Average Error: 29.5 → 1.0
Time: 7.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.42266178700124746:\\ \;\;\;\;\log \left(\left(e^{1} + 0.33333333333333337 \cdot \left(e^{1} \cdot {x}^{3}\right)\right) - 0.5 \cdot \left(e^{1} \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 2.42266178700124746:\\
\;\;\;\;\log \left(\left(e^{1} + 0.33333333333333337 \cdot \left(e^{1} \cdot {x}^{3}\right)\right) - 0.5 \cdot \left(e^{1} \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 r41454 = 1.0;
        double r41455 = eps;
        double r41456 = r41454 / r41455;
        double r41457 = r41454 + r41456;
        double r41458 = r41454 - r41455;
        double r41459 = x;
        double r41460 = r41458 * r41459;
        double r41461 = -r41460;
        double r41462 = exp(r41461);
        double r41463 = r41457 * r41462;
        double r41464 = r41456 - r41454;
        double r41465 = r41454 + r41455;
        double r41466 = r41465 * r41459;
        double r41467 = -r41466;
        double r41468 = exp(r41467);
        double r41469 = r41464 * r41468;
        double r41470 = r41463 - r41469;
        double r41471 = 2.0;
        double r41472 = r41470 / r41471;
        return r41472;
}
↓
double f(double x, double eps) {
        double r41473 = x;
        double r41474 = 2.4226617870012475;
        bool r41475 = r41473 <= r41474;
        double r41476 = 1.0;
        double r41477 = exp(r41476);
        double r41478 = 0.33333333333333337;
        double r41479 = 3.0;
        double r41480 = pow(r41473, r41479);
        double r41481 = r41477 * r41480;
        double r41482 = r41478 * r41481;
        double r41483 = r41477 + r41482;
        double r41484 = 0.5;
        double r41485 = 2.0;
        double r41486 = pow(r41473, r41485);
        double r41487 = r41477 * r41486;
        double r41488 = r41484 * r41487;
        double r41489 = r41483 - r41488;
        double r41490 = log(r41489);
        double r41491 = eps;
        double r41492 = r41476 / r41491;
        double r41493 = r41476 + r41492;
        double r41494 = r41476 - r41491;
        double r41495 = r41494 * r41473;
        double r41496 = exp(r41495);
        double r41497 = r41493 / r41496;
        double r41498 = 2.0;
        double r41499 = r41497 / r41498;
        double r41500 = r41476 + r41491;
        double r41501 = r41500 * r41473;
        double r41502 = exp(r41501);
        double r41503 = r41492 / r41502;
        double r41504 = r41503 / r41498;
        double r41505 = r41499 - r41504;
        double r41506 = r41476 / r41502;
        double r41507 = r41506 / r41498;
        double r41508 = r41505 + r41507;
        double r41509 = r41475 ? r41490 : r41508;
        return r41509;
}

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 < 2.4226617870012475

    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 1.2

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

    5. Applied add-log-exp1.2

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

    6. Applied add-log-exp1.2

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

    7. Applied add-log-exp1.2

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

    8. Applied sum-log1.2

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

    9. Applied diff-log1.2

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

    10. Simplified1.2

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

    11. Taylor expanded around 0 1.2

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

    if 2.4226617870012475 < 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 simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.42266178700124746:\\ \;\;\;\;\log \left(\left(e^{1} + 0.33333333333333337 \cdot \left(e^{1} \cdot {x}^{3}\right)\right) - 0.5 \cdot \left(e^{1} \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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))