Verified sharp bounds for the real gamma function over the entire floating-point range

Abstract

An algorithm is presented for computing verified and accurate bounds for the value of the gamma function over the entire real double precision floating-point range. It means that for every double precision floating-point number x except the poles -k for 0 ≤ k ∈ N the true value of Γ(x) is included within an almost maximally accurate interval with floating-point bounds. One motivation to write this note are some erroneous results in Matlab's gamma function. The application to interval arguments x ∈ IF, thus enclosing the range of Γ over x, is discussed as well.