if RUBY_ENGINE == 'jruby'
JRuby::Util.load_ext("org.jruby.ext.bigdecimal.BigDecimalLibrary")
class BigDecimal
def _decimal_shift(i) # :nodoc:
to_java.move_point_right(i).to_d
end
end
else
require 'bigdecimal.so'
end
class BigDecimal
module Internal # :nodoc:
# Default extra precision for intermediate calculations
# This value is currently the same as BigDecimal.double_fig, but defined separately for future changes.
EXTRA_PREC = 16
# Coerce x to BigDecimal with the specified precision.
# TODO: some methods (example: BigMath.exp) require more precision than specified to coerce.
def self.coerce_to_bigdecimal(x, prec, method_name) # :nodoc:
case x
when BigDecimal
return x
when Integer, Float
return BigDecimal(x, 0)
when Rational
return BigDecimal(x, [prec, 2 * BigDecimal.double_fig].max)
end
raise ArgumentError, "#{x.inspect} can't be coerced into BigDecimal"
end
def self.coerce_validate_prec(prec, method_name, accept_zero: false) # :nodoc:
unless Integer === prec
original = prec
# Emulate Integer.try_convert for ruby < 3.1
if prec.respond_to?(:to_int)
prec = prec.to_int
else
raise TypeError, "no implicit conversion of #{original.class} into Integer"
end
raise TypeError, "can't convert #{original.class} to Integer" unless Integer === prec
end
if accept_zero
raise ArgumentError, "Negative precision for #{method_name}" if prec < 0
else
raise ArgumentError, "Zero or negative precision for #{method_name}" if prec <= 0
end
prec
end
def self.infinity_computation_result # :nodoc:
if BigDecimal.mode(BigDecimal::EXCEPTION_ALL).anybits?(BigDecimal::EXCEPTION_INFINITY)
raise FloatDomainError, "Computation results in 'Infinity'"
end
BigDecimal::INFINITY
end
def self.nan_computation_result # :nodoc:
if BigDecimal.mode(BigDecimal::EXCEPTION_ALL).anybits?(BigDecimal::EXCEPTION_NaN)
raise FloatDomainError, "Computation results to 'NaN'"
end
BigDecimal::NAN
end
# Iteration for Newton's method with increasing precision
def self.newton_loop(prec, initial_precision: BigDecimal.double_fig / 2, safe_margin: 2) # :nodoc:
precs = []
while prec > initial_precision
precs << prec
prec = (precs.last + 1) / 2 + safe_margin
end
precs.reverse_each do |p|
yield p
end
end
# Fast and rough conversion to float for mathematical calculations.
# Bigdecimal#to_f is slow when n_significant_digits is large.
# This is because to_f internally converts BigDecimal to String
# to get the exact nearest float representation.
# TODO: Remove this workaround when BigDecimal#to_f is optimized.
def self.fast_to_f(x) # :nodoc:
x.n_significant_digits < 40 ? x.to_f : x.mult(1, 20).to_f
end
# Calculates Math.log(x.to_f) considering large or small exponent
def self.float_log(x) # :nodoc:
Math.log(fast_to_f(x._decimal_shift(-x.exponent))) + x.exponent * Math.log(10)
end
# Calculating Taylor series sum using binary splitting method
# Calculates f(x) = (x/d0)*(1+(x/d1)*(1+(x/d2)*(1+(x/d3)*(1+...))))
# x.n_significant_digits or ds.size must be small to be performant.
def self.taylor_sum_binary_splitting(x, ds, prec) # :nodoc:
fs = ds.map {|d| [0, BigDecimal(d)] }
# fs = [[a0, a1], [b0, b1], [c0, c1], ...]
# f(x) = a0/a1+(x/a1)*(1+b0/b1+(x/b1)*(1+c0/c1+(x/c1)*(1+d0/d1+(x/d1)*(1+...))))
while fs.size > 1
# Merge two adjacent fractions
# from: (1 + a0/a1 + x/a1 * (1 + b0/b1 + x/b1 * rest))
# to: (1 + (a0*b1+x*(b0+b1))/(a1*b1) + (x*x)/(a1*b1) * rest)
xn = xn ? xn.mult(xn, prec) : x
fs = fs.each_slice(2).map do |(a, b)|
b ||= [0, BigDecimal(1)._decimal_shift([xn.exponent, 0].max + 2)]
[
(a[0] * b[1]).add(xn * (b[0] + b[1]), prec),
a[1].mult(b[1], prec)
]
end
end
BigDecimal(fs[0][0]).div(fs[0][1], prec)
end
end
# call-seq:
# self ** other -> bigdecimal
#
# Returns the \BigDecimal value of +self+ raised to power +other+:
#
# b = BigDecimal('3.14')
# b ** 2 # => 0.98596e1
# b ** 2.0 # => 0.98596e1
# b ** Rational(2, 1) # => 0.98596e1
#
# Related: BigDecimal#power.
#
def **(y)
case y
when BigDecimal, Integer, Float, Rational
power(y)
when nil
raise TypeError, 'wrong argument type NilClass'
else
x, y = y.coerce(self)
x**y
end
end
# call-seq:
# power(n)
# power(n, prec)
#
# Returns the value raised to the power of n.
#
# Also available as the operator **.
#
def power(y, prec = 0)
prec = Internal.coerce_validate_prec(prec, :power, accept_zero: true)
x = self
y = Internal.coerce_to_bigdecimal(y, prec.nonzero? || n_significant_digits, :power)
return Internal.nan_computation_result if x.nan? || y.nan?
return BigDecimal(1) if y.zero?
if y.infinite?
if x < 0
return BigDecimal(0) if x < -1 && y.negative?
return BigDecimal(0) if x > -1 && y.positive?
raise Math::DomainError, 'Result undefined for negative base raised to infinite power'
elsif x < 1
return y.positive? ? BigDecimal(0) : BigDecimal::Internal.infinity_computation_result
elsif x == 1
return BigDecimal(1)
else
return y.positive? ? BigDecimal::Internal.infinity_computation_result : BigDecimal(0)
end
end
if x.infinite? && y < 0
# Computation result will be +0 or -0. Avoid overflow.
neg = x < 0 && y.frac.zero? && y % 2 == 1
return neg ? -BigDecimal(0) : BigDecimal(0)
end
if x.zero?
return BigDecimal(1) if y.zero?
return BigDecimal(0) if y > 0
if y.frac.zero? && y % 2 == 1 && x.sign == -1
return -BigDecimal::Internal.infinity_computation_result
else
return BigDecimal::Internal.infinity_computation_result
end
elsif x < 0
if y.frac.zero?
if y % 2 == 0
return (-x).power(y, prec)
else
return -(-x).power(y, prec)
end
else
raise Math::DomainError, 'Computation results in complex number'
end
elsif x == 1
return BigDecimal(1)
end
limit = BigDecimal.limit
frac_part = y.frac
if frac_part.zero? && prec.zero? && limit.zero?
# Infinite precision calculation for `x ** int` and `x.power(int)`
int_part = y.fix.to_i
int_part = -int_part if (neg = int_part < 0)
ans = BigDecimal(1)
n = 1
xn = x
while true
ans *= xn if int_part.allbits?(n)
n <<= 1
break if n > int_part
xn *= xn
# Detect overflow/underflow before consuming infinite memory
if (xn.exponent.abs - 1) * int_part / n >= 0x7FFFFFFFFFFFFFFF
return ((xn.exponent > 0) ^ neg ? BigDecimal::Internal.infinity_computation_result : BigDecimal(0)) * (int_part.even? || x > 0 ? 1 : -1)
end
end
return neg ? BigDecimal(1) / ans : ans
end
result_prec = prec.nonzero? || [x.n_significant_digits, y.n_significant_digits, BigDecimal.double_fig].max + BigDecimal.double_fig
result_prec = [result_prec, limit].min if prec.zero? && limit.nonzero?
prec2 = result_prec + BigDecimal::Internal::EXTRA_PREC
if y < 0
inv = x.power(-y, prec2)
return BigDecimal(0) if inv.infinite?
return BigDecimal::Internal.infinity_computation_result if inv.zero?
return BigDecimal(1).div(inv, result_prec)
end
if frac_part.zero? && y.exponent < Math.log(result_prec) * 5 + 20
# Use exponentiation by squaring if y is an integer and not too large
pow_prec = prec2 + y.exponent
n = 1
xn = x
ans = BigDecimal(1)
int_part = y.fix.to_i
while true
ans = ans.mult(xn, pow_prec) if int_part.allbits?(n)
n <<= 1
break if n > int_part
xn = xn.mult(xn, pow_prec)
end
ans.mult(1, result_prec)
else
if x > 1 && x.finite?
# To calculate exp(z, prec), z needs prec+max(z.exponent, 0) precision if z > 0.
# Estimate (y*log(x)).exponent
logx_exponent = x < 2 ? (x - 1).exponent : Math.log10(x.exponent).round
ylogx_exponent = y.exponent + logx_exponent
prec2 += [ylogx_exponent, 0].max
end
BigMath.exp(BigMath.log(x, prec2).mult(y, prec2), result_prec)
end
end
# Returns the square root of the value.
#
# Result has at least prec significant digits.
#
def sqrt(prec)
prec = Internal.coerce_validate_prec(prec, :sqrt, accept_zero: true)
return Internal.infinity_computation_result if infinite? == 1
raise FloatDomainError, 'sqrt of negative value' if self < 0
raise FloatDomainError, "sqrt of 'NaN'(Not a Number)" if nan?
return self if zero?
if prec == 0
limit = BigDecimal.limit
prec = n_significant_digits + BigDecimal.double_fig
prec = [limit, prec].min if limit.nonzero?
end
ex = exponent / 2
x = _decimal_shift(-2 * ex)
y = BigDecimal(Math.sqrt(BigDecimal::Internal.fast_to_f(x)), 0)
Internal.newton_loop(prec + BigDecimal::Internal::EXTRA_PREC) do |p|
y = y.add(x.div(y, p), p).div(2, p)
end
y._decimal_shift(ex).mult(1, prec)
end
end
# Core BigMath methods for BigDecimal (log, exp) are defined here.
# Other methods (sin, cos, atan) are defined in 'bigdecimal/math.rb'.
module BigMath
module_function
# call-seq:
# BigMath.log(decimal, numeric) -> BigDecimal
#
# 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.
#
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::Internal::EXTRA_PREC
# Reduce x to near 1
if x > 1.01 || x < 0.99
# log(x) = log(x/exp(logx_approx)) + logx_approx
logx_approx = BigDecimal(BigDecimal::Internal.float_log(x), 0)
x = x.div(exp(logx_approx, prec2), prec2)
else
logx_approx = BigDecimal(0)
end
# Solve exp(y) - x = 0 with Newton's method
# Repeat: y -= (exp(y) - x) / exp(y)
y = BigDecimal(BigDecimal::Internal.float_log(x), 0)
exp_additional_prec = [-(x - 1).exponent, 0].max
BigDecimal::Internal.newton_loop(prec2) do |p|
expy = exp(y, p + exp_additional_prec)
y = y.sub(expy.sub(x, p).div(expy, p), p)
end
y.add(logx_approx, prec)
end
private_class_method def _exp_binary_splitting(x, prec) # :nodoc:
return BigDecimal(1) if x.zero?
# Find k that satisfies x**k / k! < 10**(-prec)
log10 = Math.log(10)
logx = BigDecimal::Internal.float_log(x.abs)
step = (1..).bsearch { |k| Math.lgamma(k + 1)[0] - k * logx > prec * log10 }
# exp(x)-1 = x*(1+x/2*(1+x/3*(1+x/4*(1+x/5*(1+...)))))
1 + BigDecimal::Internal.taylor_sum_binary_splitting(x, [*1..step], prec)
end
# call-seq:
# BigMath.exp(decimal, numeric) -> BigDecimal
#
# 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.
#
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::Internal::EXTRA_PREC + cnt
x = x._decimal_shift(-cnt)
# Decimal form of bit-burst algorithm
# Calculate exp(x.xxxxxxxxxxxxxxxx) as
# exp(x.xx) * exp(0.00xx) * exp(0.0000xxxx) * exp(0.00000000xxxxxxxx)
x = x.mult(1, prec2)
n = 2
y = BigDecimal(1)
BigDecimal.save_limit do
BigDecimal.limit(0)
while x != 0 do
partial_x = x.truncate(n)
x -= partial_x
y = y.mult(_exp_binary_splitting(partial_x, prec2), prec2)
n *= 2
end
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
end