Propose an algorithm for computing the following expression

Propose an algorithm for computing the following expression that avoids the loss of precision: f(x) = squareroot 1/1 - x - squareroot 1/cos (x) for x ~ 0.

Solution

Consider the reduction formula ln(1 + x) = 2 ln(1 + y)

where 1 + y = 1 + x = 1 + x 1 + 1 + x .

It could be better to work with the function log1p defined by log1p(x) = ln(1 + x) to avoid loss of precision (this time in the argument rather than in the function).

Then log1p(x) = 2log1p x 1 + 1 + x avoids loss of significance when |x| is small. However, note that argument reduction for ln or log1p is more expensive than that for exp or expm1, because of the square root.