Average Error: 29.1 → 1.1
Time: 4.8s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 439.217255613824989:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{2}, 0.66666666666666674 \cdot x - 1, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \sqrt[3]{{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}^{3}}}{2}\\ \end{array}\]
\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 439.217255613824989:\\
\;\;\;\;\frac{\mathsf{fma}\left({x}^{2}, 0.66666666666666674 \cdot x - 1, 2\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \sqrt[3]{{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}^{3}}}{2}\\

\end{array}
double f(double x, double eps) {
        double r40496 = 1.0;
        double r40497 = eps;
        double r40498 = r40496 / r40497;
        double r40499 = r40496 + r40498;
        double r40500 = r40496 - r40497;
        double r40501 = x;
        double r40502 = r40500 * r40501;
        double r40503 = -r40502;
        double r40504 = exp(r40503);
        double r40505 = r40499 * r40504;
        double r40506 = r40498 - r40496;
        double r40507 = r40496 + r40497;
        double r40508 = r40507 * r40501;
        double r40509 = -r40508;
        double r40510 = exp(r40509);
        double r40511 = r40506 * r40510;
        double r40512 = r40505 - r40511;
        double r40513 = 2.0;
        double r40514 = r40512 / r40513;
        return r40514;
}
↓
double f(double x, double eps) {
        double r40515 = x;
        double r40516 = 439.217255613825;
        bool r40517 = r40515 <= r40516;
        double r40518 = 2.0;
        double r40519 = pow(r40515, r40518);
        double r40520 = 0.6666666666666667;
        double r40521 = r40520 * r40515;
        double r40522 = 1.0;
        double r40523 = r40521 - r40522;
        double r40524 = 2.0;
        double r40525 = fma(r40519, r40523, r40524);
        double r40526 = r40525 / r40524;
        double r40527 = eps;
        double r40528 = r40522 / r40527;
        double r40529 = r40522 + r40528;
        double r40530 = r40522 - r40527;
        double r40531 = r40530 * r40515;
        double r40532 = -r40531;
        double r40533 = exp(r40532);
        double r40534 = r40529 * r40533;
        double r40535 = r40528 - r40522;
        double r40536 = r40522 + r40527;
        double r40537 = r40536 * r40515;
        double r40538 = -r40537;
        double r40539 = exp(r40538);
        double r40540 = r40535 * r40539;
        double r40541 = 3.0;
        double r40542 = pow(r40540, r40541);
        double r40543 = cbrt(r40542);
        double r40544 = r40534 - r40543;
        double r40545 = r40544 / r40524;
        double r40546 = r40517 ? r40526 : r40545;
        return r40546;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 439.217255613825

    1. Initial program 38.8

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

    2. Taylor expanded around 0 1.4

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

    3. Simplified1.4

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

    4. Taylor expanded around 0 1.4

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

    5. Simplified1.4

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

    if 439.217255613825 < 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 add-cbrt-cube0.1

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

    4. Applied add-cbrt-cube42.7

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

    5. Applied cbrt-unprod42.7

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

    6. Simplified0.1

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 439.217255613824989:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{2}, 0.66666666666666674 \cdot x - 1, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \sqrt[3]{{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}^{3}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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))