Average Error: 30.5 → 1.0
Time: 6.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 174.476621284914501:\\ \;\;\;\;\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{1}{\frac{e^{\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} - 1}}}{2}\\ \end{array}\]
\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 174.476621284914501:\\
\;\;\;\;\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}\\

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

\end{array}
double f(double x, double eps) {
        double r39450 = 1.0;
        double r39451 = eps;
        double r39452 = r39450 / r39451;
        double r39453 = r39450 + r39452;
        double r39454 = r39450 - r39451;
        double r39455 = x;
        double r39456 = r39454 * r39455;
        double r39457 = -r39456;
        double r39458 = exp(r39457);
        double r39459 = r39453 * r39458;
        double r39460 = r39452 - r39450;
        double r39461 = r39450 + r39451;
        double r39462 = r39461 * r39455;
        double r39463 = -r39462;
        double r39464 = exp(r39463);
        double r39465 = r39460 * r39464;
        double r39466 = r39459 - r39465;
        double r39467 = 2.0;
        double r39468 = r39466 / r39467;
        return r39468;
}
↓
double f(double x, double eps) {
        double r39469 = x;
        double r39470 = 174.4766212849145;
        bool r39471 = r39469 <= r39470;
        double r39472 = 0.33333333333333337;
        double r39473 = 3.0;
        double r39474 = pow(r39469, r39473);
        double r39475 = r39472 * r39474;
        double r39476 = 1.0;
        double r39477 = r39475 + r39476;
        double r39478 = 0.5;
        double r39479 = 2.0;
        double r39480 = pow(r39469, r39479);
        double r39481 = r39478 * r39480;
        double r39482 = r39477 - r39481;
        double r39483 = eps;
        double r39484 = r39476 / r39483;
        double r39485 = r39476 + r39484;
        double r39486 = r39476 - r39483;
        double r39487 = r39486 * r39469;
        double r39488 = exp(r39487);
        double r39489 = r39485 / r39488;
        double r39490 = 2.0;
        double r39491 = r39489 / r39490;
        double r39492 = 1.0;
        double r39493 = r39476 + r39483;
        double r39494 = r39493 * r39469;
        double r39495 = exp(r39494);
        double r39496 = r39484 - r39476;
        double r39497 = r39495 / r39496;
        double r39498 = r39492 / r39497;
        double r39499 = r39498 / r39490;
        double r39500 = r39491 - r39499;
        double r39501 = r39471 ? r39482 : r39500;
        return r39501;
}

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 < 174.4766212849145

    1. Initial program 39.5

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

    2. Simplified39.5

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

    3. Taylor expanded around 0 1.3

      \[\leadsto \color{blue}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\]

    if 174.4766212849145 < x

    1. Initial program 0.2

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

    2. Simplified0.2

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

    4. Applied clear-num0.2

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 174.476621284914501:\\ \;\;\;\;\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{1}{\frac{e^{\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} - 1}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(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))