Average Error: 29.8 → 1.2
Time: 8.8s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 177.7594670374271:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
↓
\begin{array}{l}
\mathbf{if}\;x \le 177.7594670374271:\\
\;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\

\end{array}
double f(double x, double eps) {
        double r58324 = 1.0;
        double r58325 = eps;
        double r58326 = r58324 / r58325;
        double r58327 = r58324 + r58326;
        double r58328 = r58324 - r58325;
        double r58329 = x;
        double r58330 = r58328 * r58329;
        double r58331 = -r58330;
        double r58332 = exp(r58331);
        double r58333 = r58327 * r58332;
        double r58334 = r58326 - r58324;
        double r58335 = r58324 + r58325;
        double r58336 = r58335 * r58329;
        double r58337 = -r58336;
        double r58338 = exp(r58337);
        double r58339 = r58334 * r58338;
        double r58340 = r58333 - r58339;
        double r58341 = 2.0;
        double r58342 = r58340 / r58341;
        return r58342;
}
↓
double f(double x, double eps) {
        double r58343 = x;
        double r58344 = 177.7594670374271;
        bool r58345 = r58343 <= r58344;
        double r58346 = 3.0;
        double r58347 = pow(r58343, r58346);
        double r58348 = 0.6666666666666667;
        double r58349 = 2.0;
        double r58350 = 1.0;
        double r58351 = 2.0;
        double r58352 = pow(r58343, r58351);
        double r58353 = r58350 * r58352;
        double r58354 = r58349 - r58353;
        double r58355 = fma(r58347, r58348, r58354);
        double r58356 = r58355 / r58349;
        double r58357 = eps;
        double r58358 = r58350 / r58357;
        double r58359 = r58350 + r58358;
        double r58360 = r58350 - r58357;
        double r58361 = r58360 * r58343;
        double r58362 = -r58361;
        double r58363 = exp(r58362);
        double r58364 = r58359 * r58363;
        double r58365 = r58358 - r58350;
        double r58366 = r58350 + r58357;
        double r58367 = r58366 * r58343;
        double r58368 = exp(r58367);
        double r58369 = r58365 / r58368;
        double r58370 = r58364 - r58369;
        double r58371 = r58370 / r58349;
        double r58372 = r58345 ? r58356 : r58371;
        return r58372;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 177.7594670374271

    1. Initial program 39.5

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

    2. Taylor expanded around 0 1.6

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.6

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}}{2}\]

    if 177.7594670374271 < x

    1. Initial program 0.0

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
    2. Using strategy rm

    3. Applied exp-neg0.0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]

    4. Applied un-div-inv0.0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 177.7594670374271:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))