Average Error: 30.2 → 1.1
Time: 29.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 84.47417936124703885525377700105309486389:\\ \;\;\;\;\frac{e^{\log \left(2 + x \cdot \left(\left(x \cdot 0.6666666666666667406815349750104360282421 - 1\right) \cdot x\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x \cdot \left(\varepsilon - 1\right)} + \frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon}\right) \cdot 1 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]
\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{e^{\log \left(2 + x \cdot \left(\left(x \cdot 0.6666666666666667406815349750104360282421 - 1\right) \cdot x\right)\right)}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r34557 = 1.0;
        double r34558 = eps;
        double r34559 = r34557 / r34558;
        double r34560 = r34557 + r34559;
        double r34561 = r34557 - r34558;
        double r34562 = x;
        double r34563 = r34561 * r34562;
        double r34564 = -r34563;
        double r34565 = exp(r34564);
        double r34566 = r34560 * r34565;
        double r34567 = r34559 - r34557;
        double r34568 = r34557 + r34558;
        double r34569 = r34568 * r34562;
        double r34570 = -r34569;
        double r34571 = exp(r34570);
        double r34572 = r34567 * r34571;
        double r34573 = r34566 - r34572;
        double r34574 = 2.0;
        double r34575 = r34573 / r34574;
        return r34575;
}
↓
double f(double x, double eps) {
        double r34576 = x;
        double r34577 = 84.47417936124704;
        bool r34578 = r34576 <= r34577;
        double r34579 = 2.0;
        double r34580 = 0.6666666666666667;
        double r34581 = r34576 * r34580;
        double r34582 = 1.0;
        double r34583 = r34581 - r34582;
        double r34584 = r34583 * r34576;
        double r34585 = r34576 * r34584;
        double r34586 = r34579 + r34585;
        double r34587 = log(r34586);
        double r34588 = exp(r34587);
        double r34589 = r34588 / r34579;
        double r34590 = eps;
        double r34591 = r34590 - r34582;
        double r34592 = r34576 * r34591;
        double r34593 = exp(r34592);
        double r34594 = r34593 / r34590;
        double r34595 = r34593 + r34594;
        double r34596 = r34595 * r34582;
        double r34597 = r34582 / r34590;
        double r34598 = r34597 - r34582;
        double r34599 = r34582 + r34590;
        double r34600 = r34599 * r34576;
        double r34601 = -r34600;
        double r34602 = exp(r34601);
        double r34603 = r34598 * r34602;
        double r34604 = r34596 - r34603;
        double r34605 = r34604 / r34579;
        double r34606 = r34578 ? r34589 : r34605;
        return r34606;
}

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. Taylor expanded around 0 1.3

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

    4. Applied add-exp-log1.4

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

    5. Simplified1.4

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

    7. Applied associate-*l*1.4

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

    8. Simplified1.4

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

    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. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification1.1

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

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))