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#--
# $Release Version: 0.5 $
# $Revision: 1.1.1.1.4.1 $
##
# = mathn
#
# mathn is a library for changing the way Ruby does math. If you need
# more precise rounding with multiple division or exponentiation
# operations, then mathn is the right tool.
#
# Without mathn:
#
# 3 / 2 => 1 # Integer
#
# With mathn:
#
# 3 / 2 => 3/2 # Rational
#
# mathn features late rounding and lacks truncation of intermediate results:
#
# Without mathn:
#
# 20 / 9 * 3 * 14 / 7 * 3 / 2 # => 18
#
# With mathn:
#
# 20 / 9 * 3 * 14 / 7 * 3 / 2 # => 20
#
#
# When you require 'mathn', the libraries for Prime, CMath, Matrix and Vector
# are also loaded.
#
# == Copyright
#
# Author: Keiju ISHITSUKA (SHL Japan Inc.)
#--
# class Numeric follows to make this documentation findable in a reasonable
# location
class Numeric; end
require "cmath.rb"
require "matrix.rb"
require "prime.rb"
require "mathn/rational"
require "mathn/complex"
unless defined?(Math.exp!)
Object.instance_eval{remove_const :Math}
Math = CMath # :nodoc:
end
##
# When mathn is required, Fixnum's division and exponentiation are enhanced to
# return more precise values from mathematical expressions.
#
# 2/3*3 # => 0
# require 'mathn'
# 2/3*3 # => 2
class Fixnum
remove_method :/
##
# +/+ defines the Rational division for Fixnum.
#
# 1/3 # => (1/3)
alias / quo
alias power! ** unless method_defined? :power!
##
# Exponentiate by +other+
def ** (other)
if self < 0 && other.round != other
Complex(self, 0.0) ** other
else
power!(other)
end
end
end
##
# When mathn is required Bignum's division and exponentiation are enhanced to
# return more precise values from mathematical expressions.
class Bignum
remove_method :/
##
# +/+ defines the Rational division for Bignum.
#
# (2**72) / ((2**70) * 3) # => 4/3
alias / quo
alias power! ** unless method_defined? :power!
##
# Exponentiate by +other+
def ** (other)
if self < 0 && other.round != other
Complex(self, 0.0) ** other
else
power!(other)
end
end
end
##
# When mathn is required Rational is changed to simplify the use of Rational
# operations.
#
# Normal behaviour:
#
# Rational.new!(1,3) ** 2 # => Rational(1, 9)
# (1 / 3) ** 2 # => 0
#
# require 'mathn' behaviour:
#
# (1 / 3) ** 2 # => 1/9
class Rational
remove_method :**
##
# Exponentiate by +other+
#
# (1/3) ** 2 # => 1/9
def ** (other)
if other.kind_of?(Rational)
other2 = other
if self < 0
return Complex(self, 0.0) ** other
elsif other == 0
return Rational(1,1)
elsif self == 0
return Rational(0,1)
elsif self == 1
return Rational(1,1)
end
npd = numerator.prime_division
dpd = denominator.prime_division
if other < 0
other = -other
npd, dpd = dpd, npd
end
for elm in npd
elm[1] = elm[1] * other
if !elm[1].kind_of?(Integer) and elm[1].denominator != 1
return Float(self) ** other2
end
elm[1] = elm[1].to_i
end
for elm in dpd
elm[1] = elm[1] * other
if !elm[1].kind_of?(Integer) and elm[1].denominator != 1
return Float(self) ** other2
end
elm[1] = elm[1].to_i
end
num = Integer.from_prime_division(npd)
den = Integer.from_prime_division(dpd)
Rational(num,den)
elsif other.kind_of?(Integer)
if other > 0
num = numerator ** other
den = denominator ** other
elsif other < 0
num = denominator ** -other
den = numerator ** -other
elsif other == 0
num = 1
den = 1
end
Rational(num, den)
elsif other.kind_of?(Float)
Float(self) ** other
else
x , y = other.coerce(self)
x ** y
end
end
end
##
# When mathn is required, the Math module changes as follows:
#
# Standard Math module behaviour:
# Math.sqrt(4/9) # => 0.0
# Math.sqrt(4.0/9.0) # => 0.666666666666667
# Math.sqrt(- 4/9) # => Errno::EDOM: Numerical argument out of domain - sqrt
#
# After require 'mathn', this is changed to:
#
# require 'mathn'
# Math.sqrt(4/9) # => 2/3
# Math.sqrt(4.0/9.0) # => 0.666666666666667
# Math.sqrt(- 4/9) # => Complex(0, 2/3)
module Math
remove_method(:sqrt)
##
# Computes the square root of +a+. It makes use of Complex and
# Rational to have no rounding errors if possible.
#
# Math.sqrt(4/9) # => 2/3
# Math.sqrt(- 4/9) # => Complex(0, 2/3)
# Math.sqrt(4.0/9.0) # => 0.666666666666667
def sqrt(a)
if a.kind_of?(Complex)
abs = sqrt(a.real*a.real + a.imag*a.imag)
# if not abs.kind_of?(Rational)
# return a**Rational(1,2)
# end
x = sqrt((a.real + abs)/Rational(2))
y = sqrt((-a.real + abs)/Rational(2))
# if !(x.kind_of?(Rational) and y.kind_of?(Rational))
# return a**Rational(1,2)
# end
if a.imag >= 0
Complex(x, y)
else
Complex(x, -y)
end
elsif a.respond_to?(:nan?) and a.nan?
a
elsif a >= 0
rsqrt(a)
else
Complex(0,rsqrt(-a))
end
end
##
# Compute square root of a non negative number. This method is
# internally used by +Math.sqrt+.
def rsqrt(a)
if a.kind_of?(Float)
sqrt!(a)
elsif a.kind_of?(Rational)
rsqrt(a.numerator)/rsqrt(a.denominator)
else
src = a
max = 2 ** 32
byte_a = [src & 0xffffffff]
# ruby's bug
while (src >= max) and (src >>= 32)
byte_a.unshift src & 0xffffffff
end
answer = 0
main = 0
side = 0
for elm in byte_a
main = (main << 32) + elm
side <<= 16
if answer != 0
if main * 4 < side * side
applo = main.div(side)
else
applo = ((sqrt!(side * side + 4 * main) - side)/2.0).to_i + 1
end
else
applo = sqrt!(main).to_i + 1
end
while (x = (side + applo) * applo) > main
applo -= 1
end
main -= x
answer = (answer << 16) + applo
side += applo * 2
end
if main == 0
answer
else
sqrt!(a)
end
end
end
class << self
remove_method(:sqrt)
end
module_function :sqrt
module_function :rsqrt
end
##
# When mathn is required, Float is changed to handle Complex numbers.
class Float
alias power! **
##
# Exponentiate by +other+
def ** (other)
if self < 0 && other.round != other
Complex(self, 0.0) ** other
else
power!(other)
end
end
end