Average Error: 29.0 → 1.0
Time: 23.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.8522439107470307639857765025226399302483:\\ \;\;\;\;\frac{e^{\log \left(\left(2 - 1 \cdot \left(x \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(0.6666666666666667406815349750104360282421 \cdot x\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon - 1\right)} \cdot 1 + \left(\left(\frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon} + e^{-\left(\varepsilon \cdot x + x \cdot 1\right)}\right) - \frac{e^{-\left(\varepsilon \cdot x + x \cdot 1\right)}}{\varepsilon}\right) \cdot 1}{2}\\ \end{array}\]
\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 0.8522439107470307639857765025226399302483:\\
\;\;\;\;\frac{e^{\log \left(\left(2 - 1 \cdot \left(x \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(0.6666666666666667406815349750104360282421 \cdot x\right)\right)}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r2090528 = 1.0;
        double r2090529 = eps;
        double r2090530 = r2090528 / r2090529;
        double r2090531 = r2090528 + r2090530;
        double r2090532 = r2090528 - r2090529;
        double r2090533 = x;
        double r2090534 = r2090532 * r2090533;
        double r2090535 = -r2090534;
        double r2090536 = exp(r2090535);
        double r2090537 = r2090531 * r2090536;
        double r2090538 = r2090530 - r2090528;
        double r2090539 = r2090528 + r2090529;
        double r2090540 = r2090539 * r2090533;
        double r2090541 = -r2090540;
        double r2090542 = exp(r2090541);
        double r2090543 = r2090538 * r2090542;
        double r2090544 = r2090537 - r2090543;
        double r2090545 = 2.0;
        double r2090546 = r2090544 / r2090545;
        return r2090546;
}
↓
double f(double x, double eps) {
        double r2090547 = x;
        double r2090548 = 0.8522439107470308;
        bool r2090549 = r2090547 <= r2090548;
        double r2090550 = 2.0;
        double r2090551 = 1.0;
        double r2090552 = r2090547 * r2090547;
        double r2090553 = r2090551 * r2090552;
        double r2090554 = r2090550 - r2090553;
        double r2090555 = 0.6666666666666667;
        double r2090556 = r2090555 * r2090547;
        double r2090557 = r2090552 * r2090556;
        double r2090558 = r2090554 + r2090557;
        double r2090559 = log(r2090558);
        double r2090560 = exp(r2090559);
        double r2090561 = r2090560 / r2090550;
        double r2090562 = eps;
        double r2090563 = r2090562 - r2090551;
        double r2090564 = r2090547 * r2090563;
        double r2090565 = exp(r2090564);
        double r2090566 = r2090565 * r2090551;
        double r2090567 = r2090565 / r2090562;
        double r2090568 = r2090562 * r2090547;
        double r2090569 = r2090547 * r2090551;
        double r2090570 = r2090568 + r2090569;
        double r2090571 = -r2090570;
        double r2090572 = exp(r2090571);
        double r2090573 = r2090567 + r2090572;
        double r2090574 = r2090572 / r2090562;
        double r2090575 = r2090573 - r2090574;
        double r2090576 = r2090575 * r2090551;
        double r2090577 = r2090566 + r2090576;
        double r2090578 = r2090577 / r2090550;
        double r2090579 = r2090549 ? r2090561 : r2090578;
        return r2090579;
}

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 < 0.8522439107470308

    1. Initial program 38.6

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

    2. Taylor expanded around 0 1.2

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

    3. Simplified1.2

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

    5. Applied add-exp-log1.2

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

    6. Simplified1.2

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

    if 0.8522439107470308 < x

    1. Initial program 0.5

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

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

    4. Taylor expanded around inf 0.5

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

    5. Simplified0.5

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.8522439107470307639857765025226399302483:\\ \;\;\;\;\frac{e^{\log \left(\left(2 - 1 \cdot \left(x \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(0.6666666666666667406815349750104360282421 \cdot x\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon - 1\right)} \cdot 1 + \left(\left(\frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon} + e^{-\left(\varepsilon \cdot x + x \cdot 1\right)}\right) - \frac{e^{-\left(\varepsilon \cdot x + x \cdot 1\right)}}{\varepsilon}\right) \cdot 1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))