Average Error: 29.5 → 1.0
Time: 6.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.97201211647392571:\\ \;\;\;\;\log \left(e^{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 0.97201211647392571:\\
\;\;\;\;\log \left(e^{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \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 r42445 = 1.0;
        double r42446 = eps;
        double r42447 = r42445 / r42446;
        double r42448 = r42445 + r42447;
        double r42449 = r42445 - r42446;
        double r42450 = x;
        double r42451 = r42449 * r42450;
        double r42452 = -r42451;
        double r42453 = exp(r42452);
        double r42454 = r42448 * r42453;
        double r42455 = r42447 - r42445;
        double r42456 = r42445 + r42446;
        double r42457 = r42456 * r42450;
        double r42458 = -r42457;
        double r42459 = exp(r42458);
        double r42460 = r42455 * r42459;
        double r42461 = r42454 - r42460;
        double r42462 = 2.0;
        double r42463 = r42461 / r42462;
        return r42463;
}
↓
double f(double x, double eps) {
        double r42464 = x;
        double r42465 = 0.9720121164739257;
        bool r42466 = r42464 <= r42465;
        double r42467 = 0.33333333333333337;
        double r42468 = 3.0;
        double r42469 = pow(r42464, r42468);
        double r42470 = r42467 * r42469;
        double r42471 = 1.0;
        double r42472 = r42470 + r42471;
        double r42473 = 0.5;
        double r42474 = 2.0;
        double r42475 = pow(r42464, r42474);
        double r42476 = r42473 * r42475;
        double r42477 = r42472 - r42476;
        double r42478 = exp(r42477);
        double r42479 = log(r42478);
        double r42480 = 1.0;
        double r42481 = eps;
        double r42482 = r42471 / r42481;
        double r42483 = r42471 + r42482;
        double r42484 = r42471 - r42481;
        double r42485 = r42484 * r42464;
        double r42486 = exp(r42485);
        double r42487 = r42483 / r42486;
        double r42488 = 2.0;
        double r42489 = r42487 / r42488;
        double r42490 = r42471 + r42481;
        double r42491 = r42490 * r42464;
        double r42492 = exp(r42491);
        double r42493 = r42482 / r42492;
        double r42494 = r42493 / r42488;
        double r42495 = r42489 - r42494;
        double r42496 = r42480 * r42495;
        double r42497 = r42471 / r42492;
        double r42498 = r42497 / r42488;
        double r42499 = r42496 + r42498;
        double r42500 = r42466 ? r42479 : r42499;
        return r42500;
}

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 < 0.9720121164739257

    1. Initial program 38.6

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

    2. Simplified38.6

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

    3. Taylor expanded around 0 1.1

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

    5. Applied add-log-exp1.1

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

    6. Applied add-log-exp1.1

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

    7. Applied add-log-exp1.1

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

    8. Applied sum-log1.1

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

    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. Simplified1.1

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

    if 0.9720121164739257 < x

    1. Initial program 0.6

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

    2. Simplified0.6

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

    4. Applied div-sub0.6

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

    5. Applied div-sub0.6

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

    6. Applied associate--r-0.6

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

    8. Applied *-un-lft-identity0.6

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.97201211647392571:\\ \;\;\;\;\log \left(e^{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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))