The Mathematical-Function Computation Handbook

Abstract

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Semi-Automatic Implementation of the Complementary Error Function

“…Overall, adding the error with whichŝ approximates 61 ln(2), we deduce that t h +t approximates t(x) with an absolute error |t h + t − t| ≤ 2 −48 + 0.2583u ≤ 32.259u ≤ 1.009 • 2 −48 . for which we have |z + r | ≤ 2 −44 + 2 −48 so the error of that FP addition is bounded by 1 2 ulp(2 −44 ) = 2 −97 . z h +e approximates t(x) with an absolute error less than 0.2583u + 2 −97 ≤ 0.2584u ≤ 1.034 • 2 −55 .…”

“…Hence, what we finally compute is For x ≥ x LARGE , erfc(x) is in the subnormal domain: trying to guarantee a good relative error does not make sense. For such values, we need to focus on the absolute error, and the best possible absolute error bound is 1 2 ulp(subnormal) = 2 −1075 . The general problem is to evaluate erfc via (12) with a user-required accuracy.…”

“…z h +e approximates t(x) with an absolute error less than 0.2583u + 2 −97 ≤ 0.2584u ≤ 1.034 • 2 −55 . However, the pair (z h , e) is not necessarily a double-word, since |e| may be significantly larger than 1 2 ulp(z h ). Hence we need to "normalize" that pair as follows:…”

“…Erf and erfc have applications in statistics and finance [1], Gaussian sampling in cryptography [2], partial differential diffusion equations, etc. They are significantly more complex to implement than exponentials, logarithms and trigonometric functions.…”

“…Much work on the implementation of these functions is due to Cody [3]- [6]. A recent presentation is given by Beebe [1,Chapter 19]. Beebe summarizes that existing approaches for the approximations to erfc work on segment-by-segment basis.…”