Average Error: 29.3 → 1.0
Time: 5.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 434.16456045256211:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{2}, 0.66666666666666674 \cdot x - 1, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2}\\ \end{array}\]
\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 434.16456045256211:\\
\;\;\;\;\frac{\mathsf{fma}\left({x}^{2}, 0.66666666666666674 \cdot x - 1, 2\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\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}\\

\end{array}
double f(double x, double eps) {
        double r43445 = 1.0;
        double r43446 = eps;
        double r43447 = r43445 / r43446;
        double r43448 = r43445 + r43447;
        double r43449 = r43445 - r43446;
        double r43450 = x;
        double r43451 = r43449 * r43450;
        double r43452 = -r43451;
        double r43453 = exp(r43452);
        double r43454 = r43448 * r43453;
        double r43455 = r43447 - r43445;
        double r43456 = r43445 + r43446;
        double r43457 = r43456 * r43450;
        double r43458 = -r43457;
        double r43459 = exp(r43458);
        double r43460 = r43455 * r43459;
        double r43461 = r43454 - r43460;
        double r43462 = 2.0;
        double r43463 = r43461 / r43462;
        return r43463;
}
↓
double f(double x, double eps) {
        double r43464 = x;
        double r43465 = 434.1645604525621;
        bool r43466 = r43464 <= r43465;
        double r43467 = 2.0;
        double r43468 = pow(r43464, r43467);
        double r43469 = 0.6666666666666667;
        double r43470 = r43469 * r43464;
        double r43471 = 1.0;
        double r43472 = r43470 - r43471;
        double r43473 = 2.0;
        double r43474 = fma(r43468, r43472, r43473);
        double r43475 = r43474 / r43473;
        double r43476 = eps;
        double r43477 = r43471 / r43476;
        double r43478 = r43471 + r43477;
        double r43479 = r43471 - r43476;
        double r43480 = r43479 * r43464;
        double r43481 = -r43480;
        double r43482 = exp(r43481);
        double r43483 = r43477 - r43471;
        double r43484 = r43471 + r43476;
        double r43485 = r43484 * r43464;
        double r43486 = -r43485;
        double r43487 = exp(r43486);
        double r43488 = r43483 * r43487;
        double r43489 = -r43488;
        double r43490 = fma(r43478, r43482, r43489);
        double r43491 = r43490 / r43473;
        double r43492 = r43466 ? r43475 : r43491;
        return r43492;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 434.1645604525621

    1. Initial program 39.3

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

    2. Taylor expanded around 0 1.2

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

    3. Simplified1.2

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

    4. Taylor expanded around 0 1.2

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

    5. Simplified1.2

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

    if 434.1645604525621 < x

    1. Initial program 0.1

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

    3. Applied fma-neg0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 434.16456045256211:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{2}, 0.66666666666666674 \cdot x - 1, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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))