Average Error: 29.7 → 1.1
Time: 29.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 331.7599664969623631805006880313158035278:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}{2}\\ \end{array}\]
\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 331.7599664969623631805006880313158035278:\\
\;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r71434 = 1.0;
        double r71435 = eps;
        double r71436 = r71434 / r71435;
        double r71437 = r71434 + r71436;
        double r71438 = r71434 - r71435;
        double r71439 = x;
        double r71440 = r71438 * r71439;
        double r71441 = -r71440;
        double r71442 = exp(r71441);
        double r71443 = r71437 * r71442;
        double r71444 = r71436 - r71434;
        double r71445 = r71434 + r71435;
        double r71446 = r71445 * r71439;
        double r71447 = -r71446;
        double r71448 = exp(r71447);
        double r71449 = r71444 * r71448;
        double r71450 = r71443 - r71449;
        double r71451 = 2.0;
        double r71452 = r71450 / r71451;
        return r71452;
}
↓
double f(double x, double eps) {
        double r71453 = x;
        double r71454 = 331.75996649696236;
        bool r71455 = r71453 <= r71454;
        double r71456 = 0.6666666666666667;
        double r71457 = 3.0;
        double r71458 = pow(r71453, r71457);
        double r71459 = 2.0;
        double r71460 = fma(r71456, r71458, r71459);
        double r71461 = 1.0;
        double r71462 = 2.0;
        double r71463 = pow(r71453, r71462);
        double r71464 = r71461 * r71463;
        double r71465 = r71460 - r71464;
        double r71466 = r71465 / r71459;
        double r71467 = eps;
        double r71468 = r71461 / r71467;
        double r71469 = r71461 + r71468;
        double r71470 = r71461 - r71467;
        double r71471 = r71470 * r71453;
        double r71472 = -r71471;
        double r71473 = exp(r71472);
        double r71474 = r71468 - r71461;
        double r71475 = r71461 + r71467;
        double r71476 = r71475 * r71453;
        double r71477 = exp(r71476);
        double r71478 = r71474 / r71477;
        double r71479 = -r71478;
        double r71480 = fma(r71469, r71473, r71479);
        double r71481 = r71480 / r71459;
        double r71482 = r71455 ? r71466 : r71481;
        return r71482;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 331.75996649696236

    1. Initial program 39.4

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

    2. Taylor expanded around 0 1.4

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.4

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}}{2}\]

    if 331.75996649696236 < x

    1. Initial program 0.2

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

    3. Applied fma-neg0.2

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

    4. Simplified0.2

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 331.7599664969623631805006880313158035278:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))