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algorithm for overcoming approximation approximation error aR(i arithmetic assume bool bounding the error chapter comp completes the proof computable real functions corollary definition denote discontinuous double-precision e-arithmetic e-calculus e-continuity e-converges e-derivative e-function corresponding e-integrable e-limit e-notions e-operators error bounds evaluation example F and RF F is unstable finite number finitely computable follows ft(e given ideal function implies initial-value problem input routine integer interval analysis iterated integrals LEMMA lim f lim f(x lim f(x,y lim lim machine numbers members of R(e notation null set number set numerical analysis overcoming instability poor real input precision prove real number recurrence relation recursive functions recursive operators Remez algorithm RF(e roundoff-error bounds RX(e RY(e satisfy say f smallest stably convergent stopping criterion subroutine subroutine F sufficiently small Suppose f TF(e theorem triangle inequality truncation-error bounds Turing machine variable-precision variables yield