added statscounter code

1 files changed, 927 insertions, 0 deletions

diff --git a/vendor/bundle/ruby/3.4.0/gems/bigdecimal-4.1.2/lib/bigdecimal/math.rb b/vendor/bundle/ruby/3.4.0/gems/bigdecimal-4.1.2/lib/bigdecimal/math.rb
new file mode 100644
index 0000000..1b3357c
--- /dev/null
+++ b/vendor/bundle/ruby/3.4.0/gems/bigdecimal-4.1.2/lib/bigdecimal/math.rb
@@ -0,0 +1,927 @@
+# frozen_string_literal: true
+require 'bigdecimal'
+
+#
+#--
+# Contents:
+#   sqrt(x, prec)
+#   cbrt(x, prec)
+#   hypot(x, y, prec)
+#   sin (x, prec)
+#   cos (x, prec)
+#   tan (x, prec)
+#   asin(x, prec)
+#   acos(x, prec)
+#   atan(x, prec)
+#   atan2(y, x, prec)
+#   sinh (x, prec)
+#   cosh (x, prec)
+#   tanh (x, prec)
+#   asinh(x, prec)
+#   acosh(x, prec)
+#   atanh(x, prec)
+#   log2 (x, prec)
+#   log10(x, prec)
+#   log1p(x, prec)
+#   expm1(x, prec)
+#   erf (x, prec)
+#   erfc(x, prec)
+#   gamma(x, prec)
+#   lgamma(x, prec)
+#   frexp(x)
+#   ldexp(x, exponent)
+#   PI  (prec)
+#   E   (prec) == exp(1.0,prec)
+#
+# where:
+#   x, y ... BigDecimal number to be computed.
+#   prec ... Number of digits to be obtained.
+#++
+#
+# Provides mathematical functions.
+#
+# Example:
+#
+#   require "bigdecimal/math"
+#
+#   include BigMath
+#
+#   a = BigDecimal((PI(49)/2).to_s)
+#   puts sin(a,100) # => 0.9999999999...9999999986e0
+#
+module BigMath
+  module_function
+
+  # call-seq:
+  #   sqrt(decimal, numeric) -> BigDecimal
+  #
+  # Computes the square root of +decimal+ to the specified number of digits of
+  # precision, +numeric+.
+  #
+  #   BigMath.sqrt(BigDecimal('2'), 32).to_s
+  #   #=> "0.14142135623730950488016887242097e1"
+  #
+  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
+
+
+  # Returns [sign, reduced_x] where reduced_x is in -pi/2..pi/2
+  # and satisfies sin(x) = sign * sin(reduced_x)
+  # If add_half_pi is true, adds pi/2 to x before reduction.
+  # Precision of pi is adjusted to ensure reduced_x has the required precision.
+  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::Internal::EXTRA_PREC
+    pi_extra_prec = [x.exponent, 0].max + BigDecimal::Internal::EXTRA_PREC
+    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::Internal::EXTRA_PREC
+      elsif mod_prec + mod.exponent < prec
+        # Estimate required precision of pi
+        mod_prec = prec - mod.exponent + BigDecimal::Internal::EXTRA_PREC
+      else
+        return [div % 2 == 0 ? 1 : -1, mod.mult(1, prec)]
+      end
+    end
+  end
+
+  private_class_method def _sin_binary_splitting(x, prec) # :nodoc:
+    return x if x.zero?
+    x2 = x.mult(x, prec)
+    # Find k that satisfies x2**k / (2k+1)! < 10**(-prec)
+    log10 = Math.log(10)
+    logx = BigDecimal::Internal.float_log(x.abs)
+    step = (1..).bsearch { |k| Math.lgamma(2 * k + 1)[0] - 2 * k * logx > prec * log10 }
+    # Construct denominator sequence for binary splitting
+    # sin(x) = x*(1-x2/(2*3)*(1-x2/(4*5)*(1-x2/(6*7)*(1-x2/(8*9)*(1-...)))))
+    ds = (1..step).map {|i| -(2 * i) * (2 * i + 1) }
+    x.mult(1 + BigDecimal::Internal.taylor_sum_binary_splitting(x2, ds, prec), prec)
+  end
+
+  private_class_method def _sin_around_zero(x, prec) # :nodoc:
+    # Divide x into several parts
+    # sin(x.xxxxxxxx...) = sin(x.xx + 0.00xx + 0.0000xxxx + ...)
+    # Calculate sin of each part and restore sin(0.xxxxxxxx...) using addition theorem.
+    sin = BigDecimal(0)
+    cos = BigDecimal(1)
+    n = 2
+    while x != 0 do
+      partial_x = x.truncate(n)
+      x -= partial_x
+      s = _sin_binary_splitting(partial_x, prec)
+      c = (1 - s * s).sqrt(prec)
+      sin, cos = (sin * c).add(cos * s, prec), (cos * c).sub(sin * s, prec)
+      n *= 2
+    end
+    sin.clamp(BigDecimal(-1), BigDecimal(1))
+  end
+
+  # call-seq:
+  #   cbrt(decimal, numeric) -> BigDecimal
+  #
+  # Computes the cube root of +decimal+ to the specified number of digits of
+  # precision, +numeric+.
+  #
+  #   BigMath.cbrt(BigDecimal('2'), 32).to_s
+  #   #=> "0.12599210498948731647672106072782e1"
+  #
+  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)
+    BigDecimal::Internal.newton_loop(prec + BigDecimal::Internal::EXTRA_PREC) 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
+
+  # call-seq:
+  #   hypot(x, y, numeric) -> BigDecimal
+  #
+  # Returns sqrt(x**2 + y**2) to the specified number of digits of
+  # precision, +numeric+.
+  #
+  #   BigMath.hypot(BigDecimal('1'), BigDecimal('2'), 32).to_s
+  #   #=> "0.22360679774997896964091736687313e1"
+  #
+  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::Internal::EXTRA_PREC
+    sqrt(x.mult(x, prec2) + y.mult(y, prec2), prec)
+  end
+
+  # call-seq:
+  #   sin(decimal, numeric) -> BigDecimal
+  #
+  # 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, 32).to_s
+  #   #=> "0.70710807985947359435812921837984e0"
+  #
+  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::Internal::EXTRA_PREC
+    sign, x = _sin_periodic_reduction(x, n)
+    _sin_around_zero(x, n).mult(sign, prec)
+  end
+
+  # call-seq:
+  #   cos(decimal, numeric) -> BigDecimal
+  #
+  # 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(16), 32).to_s
+  #   #=> "-0.99999999999999999999999999999997e0"
+  #
+  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?
+    n = prec + BigDecimal::Internal::EXTRA_PREC
+    sign, x = _sin_periodic_reduction(x, n, add_half_pi: true)
+    _sin_around_zero(x, n).mult(sign, prec)
+  end
+
+  # call-seq:
+  #   tan(decimal, numeric) -> BigDecimal
+  #
+  # Computes the tangent of +decimal+ to the specified number of digits of
+  # precision, +numeric+.
+  #
+  # If +decimal+ is Infinity or NaN, returns NaN.
+  #
+  #   BigMath.tan(BigDecimal("0.0"), 4).to_s
+  #   #=> "0.0"
+  #
+  #   BigMath.tan(BigMath.PI(24) / 4, 32).to_s
+  #   #=> "0.99999999999999999999999830836025e0"
+  #
+  def tan(x, prec)
+    prec = BigDecimal::Internal.coerce_validate_prec(prec, :tan)
+    prec2 = prec + BigDecimal::Internal::EXTRA_PREC
+    sin(x, prec2).div(cos(x, prec2), prec)
+  end
+
+  # call-seq:
+  #   asin(decimal, numeric) -> BigDecimal
+  #
+  # Computes the arcsine of +decimal+ to the specified number of digits of
+  # precision, +numeric+.
+  #
+  # If +decimal+ is NaN, returns NaN.
+  #
+  #   BigMath.asin(BigDecimal('0.5'), 32).to_s
+  #   #=> "0.52359877559829887307710723054658e0"
+  #
+  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::Internal::EXTRA_PREC
+    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
+
+  # call-seq:
+  #   acos(decimal, numeric) -> BigDecimal
+  #
+  # Computes the arccosine of +decimal+ to the specified number of digits of
+  # precision, +numeric+.
+  #
+  # If +decimal+ is NaN, returns NaN.
+  #
+  #   BigMath.acos(BigDecimal('0.5'), 32).to_s
+  #   #=> "0.10471975511965977461542144610932e1"
+  #
+  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::Internal::EXTRA_PREC
+    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
+
+  # call-seq:
+  #   atan(decimal, numeric) -> BigDecimal
+  #
+  # Computes the arctangent of +decimal+ to the specified number of digits of
+  # precision, +numeric+.
+  #
+  # If +decimal+ is NaN, returns NaN.
+  #
+  #   BigMath.atan(BigDecimal('-1'), 32).to_s
+  #   #=> "-0.78539816339744830961566084581988e0"
+  #
+  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::Internal::EXTRA_PREC
+    return PI(n).div(2 * x.infinite?, prec) if x.infinite?
+
+    x = -x if neg = x < 0
+    x = BigDecimal(1).div(x, n) if inv = x < -1 || x > 1
+
+    # Solve tan(y) - x = 0 with Newton's method
+    # Repeat: y -= (tan(y) - x) * cos(y)**2
+    y = BigDecimal(Math.atan(x.to_f), 0)
+    BigDecimal::Internal.newton_loop(n) do |p|
+      s = sin(y, p)
+      c = (1 - s * s).sqrt(p)
+      y = y.sub(c * (s.sub(c * x.mult(1, p), p)), p)
+    end
+    y = PI(n) / 2 - y if inv
+    y.mult(neg ? -1 : 1, prec)
+  end
+
+  # call-seq:
+  #   atan2(decimal, decimal, numeric) -> BigDecimal
+  #
+  # Computes the arctangent of y and x to the specified number of digits of
+  # precision, +numeric+.
+  #
+  #   BigMath.atan2(BigDecimal('-1'), BigDecimal('1'), 32).to_s
+  #   #=> "-0.78539816339744830961566084581988e0"
+  #
+  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::Internal::EXTRA_PREC
+    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
+
+  # call-seq:
+  #   sinh(decimal, numeric) -> BigDecimal
+  #
+  # Computes the hyperbolic sine of +decimal+ to the specified number of digits of
+  # precision, +numeric+.
+  #
+  # If +decimal+ is NaN, returns NaN.
+  #
+  #   BigMath.sinh(BigDecimal('1'), 32).to_s
+  #   #=> "0.11752011936438014568823818505956e1"
+  #
+  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::Internal::EXTRA_PREC
+    prec2 -= x.exponent if x.exponent < 0
+    e = exp(x, prec2)
+    (e - BigDecimal(1).div(e, prec2)).div(2, prec)
+  end
+
+  # call-seq:
+  #   cosh(decimal, numeric) -> BigDecimal
+  #
+  # Computes the hyperbolic cosine of +decimal+ to the specified number of digits of
+  # precision, +numeric+.
+  #
+  # If +decimal+ is NaN, returns NaN.
+  #
+  #   BigMath.cosh(BigDecimal('1'), 32).to_s
+  #   #=> "0.15430806348152437784779056207571e1"
+  #
+  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::Internal::EXTRA_PREC
+    e = exp(x, prec2)
+    (e + BigDecimal(1).div(e, prec2)).div(2, prec)
+  end
+
+  # call-seq:
+  #   tanh(decimal, numeric) -> BigDecimal
+  #
+  # Computes the hyperbolic tangent of +decimal+ to the specified number of digits of
+  # precision, +numeric+.
+  #
+  # If +decimal+ is NaN, returns NaN.
+  #
+  #   BigMath.tanh(BigDecimal('1'), 32).to_s
+  #   #=> "0.76159415595576488811945828260479e0"
+  #
+  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::Internal::EXTRA_PREC + [-x.exponent, 0].max
+    e = exp(x, prec2)
+    einv = BigDecimal(1).div(e, prec2)
+    (e - einv).div(e + einv, prec)
+  end
+
+  # call-seq:
+  #   asinh(decimal, numeric) -> BigDecimal
+  #
+  # Computes the inverse hyperbolic sine of +decimal+ to the specified number of digits of
+  # precision, +numeric+.
+  #
+  # If +decimal+ is NaN, returns NaN.
+  #
+  #   BigMath.asinh(BigDecimal('1'), 32).to_s
+  #   #=> "0.88137358701954302523260932497979e0"
+  #
+  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::Internal::EXTRA_PREC
+    log(x + sqrt(x**2 + 1, sqrt_prec), prec)
+  end
+
+  # call-seq:
+  #   acosh(decimal, numeric) -> BigDecimal
+  #
+  # Computes the inverse hyperbolic cosine of +decimal+ to the specified number of digits of
+  # precision, +numeric+.
+  #
+  # If +decimal+ is NaN, returns NaN.
+  #
+  #   BigMath.acosh(BigDecimal('2'), 32).to_s
+  #   #=> "0.1316957896924816708625046347308e1"
+  #
+  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::Internal::EXTRA_PREC), prec)
+  end
+
+  # call-seq:
+  #   atanh(decimal, numeric) -> BigDecimal
+  #
+  # Computes the inverse hyperbolic tangent of +decimal+ to the specified number of digits of
+  # precision, +numeric+.
+  #
+  # If +decimal+ is NaN, returns NaN.
+  #
+  #   BigMath.atanh(BigDecimal('0.5'), 32).to_s
+  #   #=> "0.54930614433405484569762261846126e0"
+  #
+  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::Internal::EXTRA_PREC
+    (log(x + 1, prec2) - log(1 - x, prec2)).div(2, prec)
+  end
+
+  # call-seq:
+  #   BigMath.log2(decimal, numeric)    -> BigDecimal
+  #
+  # Computes the base 2 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.
+  #
+  #   BigMath.log2(BigDecimal('3'), 32).to_s
+  #   #=> "0.15849625007211561814537389439478e1"
+  #
+  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::Internal::EXTRA_PREC * 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::Internal::EXTRA_PREC - (v.exponent < 0 ? v.exponent : 0), BigDecimal::ROUND_HALF_UP)
+    v.mult(1, prec)
+  end
+
+  # call-seq:
+  #   BigMath.log10(decimal, numeric)    -> BigDecimal
+  #
+  # Computes the base 10 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.
+  #
+  #   BigMath.log10(BigDecimal('3'), 32).to_s
+  #   #=> "0.47712125471966243729502790325512e0"
+  #
+  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::Internal::EXTRA_PREC * 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::Internal::EXTRA_PREC - (v.exponent < 0 ? v.exponent : 0), BigDecimal::ROUND_HALF_UP)
+    v.mult(1, prec)
+  end
+
+  # call-seq:
+  #   BigMath.log1p(decimal, numeric)    -> BigDecimal
+  #
+  # Computes log(1 + decimal) to the specified number of digits of precision, +numeric+.
+  #
+  #   BigMath.log1p(BigDecimal('0.1'), 32).to_s
+  #   #=> "0.95310179804324860043952123280765e-1"
+  #
+  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
+
+  # call-seq:
+  #   BigMath.expm1(decimal, numeric)    -> BigDecimal
+  #
+  # Computes exp(decimal) - 1 to the specified number of digits of precision, +numeric+.
+  #
+  #   BigMath.expm1(BigDecimal('0.1'), 32).to_s
+  #   #=> "0.10517091807564762481170782649025e0"
+  #
+  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::Internal::EXTRA_PREC
+    elsif x < 1
+      exp_prec = prec - x.exponent + BigDecimal::Internal::EXTRA_PREC
+    else
+      exp_prec = prec
+    end
+
+    return BigDecimal(-1) if exp_prec <= 0
+
+    exp(x, exp_prec).sub(1, prec)
+  end
+
+  # call-seq:
+  #   erf(decimal, numeric) -> BigDecimal
+  #
+  # Computes the error function of +decimal+ to the specified number of digits of
+  # precision, +numeric+.
+  #
+  # If +decimal+ is NaN, returns NaN.
+  #
+  #   BigMath.erf(BigDecimal('1'), 32).to_s
+  #   #=> "0.84270079294971486934122063508261e0"
+  #
+  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)
+      return BigDecimal(1).sub(erfc, prec) if erfc
+    end
+
+    prec2 = prec + BigDecimal::Internal::EXTRA_PREC
+    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
+
+  # call-seq:
+  #   erfc(decimal, numeric) -> BigDecimal
+  #
+  # Computes the complementary error function of +decimal+ to the specified number of digits of
+  # precision, +numeric+.
+  #
+  # If +decimal+ is NaN, returns NaN.
+  #
+  #   BigMath.erfc(BigDecimal('10'), 32).to_s
+  #   #=> "0.20884875837625447570007862949578e-44"
+  #
+  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::Internal::EXTRA_PREC), prec) if x < 0.5
+    return BigDecimal::Internal.underflow_computation_result 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)/x/sqrt(pi)) extra digits
+    log10 = 2.302585092994046
+    xf = x.to_f
+    high_prec = prec + BigDecimal::Internal::EXTRA_PREC + ((xf**2 + Math.log(xf) + Math.log(Math::PI)/2) / log10).ceil
+    BigDecimal(1).sub(erf(x, high_prec), prec)
+  end
+
+  # Calculates erf(x + a)
+  private_class_method def _erf_taylor(x, a, erf_a, prec) # :nodoc:
+    return erf_a if x.zero?
+    # Let f(x+a) = erf(x+a)*exp((x+a)**2)*sqrt(pi)/2
+    #            = c0 + c1*x + c2*x**2 + c3*x**3 + c4*x**4 + ...
+    # f'(x+a) = 1+2*(x+a)*f(x+a)
+    # f'(x+a) = c1 + 2*c2*x + 3*c3*x**2 + 4*c4*x**3 + 5*c5*x**4 + ...
+    #         = 1+2*(x+a)*(c0 + c1*x + c2*x**2 + c3*x**3 + c4*x**4 + ...)
+    # therefore,
+    # c0 = f(a)
+    # c1 = 2 * a * c0 + 1
+    # c2 = (2 * c0 + 2 * a * c1) / 2
+    # c3 = (2 * c1 + 2 * a * c2) / 3
+    # c4 = (2 * c2 + 2 * a * c3) / 4
+    #
+    # All coefficients are positive when a >= 0
+
+    scale = BigDecimal(2).div(sqrt(PI(prec), prec), prec)
+    c_prev = erf_a.div(scale.mult(exp(-a*a, prec), prec), prec)
+    c_next = (2 * a * c_prev).add(1, prec).mult(x, prec)
+    sum = c_prev.add(c_next, prec)
+
+    2.step do |k|
+      cn = (c_prev.mult(x, prec) + a * c_next).mult(2, prec).mult(x, prec).div(k, prec)
+      sum = sum.add(cn, prec)
+      c_prev, c_next = c_next, cn
+      break if [c_prev, c_next].all? { |c| c.zero?  || (c.exponent < sum.exponent - prec) }
+    end
+    value = sum.mult(scale.mult(exp(-(x + a).mult(x + a, prec), prec), prec), prec)
+    value > 1 ? BigDecimal(1) : value
+  end
+
+  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::Internal::EXTRA_PREC
+    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)
+    # To calculate `exp(x2, prec)`, x2 needs extra log10(x**2) digits of precision
+    x2 = x.mult(x, prec + (2 * Math.log10(xf)).ceil)
+    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
+
+  # call-seq:
+  #   BigMath.gamma(decimal, numeric)    -> BigDecimal
+  #
+  # Computes the gamma function of +decimal+ to the specified number of
+  # digits of precision, +numeric+.
+  #
+  #   BigMath.gamma(BigDecimal('0.5'), 32).to_s
+  #   #=> "0.17724538509055160272981674833411e1"
+  #
+  def gamma(x, prec)
+    prec = BigDecimal::Internal.coerce_validate_prec(prec, :gamma)
+    x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :gamma)
+    prec2 = prec + BigDecimal::Internal::EXTRA_PREC
+    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
+
+  # call-seq:
+  #   BigMath.lgamma(decimal, numeric)    -> [BigDecimal, Integer]
+  #
+  # Computes the natural logarithm of the absolute value of the gamma function
+  # of +decimal+ to the specified number of digits of precision, +numeric+ and its sign.
+  #
+  #   BigMath.lgamma(BigDecimal('0.5'), 32)
+  #   #=> [0.57236494292470008707171367567653e0, 1]
+  #
+  def lgamma(x, prec)
+    prec = BigDecimal::Internal.coerce_validate_prec(prec, :lgamma)
+    x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :lgamma)
+    prec2 = prec + BigDecimal::Internal::EXTRA_PREC
+    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::Internal::EXTRA_PREC
+
+        # 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
+
+  # Returns sum part: sqrt(2*pi) and c[k]/(x+k) terms of Spouge's approximation
+  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) - BigDecimal::Internal.float_log(x_low_prec.add(k, low_prec)) / log10f
+    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
+
+  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
+
+  # Returns sin(pi * x), for gamma reflection formula calculation
+  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
+
+  # call-seq:
+  #   frexp(x) -> [BigDecimal, Integer]
+  #
+  # Decomposes +x+ into a normalized fraction and an integral power of ten.
+  #
+  #   BigMath.frexp(BigDecimal(123.456))
+  #   #=> [0.123456e0, 3]
+  #
+  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
+
+  # call-seq:
+  #   ldexp(fraction, exponent) -> BigDecimal
+  #
+  # Inverse of +frexp+.
+  # Returns the value of fraction * 10**exponent.
+  #
+  #   BigMath.ldexp(BigDecimal("0.123456e0"), 3)
+  #   #=> 0.123456e3
+  #
+  def ldexp(x, exponent)
+    x = BigDecimal::Internal.coerce_to_bigdecimal(x, 0, :ldexp)
+    x.finite? ? x._decimal_shift(exponent) : x
+  end
+
+  # call-seq:
+  #   PI(numeric) -> BigDecimal
+  #
+  # Computes the value of pi to the specified number of digits of precision,
+  # +numeric+.
+  #
+  #   BigMath.PI(32).to_s
+  #   #=> "0.31415926535897932384626433832795e1"
+  #
+  def PI(prec)
+    # Gauss–Legendre algorithm
+    prec = BigDecimal::Internal.coerce_validate_prec(prec, :PI)
+    n = prec + BigDecimal::Internal::EXTRA_PREC
+    a = BigDecimal(1)
+    b = BigDecimal(0.5, 0).sqrt(n)
+    s = BigDecimal(0.25, 0)
+    t = 1
+    while a != b && (a - b).exponent > 1 - n
+      c = (a - b).div(2, n)
+      a, b = (a + b).div(2, n), (a * b).sqrt(n)
+      s = s.sub(c * c * t, n)
+      t *= 2
+    end
+    (a * b).div(s, prec)
+  end
+
+  # call-seq:
+  #   E(numeric) -> BigDecimal
+  #
+  # Computes e (the base of natural logarithms) to the specified number of
+  # digits of precision, +numeric+.
+  #
+  #   BigMath.E(32).to_s
+  #   #=> "0.27182818284590452353602874713527e1"
+  #
+  def E(prec)
+    prec = BigDecimal::Internal.coerce_validate_prec(prec, :E)
+    exp(1, prec)
+  end
+end