gauss_logarithm_approx.sf

#!/usr/bin/ruby

# Approximation for the natural logarithm of a given
# number, using a formula due to Carl Friedrich Gauss.

# The formula is:
#   log(x) ≈  π / (2 * M(1, 2^(2-P) / x)) - P * log(2)

# See also:
#   https://en.wikipedia.org/wiki/Logarithm#Arithmetic.E2.80.93geometric_mean_approximation

define π = Num.pi
define log2 = log(2)

func M(a, g, p) {       # p < 0
    loop {
        var a1 = ((a+g)/2)
        var g1 = sqrt(a*g)

        if (approx_eq(a, a1, p) && approx_eq(g, g1, p)) {
            return a
        }

        (a, g) = (a1, g1)
    }
}

func gauss_log(x, p) {
    π / 2*M(1, 2**(2-p) / x, -p) - p*log2
}

say gauss_log(        2, 100)        # 0.69314718055994530941723212145817656807550013436026
say gauss_log(       -2, 100)        # 0.69314718055994363166115731719424653785695888288578-3.14159265358979323846264338327950288419716939937511i
say gauss_log(    -2.34, 100)        # 0.85015092936960838246915232323588813675500496603394-3.14159265358979323846264338327950288419716939937511i
say gauss_log(   3 + 4i, 100)        # 1.60943791243409869684468452896225760930706010279404+0.92729521800161223242851246292242880405707410857224i
say gauss_log(-42 - 23i, 100)        # 3.86880814142895054249561036425701508230238930773225-2.64057926678620379072688927950864468829687609303423i