Average Error: 29.1 → 0.0
Time: 15.0s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.12853531169107207:\\ \;\;\;\;e^{\log \left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\\ \mathbf{elif}\;-2 \cdot x \le 0.0023969580591003724:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{3}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.12853531169107207:\\
\;\;\;\;e^{\log \left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\\

\mathbf{elif}\;-2 \cdot x \le 0.0023969580591003724:\\
\;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{3}\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r719445 = 2.0;
        double r719446 = 1.0;
        double r719447 = -2.0;
        double r719448 = x;
        double r719449 = r719447 * r719448;
        double r719450 = exp(r719449);
        double r719451 = r719446 + r719450;
        double r719452 = r719445 / r719451;
        double r719453 = r719452 - r719446;
        return r719453;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r719454 = -2.0;
        double r719455 = x;
        double r719456 = r719454 * r719455;
        double r719457 = -0.12853531169107207;
        bool r719458 = r719456 <= r719457;
        double r719459 = 2.0;
        double r719460 = exp(r719456);
        double r719461 = 1.0;
        double r719462 = r719460 + r719461;
        double r719463 = r719459 / r719462;
        double r719464 = r719463 - r719461;
        double r719465 = cbrt(r719464);
        double r719466 = r719465 * r719465;
        double r719467 = log(r719466);
        double r719468 = exp(r719467);
        double r719469 = r719468 * r719465;
        double r719470 = 0.0023969580591003724;
        bool r719471 = r719456 <= r719470;
        double r719472 = 5.0;
        double r719473 = pow(r719455, r719472);
        double r719474 = 0.13333333333333333;
        double r719475 = r719473 * r719474;
        double r719476 = r719475 + r719455;
        double r719477 = r719455 * r719455;
        double r719478 = r719455 * r719477;
        double r719479 = -0.3333333333333333;
        double r719480 = r719478 * r719479;
        double r719481 = r719476 + r719480;
        double r719482 = r719471 ? r719481 : r719469;
        double r719483 = r719458 ? r719469 : r719482;
        return r719483;
}

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 2 regimes

  2. if (* -2 x) < -0.12853531169107207 or 0.0023969580591003724 < (* -2 x)

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.0

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

    5. Applied add-exp-log0.0

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

    if -0.12853531169107207 < (* -2 x) < 0.0023969580591003724

    1. Initial program 59.0

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left({x}^{5} \cdot \frac{2}{15} + x\right) + \frac{-1}{3} \cdot \left(x \cdot \left(x \cdot x\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.12853531169107207:\\ \;\;\;\;e^{\log \left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\\ \mathbf{elif}\;-2 \cdot x \le 0.0023969580591003724:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{3}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))