Accurate x - log(1+x)

[Albert Chan]

We can use x_sub_log1p(), to get accurate (e^x - 1 - x), for small x

y = x - log(1+x)
e^y = e^x / (1+x)
(e^y - 1) = (e^x - 1 - x) / (1+x)

(e^x - 1 - x) = (e^y - 1) * (1+x)

Code:

function expm1_sub_x(x)
    local z = expm1(x_sub_log1p(x))
    return z + z*x
end

lua> x = 1/1024
lua> expm1(x) - x      -- error = 967 ULP
4.769924165351404e-007
lua> expm1_sub_x(x) -- error = -1 ULP
4.769924165352429e-007

Comment: next post expm1_sub(x) take care of big x too.
It is also more consistent throughout, func_sub(x) = func(x) - x