  y = pi / 2 - y if inv
  y = -y if neg
  y
end

Computes the arctangent of decimal to the specified number of digits of precision, numeric.

If decimal is NaN, returns NaN.

BigMath.atan(BigDecimal('-1'), 16).to_s
#=> "-0.785398163397448309615660845819878471907514682065e0"

cos(decimal, numeric) → BigDecimal

Source

# File vendor/bundle/ruby/3.4.0/gems/bigdecimal-3.2.3/lib/bigdecimal/math.rb, line 103
def cos(x, prec)
  raise ArgumentError, "Zero or negative precision for cos" if prec <= 0
  return BigDecimal("NaN") if x.infinite? || x.nan?
  n    = prec + BigDecimal.double_fig
  one  = BigDecimal("1")
  two  = BigDecimal("2")
  x = -x if x < 0
  if x > 6
    twopi = two * BigMath.PI(prec + x.exponent)
    if x > 30
      x %= twopi
    else
      x -= twopi while x > twopi
    end
  end
  x1 = one
  x2 = x.mult(x,n)
  sign = 1
  y = one
  d = y
  i = BigDecimal("0")
  z = one
  while d.nonzero? && ((m = n - (y.exponent - d.exponent).abs) > 0)
    m = BigDecimal.double_fig if m < BigDecimal.double_fig
    sign = -sign
    x1  = x2.mult(x1,n)
    i  += two
    z  *= (i-one) * i
    d   = sign * x1.div(z,m)
    y  += d
  end
  y < -1 ? BigDecimal("-1") : y > 1 ? BigDecimal("1") : y
end

Computes the cosine of decimal to the specified number of digits of precision, numeric.

If decimal is Infinity or NaN, returns NaN.

BigMath.cos(BigMath.PI(4), 16).to_s
#=> "-0.999999999999999999999999999999856613163740061349e0"

sin(decimal, numeric) → BigDecimal

Source

# File vendor/bundle/ruby/3.4.0/gems/bigdecimal-3.2.3/lib/bigdecimal/math.rb, line 57
def sin(x, prec)
  raise ArgumentError, "Zero or negative precision for sin" if prec <= 0
  return BigDecimal("NaN") if x.infinite? || x.nan?
  n    = prec + BigDecimal.double_fig
  one  = BigDecimal("1")
  two  = BigDecimal("2")
  x = -x if neg = x < 0
  if x > 6
    twopi = two * BigMath.PI(prec + x.exponent)
    if x > 30
      x %= twopi
    else
      x -= twopi while x > twopi
    end
  end
  x1   = x
  x2   = x.mult(x,n)
  sign = 1
  y    = x
  d    = y
  i    = one
  z    = one
  while d.nonzero? && ((m = n - (y.exponent - d.exponent).abs) > 0)
    m = BigDecimal.double_fig if m < BigDecimal.double_fig
    sign = -sign
    x1  = x2.mult(x1,n)
    i  += two
    z  *= (i-one) * i
    d   = sign * x1.div(z,m)
    y  += d
  end
  y = BigDecimal("1") if y > 1
  neg ? -y : y
end

Computes the sine of decimal to the specified number of digits of precision, numeric.

If decimal is Infinity or NaN, returns NaN.

BigMath.sin(BigMath.PI(5)/4, 5).to_s
#=> "0.70710678118654752440082036563292800375e0"

sqrt(decimal, numeric) → BigDecimal

Source

# File vendor/bundle/ruby/3.4.0/gems/bigdecimal-3.2.3/lib/bigdecimal/math.rb, line 42
def sqrt(x, prec)
  x.sqrt(prec)
end

Computes the square root of decimal to the specified number of digits of precision, numeric.

BigMath.sqrt(BigDecimal('2'), 16).to_s
#=> "0.1414213562373095048801688724e1"