Average Error: 30.1 → 0.9
Time: 24.9s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 31.67008507871663525179428688716143369675:\\ \;\;\;\;\frac{2 + {x}^{2} \cdot \left(x \cdot 0.6666666666666667406815349750104360282421 - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\varepsilon} + 1}{e^{x \cdot \left(1 - \varepsilon\right)}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 31.67008507871663525179428688716143369675:\\
\;\;\;\;\frac{2 + {x}^{2} \cdot \left(x \cdot 0.6666666666666667406815349750104360282421 - 1\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r31350 = 1.0;
        double r31351 = eps;
        double r31352 = r31350 / r31351;
        double r31353 = r31350 + r31352;
        double r31354 = r31350 - r31351;
        double r31355 = x;
        double r31356 = r31354 * r31355;
        double r31357 = -r31356;
        double r31358 = exp(r31357);
        double r31359 = r31353 * r31358;
        double r31360 = r31352 - r31350;
        double r31361 = r31350 + r31351;
        double r31362 = r31361 * r31355;
        double r31363 = -r31362;
        double r31364 = exp(r31363);
        double r31365 = r31360 * r31364;
        double r31366 = r31359 - r31365;
        double r31367 = 2.0;
        double r31368 = r31366 / r31367;
        return r31368;
}
↓
double f(double x, double eps) {
        double r31369 = x;
        double r31370 = 31.670085078716635;
        bool r31371 = r31369 <= r31370;
        double r31372 = 2.0;
        double r31373 = 2.0;
        double r31374 = pow(r31369, r31373);
        double r31375 = 0.6666666666666667;
        double r31376 = r31369 * r31375;
        double r31377 = 1.0;
        double r31378 = r31376 - r31377;
        double r31379 = r31374 * r31378;
        double r31380 = r31372 + r31379;
        double r31381 = r31380 / r31372;
        double r31382 = eps;
        double r31383 = r31377 / r31382;
        double r31384 = r31383 + r31377;
        double r31385 = r31377 - r31382;
        double r31386 = r31369 * r31385;
        double r31387 = exp(r31386);
        double r31388 = r31384 / r31387;
        double r31389 = r31383 - r31377;
        double r31390 = r31377 + r31382;
        double r31391 = r31390 * r31369;
        double r31392 = exp(r31391);
        double r31393 = r31389 / r31392;
        double r31394 = r31388 - r31393;
        double r31395 = r31394 / r31372;
        double r31396 = r31371 ? r31381 : r31395;
        return r31396;
}

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 < 31.670085078716635

    1. Initial program 39.8

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

    2. Simplified39.8

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

    3. Taylor expanded around 0 1.0

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

    4. Simplified1.0

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

    if 31.670085078716635 < 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

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 31.67008507871663525179428688716143369675:\\ \;\;\;\;\frac{2 + {x}^{2} \cdot \left(x \cdot 0.6666666666666667406815349750104360282421 - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\varepsilon} + 1}{e^{x \cdot \left(1 - \varepsilon\right)}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))