Average Error: 29.7 → 0.1
Time: 4.8s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -12.329649134843827:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 5.30640056390034621 \cdot 10^{-6}:\\ \;\;\;\;1 \cdot x - \left(4.996 \cdot 10^{-16} \cdot {x}^{4} + 0.33333333333333348 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}{\left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \left(\frac{\frac{2}{1 + e^{-2 \cdot x}}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}}} + 1 \cdot 1\right)}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -12.329649134843827:\\
\;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\

\mathbf{elif}\;-2 \cdot x \le 5.30640056390034621 \cdot 10^{-6}:\\
\;\;\;\;1 \cdot x - \left(4.996 \cdot 10^{-16} \cdot {x}^{4} + 0.33333333333333348 \cdot {x}^{3}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}{\left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \left(\frac{\frac{2}{1 + e^{-2 \cdot x}}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}}} + 1 \cdot 1\right)}\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r50589 = 2.0;
        double r50590 = 1.0;
        double r50591 = -2.0;
        double r50592 = x;
        double r50593 = r50591 * r50592;
        double r50594 = exp(r50593);
        double r50595 = r50590 + r50594;
        double r50596 = r50589 / r50595;
        double r50597 = r50596 - r50590;
        return r50597;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r50598 = -2.0;
        double r50599 = x;
        double r50600 = r50598 * r50599;
        double r50601 = -12.329649134843827;
        bool r50602 = r50600 <= r50601;
        double r50603 = 2.0;
        double r50604 = 1.0;
        double r50605 = exp(r50600);
        double r50606 = r50604 + r50605;
        double r50607 = r50603 / r50606;
        double r50608 = r50607 * r50607;
        double r50609 = r50604 * r50604;
        double r50610 = r50608 - r50609;
        double r50611 = r50607 + r50604;
        double r50612 = r50610 / r50611;
        double r50613 = 5.306400563900346e-06;
        bool r50614 = r50600 <= r50613;
        double r50615 = r50604 * r50599;
        double r50616 = 4.996003610813204e-16;
        double r50617 = 4.0;
        double r50618 = pow(r50599, r50617);
        double r50619 = r50616 * r50618;
        double r50620 = 0.3333333333333335;
        double r50621 = 3.0;
        double r50622 = pow(r50599, r50621);
        double r50623 = r50620 * r50622;
        double r50624 = r50619 + r50623;
        double r50625 = r50615 - r50624;
        double r50626 = r50608 * r50608;
        double r50627 = r50609 * r50609;
        double r50628 = r50626 - r50627;
        double r50629 = cbrt(r50606);
        double r50630 = r50629 * r50629;
        double r50631 = r50607 / r50630;
        double r50632 = r50603 / r50629;
        double r50633 = r50631 * r50632;
        double r50634 = r50633 + r50609;
        double r50635 = r50611 * r50634;
        double r50636 = r50628 / r50635;
        double r50637 = r50614 ? r50625 : r50636;
        double r50638 = r50602 ? r50612 : r50637;
        return r50638;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* -2.0 x) < -12.329649134843827

    1. Initial program 0.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
    2. Using strategy rm

    3. Applied flip--0.0

      \[\leadsto \color{blue}{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}}\]

    if -12.329649134843827 < (* -2.0 x) < 5.306400563900346e-06

    1. Initial program 59.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
    2. Using strategy rm

    3. Applied flip--59.0

      \[\leadsto \color{blue}{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}}\]

    4. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{1 \cdot x - \left(4.996 \cdot 10^{-16} \cdot {x}^{4} + 0.33333333333333348 \cdot {x}^{3}\right)}\]

    if 5.306400563900346e-06 < (* -2.0 x)

    1. Initial program 0.1

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
    2. Using strategy rm

    3. Applied flip--0.1

      \[\leadsto \color{blue}{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}}\]
    4. Using strategy rm

    5. Applied flip--0.1

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

    6. Applied associate-/l/0.1

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

    8. Applied add-cube-cbrt0.1

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

    9. Applied *-un-lft-identity0.1

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

    10. Applied times-frac0.1

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

    11. Applied associate-*r*0.1

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

    12. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -12.329649134843827:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 5.30640056390034621 \cdot 10^{-6}:\\ \;\;\;\;1 \cdot x - \left(4.996 \cdot 10^{-16} \cdot {x}^{4} + 0.33333333333333348 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}{\left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \left(\frac{\frac{2}{1 + e^{-2 \cdot x}}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}}} + 1 \cdot 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))