Average Error: 29.7 → 1.1
Time: 29.8s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 331.7599664969623631805006880313158035278:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}{2}\\ \end{array}\]
\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 r71445 = 1.0;
        double r71446 = eps;
        double r71447 = r71445 / r71446;
        double r71448 = r71445 + r71447;
        double r71449 = r71445 - r71446;
        double r71450 = x;
        double r71451 = r71449 * r71450;
        double r71452 = -r71451;
        double r71453 = exp(r71452);
        double r71454 = r71448 * r71453;
        double r71455 = r71447 - r71445;
        double r71456 = r71445 + r71446;
        double r71457 = r71456 * r71450;
        double r71458 = -r71457;
        double r71459 = exp(r71458);
        double r71460 = r71455 * r71459;
        double r71461 = r71454 - r71460;
        double r71462 = 2.0;
        double r71463 = r71461 / r71462;
        return r71463;
}
↓
double f(double x, double eps) {
        double r71464 = x;
        double r71465 = 331.75996649696236;
        bool r71466 = r71464 <= r71465;
        double r71467 = 0.6666666666666667;
        double r71468 = 3.0;
        double r71469 = pow(r71464, r71468);
        double r71470 = 2.0;
        double r71471 = fma(r71467, r71469, r71470);
        double r71472 = 1.0;
        double r71473 = 2.0;
        double r71474 = pow(r71464, r71473);
        double r71475 = r71472 * r71474;
        double r71476 = r71471 - r71475;
        double r71477 = r71476 / r71470;
        double r71478 = eps;
        double r71479 = r71472 / r71478;
        double r71480 = r71472 + r71479;
        double r71481 = r71472 - r71478;
        double r71482 = r71481 * r71464;
        double r71483 = -r71482;
        double r71484 = exp(r71483);
        double r71485 = r71479 - r71472;
        double r71486 = r71472 + r71478;
        double r71487 = r71486 * r71464;
        double r71488 = exp(r71487);
        double r71489 = r71485 / r71488;
        double r71490 = -r71489;
        double r71491 = fma(r71480, r71484, r71490);
        double r71492 = r71491 / r71470;
        double r71493 = r71466 ? r71477 : r71492;
        return r71493;
}

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))