Module: BigMath

# File 'lib/bigdecimal/math.rb', line 690

private_class_method def _erfc_asymptotic(x, prec) # :nodoc:
  # Let f(x) = erfc(x)*sqrt(pi)*exp(x**2)/2
  # f(x) satisfies the following differential equation:
  # 2*x*f(x) = f'(x) + 1
  # From the above equation, we can derive the following asymptotic expansion:
  # f(x) = (0..kmax).sum { (-1)**k * (2*k)! / 4**k / k! / x**(2*k)) } / x

  # This asymptotic expansion does not converge.
  # But if there is a k that satisfies (2*k)! / 4**k / k! / x**(2*k) < 10**(-prec),
  # It is enough to calculate erfc within the given precision.
  # Using Stirling's approximation, we can simplify this condition to:
  # sqrt(2)/2 + k*log(k) - k - 2*k*log(x) < -prec*log(10)
  # and the left side is minimized when k = x**2.
  prec += BigDecimal.double_fig
  xf = x.to_f
  kmax = (1..(xf ** 2).floor).bsearch do |k|
    Math.log(2) / 2 + k * Math.log(k) - k - 2 * k * Math.log(xf) < -prec * Math.log(10)
  end
  return unless kmax

  sum = BigDecimal(1)
  x2 = x.mult(x, prec)
  d = BigDecimal(1)
  (1..kmax).each do |k|
    d = d.div(x2, prec).mult(1 - 2 * k, prec).div(2, prec)
    sum = sum.add(d, prec)
  end
  sum.div(exp(x2, prec).mult(PI(prec).sqrt(prec), prec), prec).div(x, prec)
end

# File 'lib/bigdecimal.rb', line 310

private_class_method def _exp_taylor(x, prec) # :nodoc:
  xn = BigDecimal(1)
  y = BigDecimal(1)
  1.step do |i|
    n = prec + xn.exponent
    break if n <= 0 || xn.zero?
    xn = xn.mult(x, n).div(i, n)
    y = y.add(xn, prec)
  end
  y
end

# File 'lib/bigdecimal/math.rb', line 842

private_class_method def _gamma_positive_integer(x, prec) # :nodoc:
  return x if x == 1
  numbers = (1..x - 1).map {|i| BigDecimal(i) }
  while numbers.size > 1
    numbers = numbers.each_slice(2).map {|a, b| b ? a.mult(b, prec) : a }
  end
  numbers.first
end

# File 'lib/bigdecimal/math.rb', line 802

private_class_method def _gamma_spouge_sum_part(x, prec) # :nodoc:
  x -= 1
  # Spouge's approximation
  # x! = (x + a)**(x + 0.5) * exp(-x - a) * (sqrt(2 * pi)  + (1..a - 1).sum{|k| c[k] / (x + k) } + epsilon)
  # where c[k] = (-1)**k * (a - k)**(k - 0.5) * exp(a - k) / (k - 1)!
  # and epsilon is bounded by a**(-0.5) * (2 * pi) ** (-a - 0.5)

  # Estimate required a for given precision
  a = (prec / Math.log10(2 * Math::PI)).ceil

  # Calculate exponent of c[k] in low precision to estimate required precision
  low_prec = 16
  log10f = Math.log(10)
  x_low_prec = x.mult(1, low_prec)
  loggamma_k = 0
  ck_exponents = (1..a-1).map do |k|
    loggamma_k += Math.log10(k - 1) if k > 1
    -loggamma_k - k / log10f + (k - 0.5) * Math.log10(a - k) - BigMath.log10(x_low_prec.add(k, low_prec), low_prec)
  end

  # Estimate exponent of sum by Stirling's approximation
  approx_sum_exponent = x < 1 ? -Math.log10(a) / 2 : Math.log10(2 * Math::PI) / 2 + x_low_prec.add(0.5, low_prec) * Math.log10(x_low_prec / x_low_prec.add(a, low_prec))

  # Determine required precision of c[k]
  prec2 = [ck_exponents.max.ceil - approx_sum_exponent.floor, 0].max + prec

  einv = BigMath.exp(-1, prec2)
  sum = (PI(prec) * 2).sqrt(prec).mult(BigMath.exp(-a, prec), prec)
  y = BigDecimal(1)
  (1..a - 1).each do |k|
    # c[k] = (-1)**k * (a - k)**(k - 0.5) * exp(-k) / (k-1)! / (x + k)
    y = y.div(1 - k, prec2) if k > 1
    y = y.mult(einv, prec2)
    z = y.mult(BigDecimal((a - k) ** k), prec2).div(BigDecimal(a - k).sqrt(prec2).mult(x.add(k, prec2), prec2), prec2)
    # sum += c[k] / (x + k)
    sum = sum.add(z, prec2)
  end
  [a, sum]
end

# File 'lib/bigdecimal/math.rb', line 75

private_class_method def _sin_periodic_reduction(x, prec, add_half_pi: false) # :nodoc:
  return [1, x] if -Math::PI/2 <= x && x <= Math::PI/2 && !add_half_pi

  mod_prec = prec + BigDecimal.double_fig
  pi_extra_prec = [x.exponent, 0].max + BigDecimal.double_fig
  while true
    pi = PI(mod_prec + pi_extra_prec)
    half_pi = pi / 2
    div, mod = (add_half_pi ? x + pi : x + half_pi).divmod(pi)
    mod -= half_pi
    if mod.zero? || mod_prec + mod.exponent <= 0
      # mod is too small to estimate required pi precision
      mod_prec = mod_prec * 3 / 2 + BigDecimal.double_fig
    elsif mod_prec + mod.exponent < prec
      # Estimate required precision of pi
      mod_prec = prec - mod.exponent + BigDecimal.double_fig
    else
      return [div % 2 == 0 ? 1 : -1, mod.mult(1, prec)]
    end
  end
end

# File 'lib/bigdecimal/math.rb', line 852

private_class_method def _sinpix(x, pi, prec) # :nodoc:
  x = x % 2
  sign = x > 1 ? -1 : 1
  x %= 1
  x = 1 - x if x > 0.5 # to avoid sin(pi*x) loss of precision for x close to 1
  sign * sin(x.mult(pi, prec), prec)
end

# File 'lib/bigdecimal/math.rb', line 256

def acos(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :acos)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :acos)
  raise Math::DomainError, "Out of domain argument for acos" if x < -1 || x > 1
  return BigDecimal::Internal.nan_computation_result if x.nan?

  prec2 = prec + BigDecimal.double_fig
  return (PI(prec2) / 2).sub(asin(x, prec2), prec) if x < 0
  return PI(prec2).div(2, prec) if x.zero?

  sin = (1 - x**2).sqrt(prec2)
  atan(sin.div(x, prec2), prec)
end

# File 'lib/bigdecimal/math.rb', line 446

def acosh(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :acosh)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :acosh)
  raise Math::DomainError, "Out of domain argument for acosh" if x < 1
  return BigDecimal::Internal.infinity_computation_result if x.infinite?
  return BigDecimal::Internal.nan_computation_result if x.nan?

  log(x + sqrt(x**2 - 1, prec + BigDecimal.double_fig), prec)
end

# File 'lib/bigdecimal/math.rb', line 230

def asin(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :asin)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :asin)
  raise Math::DomainError, "Out of domain argument for asin" if x < -1 || x > 1
  return BigDecimal::Internal.nan_computation_result if x.nan?

  prec2 = prec + BigDecimal.double_fig
  cos = (1 - x**2).sqrt(prec2)
  if cos.zero?
    PI(prec2).div(x > 0 ? 2 : -2, prec)
  else
    atan(x.div(cos, prec2), prec)
  end
end

# File 'lib/bigdecimal/math.rb', line 424

def asinh(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :asinh)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :asinh)
  return BigDecimal::Internal.nan_computation_result if x.nan?
  return BigDecimal::Internal.infinity_computation_result * x.infinite? if x.infinite?
  return -asinh(-x, prec) if x < 0

  sqrt_prec = prec + [-x.exponent, 0].max + BigDecimal.double_fig
  log(x + sqrt(x**2 + 1, sqrt_prec), prec)
end

# File 'lib/bigdecimal/math.rb', line 281

def atan(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :atan)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :atan)
  return BigDecimal::Internal.nan_computation_result if x.nan?
  n = prec + BigDecimal.double_fig
  pi = PI(n)
  x = -x if neg = x < 0
  return pi.div(neg ? -2 : 2, prec) if x.infinite?
  return pi.div(neg ? -4 : 4, prec) if x.round(n) == 1
  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.mult(1, prec)
end

# File 'lib/bigdecimal/math.rb', line 319

def atan2(y, x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :atan2)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :atan2)
  y = BigDecimal::Internal.coerce_to_bigdecimal(y, prec, :atan2)
  return BigDecimal::Internal.nan_computation_result if x.nan? || y.nan?

  if x.infinite? || y.infinite?
    one = BigDecimal(1)
    zero = BigDecimal(0)
    x = x.infinite? ? (x > 0 ? one : -one) : zero
    y = y.infinite? ? (y > 0 ? one : -one) : y.sign * zero
  end

  return x.sign >= 0 ? BigDecimal(0) : y.sign * PI(prec) if y.zero?

  y = -y if neg = y < 0
  xlarge = y.abs < x.abs
  prec2 = prec + BigDecimal.double_fig
  if x > 0
    v = xlarge ? atan(y.div(x, prec2), prec) : PI(prec2) / 2 - atan(x.div(y, prec2), prec2)
  else
    v = xlarge ? PI(prec2) - atan(-y.div(x, prec2), prec2) : PI(prec2) / 2 + atan(x.div(-y, prec2), prec2)
  end
  v.mult(neg ? -1 : 1, prec)
end

# File 'lib/bigdecimal/math.rb', line 467

def atanh(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :atanh)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :atanh)
  raise Math::DomainError, "Out of domain argument for atanh" if x < -1 || x > 1
  return BigDecimal::Internal.nan_computation_result if x.nan?
  return BigDecimal::Internal.infinity_computation_result if x == 1
  return -BigDecimal::Internal.infinity_computation_result if x == -1

  prec2 = prec + BigDecimal.double_fig
  (log(x + 1, prec2) - log(1 - x, prec2)).div(2, prec)
end

# File 'lib/bigdecimal/math.rb', line 106

def cbrt(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :cbrt)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :cbrt)
  return BigDecimal::Internal.nan_computation_result if x.nan?
  return BigDecimal::Internal.infinity_computation_result * x.infinite? if x.infinite?
  return BigDecimal(0) if x.zero?

  x = -x if neg = x < 0
  ex = x.exponent / 3
  x = x._decimal_shift(-3 * ex)
  y = BigDecimal(Math.cbrt(x.to_f), 0)
  precs = [prec + BigDecimal.double_fig]
  precs << 2 + precs.last / 2 while precs.last > BigDecimal.double_fig
  precs.reverse_each do |p|
    y = (2 * y + x.div(y, p).div(y, p)).div(3, p)
  end
  y._decimal_shift(ex).mult(neg ? -1 : 1, prec)
end

# File 'lib/bigdecimal/math.rb', line 192

def cos(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :cos)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :cos)
  return BigDecimal::Internal.nan_computation_result if x.infinite? || x.nan?
  sign, x = _sin_periodic_reduction(x, prec + BigDecimal.double_fig, add_half_pi: true)
  sign * sin(x, prec)
end

# File 'lib/bigdecimal/math.rb', line 379

def cosh(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :cosh)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :cosh)
  return BigDecimal::Internal.nan_computation_result if x.nan?
  return BigDecimal::Internal.infinity_computation_result if x.infinite?

  prec2 = prec + BigDecimal.double_fig
  e = exp(x, prec2)
  (e + BigDecimal(1).div(e, prec2)).div(2, prec)
end

# File 'lib/bigdecimal/math.rb', line 944

def E(prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :E)
  exp(1, prec)
end

# File 'lib/bigdecimal/math.rb', line 597

def erf(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :erf)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :erf)
  return BigDecimal::Internal.nan_computation_result if x.nan?
  return BigDecimal(x.infinite?) if x.infinite?
  return BigDecimal(0) if x == 0
  return -erf(-x, prec) if x < 0
  return BigDecimal(1) if x > 5000000000 # erf(5000000000) > 1 - 1e-10000000000000000000

  if x > 8
    xf = x.to_f
    log10_erfc = -xf ** 2 / Math.log(10) - Math.log10(xf * Math::PI ** 0.5)
    erfc_prec = [prec + log10_erfc.ceil, 1].max
    erfc = _erfc_asymptotic(x, erfc_prec)
    if erfc
      # Workaround for https://github.com/ruby/bigdecimal/issues/464
      return BigDecimal(1) if erfc.exponent < -prec - BigDecimal.double_fig

      return BigDecimal(1).sub(erfc, prec)
    end
  end

  prec2 = prec + BigDecimal.double_fig
  x_smallprec = x.mult(1, Integer.sqrt(prec2) / 2)
  # Taylor series of x with small precision is fast
  erf1 = _erf_taylor(x_smallprec, BigDecimal(0), BigDecimal(0), prec2)
  # Taylor series converges quickly for small x
  _erf_taylor(x - x_smallprec, x_smallprec, erf1, prec2).mult(1, prec)
end

# File 'lib/bigdecimal/math.rb', line 638

def erfc(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :erfc)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :erfc)
  return BigDecimal::Internal.nan_computation_result if x.nan?
  return BigDecimal(1 - x.infinite?) if x.infinite?
  return BigDecimal(1).sub(erf(x, prec + BigDecimal.double_fig), prec) if x < 0
  return BigDecimal(0) if x > 5000000000 # erfc(5000000000) < 1e-10000000000000000000 (underflow)

  if x >= 8
    y = _erfc_asymptotic(x, prec)
    return y.mult(1, prec) if y
  end

  # erfc(x) = 1 - erf(x) < exp(-x**2)/x/sqrt(pi)
  # Precision of erf(x) needs about log10(exp(-x**2)) extra digits
  log10 = 2.302585092994046
  high_prec = prec + BigDecimal.double_fig + (x.ceil**2 / log10).ceil
  BigDecimal(1).sub(erf(x, high_prec), prec)
end

# File 'lib/bigdecimal.rb', line 332

def exp(x, prec)
  prec = BigDecimal::Internal.coerce_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?

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

  # Calculation of exp(small_prec) is fast because calculation of x**n is fast
  # Calculation of exp(small_abs) converges fast.
  # exp(x) = exp(small_prec_part + small_abs_part) = exp(small_prec_part) * exp(small_abs_part)
  x_small_prec = x.round(Integer.sqrt(prec2))
  y = _exp_taylor(x_small_prec, prec2).mult(_exp_taylor(x.sub(x_small_prec, prec2), prec2), prec2)

  # 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

# File 'lib/bigdecimal/math.rb', line 559

def expm1(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :expm1)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :expm1)
  return BigDecimal(-1) if x.infinite? == -1

  exp_prec = prec
  if x < -1
    # log10(exp(x)) = x * log10(e)
    lg_e = 0.4342944819032518
    exp_prec = prec + (lg_e * x).ceil + BigDecimal.double_fig
  elsif x < 1
    exp_prec = prec - x.exponent + BigDecimal.double_fig
  else
    exp_prec = prec
  end

  return BigDecimal(-1) if exp_prec <= 0

  exp = exp(x, exp_prec)

  if exp.exponent > prec + BigDecimal.double_fig
    # Workaroudn for https://github.com/ruby/bigdecimal/issues/464
    exp
  else
    exp.sub(1, prec)
  end
end

# File 'lib/bigdecimal/math.rb', line 868

def frexp(x)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, 0, :frexp)
  return [x, 0] unless x.finite?

  exponent = x.exponent
  [x._decimal_shift(-exponent), exponent]
end

# File 'lib/bigdecimal/math.rb', line 729

def gamma(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :gamma)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :gamma)
  prec2 = prec + BigDecimal.double_fig
  if x < 0.5
    raise Math::DomainError, 'Numerical argument is out of domain - gamma' if x.frac.zero?

    # Euler's reflection formula: gamma(z) * gamma(1-z) = pi/sin(pi*z)
    pi = PI(prec2)
    sin = _sinpix(x, pi, prec2)
    return pi.div(gamma(1 - x, prec2).mult(sin, prec2), prec)
  elsif x.frac.zero? && x < 1000 * prec
    return _gamma_positive_integer(x, prec2).mult(1, prec)
  end

  a, sum = _gamma_spouge_sum_part(x, prec2)
  (x + (a - 1)).power(x - 0.5, prec2).mult(BigMath.exp(1 - x, prec2), prec2).mult(sum, prec)
end

# File 'lib/bigdecimal/math.rb', line 134

def hypot(x, y, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :hypot)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :hypot)
  y = BigDecimal::Internal.coerce_to_bigdecimal(y, prec, :hypot)
  return BigDecimal::Internal.nan_computation_result if x.nan? || y.nan?
  return BigDecimal::Internal.infinity_computation_result if x.infinite? || y.infinite?
  prec2 = prec + BigDecimal.double_fig
  sqrt(x.mult(x, prec2) + y.mult(y, prec2), prec)
end

# File 'lib/bigdecimal/math.rb', line 885

def ldexp(x, exponent)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, 0, :ldexp)
  x.finite? ? x._decimal_shift(exponent) : x
end

# File 'lib/bigdecimal/math.rb', line 757

def lgamma(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :lgamma)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :lgamma)
  prec2 = prec + BigDecimal.double_fig
  if x < 0.5
    return [BigDecimal::INFINITY, 1] if x.frac.zero?

    loop do
      # Euler's reflection formula: gamma(z) * gamma(1-z) = pi/sin(pi*z)
      pi = PI(prec2)
      sin = _sinpix(x, pi, prec2)
      log_gamma = BigMath.log(pi, prec2).sub(lgamma(1 - x, prec2).first + BigMath.log(sin.abs, prec2), prec)
      return [log_gamma, sin > 0 ? 1 : -1] if prec2 + log_gamma.exponent > prec + BigDecimal.double_fig

      # Retry with higher precision if loss of significance is too large
      prec2 = prec2 * 3 / 2
    end
  elsif x.frac.zero? && x < 1000 * prec
    log_gamma = BigMath.log(_gamma_positive_integer(x, prec2), prec)
    [log_gamma, 1]
  else
    # if x is close to 1 or 2, increase precision to reduce loss of significance
    diff1_exponent = (x - 1).exponent
    diff2_exponent = (x - 2).exponent
    extremely_near_one = diff1_exponent < -prec2
    extremely_near_two = diff2_exponent < -prec2

    if extremely_near_one || extremely_near_two
      # If x is extreamely close to base = 1 or 2, linear interpolation is accurate enough.
      # Taylor expansion at x = base is: (x - base) * digamma(base) + (x - base) ** 2 * trigamma(base) / 2 + ...
      # And we can ignore (x - base) ** 2 and higher order terms.
      base = extremely_near_one ? 1 : 2
      d = BigDecimal(1)._decimal_shift(1 - prec2)
      log_gamma_d, sign = lgamma(base + d, prec2)
      return [log_gamma_d.mult(x - base, prec2).div(d, prec), sign]
    end

    prec2 += [-diff1_exponent, -diff2_exponent, 0].max
    a, sum = _gamma_spouge_sum_part(x, prec2)
    log_gamma = BigMath.log(sum, prec2).add((x - 0.5).mult(BigMath.log(x.add(a - 1, prec2), prec2), prec2) + 1 - x, prec)
    [log_gamma, 1]
  end
end

# File 'lib/bigdecimal.rb', line 255

def log(x, prec)
  prec = BigDecimal::Internal.coerce_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, 'Negative argument for log' if x < 0
  return -BigDecimal::Internal.infinity_computation_result if x.zero?
  return BigDecimal::Internal.infinity_computation_result if x.infinite?
  return BigDecimal(0) if x == 1

  prec2 = prec + BigDecimal.double_fig
  BigDecimal.save_limit do
    BigDecimal.limit(0)
    if x > 10 || x < 0.1
      log10 = log(BigDecimal(10), prec2)
      exponent = x.exponent
      x = x._decimal_shift(-exponent)
      if x < 0.3
        x *= 10
        exponent -= 1
      end
      return (log10 * exponent).add(log(x, prec2), prec)
    end

    x_minus_one_exponent = (x - 1).exponent

    # log(x) = log(sqrt(sqrt(sqrt(sqrt(x))))) * 2**sqrt_steps
    sqrt_steps = [Integer.sqrt(prec2) + 3 * x_minus_one_exponent, 0].max

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

    sqrt_steps.times do
      x = x.sqrt(sqrt_prec)
    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, sqrt_prec)
    y = x
    x2 = x.mult(x, prec2)
    1.step do |i|
      n = prec2 + 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), prec2)
    end

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

# File 'lib/bigdecimal/math.rb', line 522

def log10(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :log10)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :log10)
  return BigDecimal::Internal.nan_computation_result if x.nan?
  return BigDecimal::Internal.infinity_computation_result if x.infinite? == 1

  prec2 = prec + BigDecimal.double_fig * 3 / 2
  v = log(x, prec2).div(log(BigDecimal(10), prec2), prec2)
  # Perform half-up rounding to calculate log10(10**n)==n correctly in every rounding mode
  v = v.round(prec + BigDecimal.double_fig - (v.exponent < 0 ? v.exponent : 0), BigDecimal::ROUND_HALF_UP)
  v.mult(1, prec)
end

# File 'lib/bigdecimal/math.rb', line 543

def log1p(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :log1p)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :log1p)
  raise Math::DomainError, 'Out of domain argument for log1p' if x < -1

  return log(x + 1, prec)
end

# File 'lib/bigdecimal/math.rb', line 494

def log2(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :log2)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :log2)
  return BigDecimal::Internal.nan_computation_result if x.nan?
  return BigDecimal::Internal.infinity_computation_result if x.infinite? == 1

  prec2 = prec + BigDecimal.double_fig * 3 / 2
  v = log(x, prec2).div(log(BigDecimal(2), prec2), prec2)
  # Perform half-up rounding to calculate log2(2**n)==n correctly in every rounding mode
  v = v.round(prec + BigDecimal.double_fig - (v.exponent < 0 ? v.exponent : 0), BigDecimal::ROUND_HALF_UP)
  v.mult(1, prec)
end

# File 'lib/bigdecimal/math.rb', line 899

def PI(prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :PI)
  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.mult(1, prec)
end

# File 'lib/bigdecimal/math.rb', line 155

def sin(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :sin)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :sin)
  return BigDecimal::Internal.nan_computation_result if x.infinite? || x.nan?
  n    = prec + BigDecimal.double_fig
  one  = BigDecimal("1")
  two  = BigDecimal("2")
  sign, x = _sin_periodic_reduction(x, n)
  x1   = x
  x2   = x.mult(x,n)
  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
    x1  = -x2.mult(x1,n)
    i  += two
    z  *= (i-one) * i
    d   = x1.div(z,m)
    y  += d
  end
  y = BigDecimal("1") if y > 1
  y.mult(sign, prec)
end

# File 'lib/bigdecimal/math.rb', line 356

def sinh(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :sinh)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :sinh)
  return BigDecimal::Internal.nan_computation_result if x.nan?
  return BigDecimal::Internal.infinity_computation_result * x.infinite? if x.infinite?

  prec2 = prec + BigDecimal.double_fig
  prec2 -= x.exponent if x.exponent < 0
  e = exp(x, prec2)
  (e - BigDecimal(1).div(e, prec2)).div(2, prec)
end

# File 'lib/bigdecimal/math.rb', line 64

def sqrt(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :sqrt)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :sqrt)
  x.sqrt(prec)
end

# File 'lib/bigdecimal/math.rb', line 214

def tan(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :tan)
  sin(x, prec + BigDecimal.double_fig).div(cos(x, prec + BigDecimal.double_fig), prec)
end

# File 'lib/bigdecimal/math.rb', line 401

def tanh(x, prec)
  prec = BigDecimal::Internal.coerce_validate_prec(prec, :tanh)
  x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :tanh)
  return BigDecimal::Internal.nan_computation_result if x.nan?
  return BigDecimal(x.infinite?) if x.infinite?

  prec2 = prec + BigDecimal.double_fig + [-x.exponent, 0].max
  e = exp(x, prec2)
  einv = BigDecimal(1).div(e, prec2)
  (e - einv).div(e + einv, prec)
end