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#
#--
# Contents:
# sqrt(x, prec)
# sin (x, prec)
# cos (x, prec)
# atan(x, prec) Note: |x|<1, x=0.9999 may not converge.
# exp (x, prec)
# log (x, prec)
# PI (prec)
# E (prec) == exp(1.0,prec)
#
# where:
# x ... BigDecimal number to be computed.
# |x| must be small enough to get convergence.
# prec ... Number of digits to be obtained.
#++
#
# Provides mathematical functions.
#
# Example:
#
# require "bigdecimal"
# require "bigdecimal/math"
#
# include BigMath
#
# a = BigDecimal((PI(100)/2).to_s)
# puts sin(a,100) # -> 0.10000000000000000000......E1
#
module BigMath
module_function
# Computes the square root of x to the specified number of digits of
# precision.
#
# BigDecimal.new('2').sqrt(16).to_s
# -> "0.14142135623730950488016887242096975E1"
#
def sqrt(x,prec)
x.sqrt(prec)
end
# Computes the sine of x to the specified number of digits of precision.
#
# If x is infinite or NaN, returns NaN.
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 > (twopi = two * BigMath.PI(prec))
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
neg ? -y : y
end
# Computes the cosine of x to the specified number of digits of precision.
#
# If x is infinite or NaN, returns NaN.
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 > (twopi = two * BigMath.PI(prec))
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
end
# Computes the arctangent of x to the specified number of digits of precision.
#
# If x is infinite or NaN, returns NaN.
def atan(x, prec)
raise ArgumentError, "Zero or negative precision for atan" if prec <= 0
return BigDecimal("NaN") if x.infinite? || x.nan?
n = prec + BigDecimal.double_fig
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
end
# Computes the value of e (the base of natural logarithms) raised to the
# power of x, to the specified number of digits of precision.
#
# If x is infinite or NaN, returns NaN.
#
# BigMath::exp(BigDecimal.new('1'), 10).to_s
# -> "0.271828182845904523536028752390026306410273E1"
def exp(x, prec)
raise ArgumentError, "Zero or negative precision for exp" if prec <= 0
return BigDecimal("NaN") if x.infinite? || x.nan?
n = prec + BigDecimal.double_fig
one = BigDecimal("1")
x = -x if neg = x < 0
x1 = one
y = one
d = y
z = one
i = 0
while d.nonzero? && ((m = n - (y.exponent - d.exponent).abs) > 0)
m = BigDecimal.double_fig if m < BigDecimal.double_fig
x1 = x1.mult(x,n)
i += 1
z *= i
d = x1.div(z,m)
y += d
end
if neg
one.div(y, prec)
else
y.round(prec - y.exponent)
end
end
# Computes the natural logarithm of x to the specified number of digits
# of precision.
#
# Returns x if x is infinite or NaN.
#
def log(x, prec)
raise ArgumentError, "Zero or negative argument for log" if x <= 0 || prec <= 0
return x if x.infinite? || x.nan?
one = BigDecimal("1")
two = BigDecimal("2")
n = prec + BigDecimal.double_fig
if (expo = x.exponent) < 0
x = x.mult(BigDecimal("1E#{-expo}"), n)
end
x = (x - one).div(x + one,n)
x2 = x.mult(x,n)
y = x
d = y
i = one
while d.nonzero? && ((m = n - (y.exponent - d.exponent).abs) > 0)
m = BigDecimal.double_fig if m < BigDecimal.double_fig
x = x2.mult(x,n)
i += two
d = x.div(i,m)
y += d
end
y *= two
if expo < 0
y += log(BigDecimal("10"),prec) * BigDecimal(expo.to_s)
end
y
end
# Computes the value of pi to the specified number of digits of precision.
def PI(prec)
raise ArgumentError, "Zero or negative argument 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
w = 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
w = 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 e (the base of natural logarithms) to the specified number of
# digits of precision.
def E(prec)
raise ArgumentError, "Zero or negative precision for E" if prec <= 0
n = prec + BigDecimal.double_fig
one = BigDecimal("1")
y = one
d = y
z = one
i = 0
while d.nonzero? && ((m = n - (y.exponent - d.exponent).abs) > 0)
m = BigDecimal.double_fig if m < BigDecimal.double_fig
i += 1
z *= i
d = one.div(z,m)
y += d
end
y
end
end
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