Average Error: 29.5 → 1.0
Time: 29.1s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 171.1161917239734:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - {e}^{\left(\left(\left(-\varepsilon\right) + -1\right) \cdot x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]
\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 171.1161917239734:\\
\;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) \cdot x}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1194477 = 1.0;
        double r1194478 = eps;
        double r1194479 = r1194477 / r1194478;
        double r1194480 = r1194477 + r1194479;
        double r1194481 = r1194477 - r1194478;
        double r1194482 = x;
        double r1194483 = r1194481 * r1194482;
        double r1194484 = -r1194483;
        double r1194485 = exp(r1194484);
        double r1194486 = r1194480 * r1194485;
        double r1194487 = r1194479 - r1194477;
        double r1194488 = r1194477 + r1194478;
        double r1194489 = r1194488 * r1194482;
        double r1194490 = -r1194489;
        double r1194491 = exp(r1194490);
        double r1194492 = r1194487 * r1194491;
        double r1194493 = r1194486 - r1194492;
        double r1194494 = 2.0;
        double r1194495 = r1194493 / r1194494;
        return r1194495;
}
↓
double f(double x, double eps) {
        double r1194496 = x;
        double r1194497 = 171.1161917239734;
        bool r1194498 = r1194496 <= r1194497;
        double r1194499 = 2.0;
        double r1194500 = r1194496 * r1194496;
        double r1194501 = r1194499 - r1194500;
        double r1194502 = 0.6666666666666666;
        double r1194503 = r1194500 * r1194502;
        double r1194504 = r1194503 * r1194496;
        double r1194505 = r1194501 + r1194504;
        double r1194506 = r1194505 / r1194499;
        double r1194507 = 1.0;
        double r1194508 = eps;
        double r1194509 = r1194507 / r1194508;
        double r1194510 = r1194509 + r1194507;
        double r1194511 = -r1194496;
        double r1194512 = r1194507 - r1194508;
        double r1194513 = r1194511 * r1194512;
        double r1194514 = exp(r1194513);
        double r1194515 = r1194510 * r1194514;
        double r1194516 = exp(1.0);
        double r1194517 = -r1194508;
        double r1194518 = -1.0;
        double r1194519 = r1194517 + r1194518;
        double r1194520 = r1194519 * r1194496;
        double r1194521 = pow(r1194516, r1194520);
        double r1194522 = r1194509 - r1194507;
        double r1194523 = r1194521 * r1194522;
        double r1194524 = r1194515 - r1194523;
        double r1194525 = r1194524 / r1194499;
        double r1194526 = r1194498 ? r1194506 : r1194525;
        return r1194526;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 171.1161917239734

    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 1.2

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

    3. Simplified1.2

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

    4. Taylor expanded around -inf 1.2

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

    5. Simplified1.2

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

    if 171.1161917239734 < x

    1. Initial program 0.3

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

    3. Applied *-un-lft-identity0.3

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

    4. Applied exp-prod0.3

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

    5. Simplified0.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 171.1161917239734:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - {e}^{\left(\left(\left(-\varepsilon\right) + -1\right) \cdot x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))