Average Error: 30.2 → 1.1
Time: 21.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 84.47417936124703885525377700105309486389:\\ \;\;\;\;\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 84.47417936124703885525377700105309486389:\\
\;\;\;\;\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r32522 = 1.0;
        double r32523 = eps;
        double r32524 = r32522 / r32523;
        double r32525 = r32522 + r32524;
        double r32526 = r32522 - r32523;
        double r32527 = x;
        double r32528 = r32526 * r32527;
        double r32529 = -r32528;
        double r32530 = exp(r32529);
        double r32531 = r32525 * r32530;
        double r32532 = r32524 - r32522;
        double r32533 = r32522 + r32523;
        double r32534 = r32533 * r32527;
        double r32535 = -r32534;
        double r32536 = exp(r32535);
        double r32537 = r32532 * r32536;
        double r32538 = r32531 - r32537;
        double r32539 = 2.0;
        double r32540 = r32538 / r32539;
        return r32540;
}
↓
double f(double x, double eps) {
        double r32541 = x;
        double r32542 = 84.47417936124704;
        bool r32543 = r32541 <= r32542;
        double r32544 = 2.0;
        double r32545 = r32541 * r32541;
        double r32546 = 0.6666666666666667;
        double r32547 = r32546 * r32541;
        double r32548 = 1.0;
        double r32549 = r32547 - r32548;
        double r32550 = r32545 * r32549;
        double r32551 = r32544 + r32550;
        double r32552 = r32551 / r32544;
        double r32553 = eps;
        double r32554 = r32548 - r32553;
        double r32555 = r32554 * r32541;
        double r32556 = -r32555;
        double r32557 = exp(r32556);
        double r32558 = r32548 / r32553;
        double r32559 = r32558 + r32548;
        double r32560 = r32557 * r32559;
        double r32561 = r32558 - r32548;
        double r32562 = r32548 + r32553;
        double r32563 = r32562 * r32541;
        double r32564 = exp(r32563);
        double r32565 = r32561 / r32564;
        double r32566 = r32560 - r32565;
        double r32567 = r32566 / r32544;
        double r32568 = r32543 ? r32552 : r32567;
        return r32568;
}

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 < 84.47417936124704

    1. Initial program 39.6

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

    2. Simplified39.6

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

    3. Taylor expanded around 0 1.3

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

    4. Simplified1.3

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

    if 84.47417936124704 < 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. Simplified0.3

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

    3. Taylor expanded around inf 0.3

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

    4. Simplified0.3

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 84.47417936124703885525377700105309486389:\\ \;\;\;\;\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))