By LJ SEGERLIND

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E. when F is known on the complex plane. The inverse function f (t) is obtained as a Laguerre expansion: f (t) = eσ t ∞ ∑ ck e−bt Lk (2bt), k=0 ck = Φ (k) (0) k! (2) 46 Salvatore Cuomo, Luisa D’Amore, Mariarosaria Rizzardi, and Almerico Murli where Lk (2bt) is the Laguerre polynomial of degree k, σ > σ0 and b are parameters. The ck values are McLaurin’s coefficients of the function Φ obtained from F.

Future Generation Computer Systems 21, 1401–1417 (2005) 12. 1 user’s guide. Technical report 300, INRIA (2004). html 13. : DAG reversal is NP-complete. J. Discr. Alg. (2008). To appear. 14. : A differentiation-enabled Fortran 95 compiler. ACM Transactions on Mathematical Software 31(4), 458–474 (2005) 15. : OpenAD/F: A modular, open-source tool for automatic differentiation of Fortran codes. ACM Transactions on Mathematical Software 34(4) (2008). To appear. 22 Uwe Naumann A Reference Code for Result Checkpointing subroutine f0 (x,y) double precision x,y call f1 (x,y) y=sin (y) end subroutine f0 subroutine f3 (x,y) double precision x,y call f4 (x,y) y=sin (y) end subroutine f3 subroutine f1 (x,y) double precision x,y call f2 (x,y) y=sin (y) end subroutine f1 subroutine f4 (x,y) double precision x,y call f5 (x,y) y=sin (y) end subroutine f4 subroutine f2 (x,y) double precision x,y call f3 (x,y) y=sin (y) end subroutine f2 subroutine f5 (x,y) double precision x,y integer i y=0 do 10 i=1,10000000 y=y+x 10 continue end subroutine f5 0 0 1 1 1 2 2 2 2 3 3 3 4 4 4 5 5 3 4 5 5 Fig.

24 Emmanuel M. Tadjouddine In principle, AD preserves the semantics of the input code provided this has not been altered prior to AD transformation. Given this semi-automatic usage of AD, can we trust AD for safety-critical applications? Although the chain rule of calculus and the analyses used in AD are proved correct, the correctness of the AD generated code is tricky to establish. First, AD may locally replace some part B of the input code by B that is not observationally equivalent to B even though both are semantically equivalent in that particular context.