Average Error: 29.6 → 1.1
Time: 10.8s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.818513394003644867424895892327185720205:\\ \;\;\;\;0.3333333333333333703407674875052180141211 \cdot {x}^{3} + \left(1 - \left(\sqrt[3]{0.5 \cdot {x}^{2}} \cdot \sqrt[3]{0.5 \cdot {x}^{2}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{0.5 \cdot {x}^{2}}} \cdot \sqrt[3]{\sqrt[3]{0.5 \cdot {x}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{0.5 \cdot {x}^{2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 1.818513394003644867424895892327185720205:\\
\;\;\;\;0.3333333333333333703407674875052180141211 \cdot {x}^{3} + \left(1 - \left(\sqrt[3]{0.5 \cdot {x}^{2}} \cdot \sqrt[3]{0.5 \cdot {x}^{2}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{0.5 \cdot {x}^{2}}} \cdot \sqrt[3]{\sqrt[3]{0.5 \cdot {x}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{0.5 \cdot {x}^{2}}}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r46326 = 1.0;
        double r46327 = eps;
        double r46328 = r46326 / r46327;
        double r46329 = r46326 + r46328;
        double r46330 = r46326 - r46327;
        double r46331 = x;
        double r46332 = r46330 * r46331;
        double r46333 = -r46332;
        double r46334 = exp(r46333);
        double r46335 = r46329 * r46334;
        double r46336 = r46328 - r46326;
        double r46337 = r46326 + r46327;
        double r46338 = r46337 * r46331;
        double r46339 = -r46338;
        double r46340 = exp(r46339);
        double r46341 = r46336 * r46340;
        double r46342 = r46335 - r46341;
        double r46343 = 2.0;
        double r46344 = r46342 / r46343;
        return r46344;
}
↓
double f(double x, double eps) {
        double r46345 = x;
        double r46346 = 1.8185133940036449;
        bool r46347 = r46345 <= r46346;
        double r46348 = 0.33333333333333337;
        double r46349 = 3.0;
        double r46350 = pow(r46345, r46349);
        double r46351 = r46348 * r46350;
        double r46352 = 1.0;
        double r46353 = 0.5;
        double r46354 = 2.0;
        double r46355 = pow(r46345, r46354);
        double r46356 = r46353 * r46355;
        double r46357 = cbrt(r46356);
        double r46358 = r46357 * r46357;
        double r46359 = cbrt(r46357);
        double r46360 = r46359 * r46359;
        double r46361 = r46360 * r46359;
        double r46362 = r46358 * r46361;
        double r46363 = r46352 - r46362;
        double r46364 = r46351 + r46363;
        double r46365 = eps;
        double r46366 = r46352 / r46365;
        double r46367 = r46352 + r46366;
        double r46368 = r46352 - r46365;
        double r46369 = r46368 * r46345;
        double r46370 = exp(r46369);
        double r46371 = r46367 / r46370;
        double r46372 = 2.0;
        double r46373 = r46371 / r46372;
        double r46374 = r46352 + r46365;
        double r46375 = r46374 * r46345;
        double r46376 = exp(r46375);
        double r46377 = r46366 / r46376;
        double r46378 = r46377 / r46372;
        double r46379 = r46373 - r46378;
        double r46380 = r46352 / r46376;
        double r46381 = r46380 / r46372;
        double r46382 = r46379 + r46381;
        double r46383 = r46347 ? r46364 : r46382;
        return r46383;
}

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 < 1.8185133940036449

    1. Initial program 39.2

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

    2. Simplified39.2

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

    3. Taylor expanded around 0 1.3

      \[\leadsto \color{blue}{\left(0.3333333333333333703407674875052180141211 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\]
    4. Using strategy rm

    5. Applied associate--l+1.3

      \[\leadsto \color{blue}{0.3333333333333333703407674875052180141211 \cdot {x}^{3} + \left(1 - 0.5 \cdot {x}^{2}\right)}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt1.3

      \[\leadsto 0.3333333333333333703407674875052180141211 \cdot {x}^{3} + \left(1 - \color{blue}{\left(\sqrt[3]{0.5 \cdot {x}^{2}} \cdot \sqrt[3]{0.5 \cdot {x}^{2}}\right) \cdot \sqrt[3]{0.5 \cdot {x}^{2}}}\right)\]
    8. Using strategy rm

    9. Applied add-cube-cbrt1.3

      \[\leadsto 0.3333333333333333703407674875052180141211 \cdot {x}^{3} + \left(1 - \left(\sqrt[3]{0.5 \cdot {x}^{2}} \cdot \sqrt[3]{0.5 \cdot {x}^{2}}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{0.5 \cdot {x}^{2}}} \cdot \sqrt[3]{\sqrt[3]{0.5 \cdot {x}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{0.5 \cdot {x}^{2}}}\right)}\right)\]

    if 1.8185133940036449 < 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. Simplified0.5

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

    4. Applied div-sub0.5

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

    5. Applied div-sub0.5

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

    6. Applied associate--r-0.5

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.818513394003644867424895892327185720205:\\ \;\;\;\;0.3333333333333333703407674875052180141211 \cdot {x}^{3} + \left(1 - \left(\sqrt[3]{0.5 \cdot {x}^{2}} \cdot \sqrt[3]{0.5 \cdot {x}^{2}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{0.5 \cdot {x}^{2}}} \cdot \sqrt[3]{\sqrt[3]{0.5 \cdot {x}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{0.5 \cdot {x}^{2}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(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))