Average Error: 29.1 → 0.0
Time: 17.2s
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 r920463 = 2.0;
        double r920464 = 1.0;
        double r920465 = -2.0;
        double r920466 = x;
        double r920467 = r920465 * r920466;
        double r920468 = exp(r920467);
        double r920469 = r920464 + r920468;
        double r920470 = r920463 / r920469;
        double r920471 = r920470 - r920464;
        return r920471;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r920472 = -2.0;
        double r920473 = x;
        double r920474 = r920472 * r920473;
        double r920475 = -0.12853531169107207;
        bool r920476 = r920474 <= r920475;
        double r920477 = 2.0;
        double r920478 = exp(r920474);
        double r920479 = 1.0;
        double r920480 = r920478 + r920479;
        double r920481 = r920477 / r920480;
        double r920482 = r920481 - r920479;
        double r920483 = cbrt(r920482);
        double r920484 = r920483 * r920483;
        double r920485 = log(r920484);
        double r920486 = exp(r920485);
        double r920487 = r920486 * r920483;
        double r920488 = 0.0023969580591003724;
        bool r920489 = r920474 <= r920488;
        double r920490 = 5.0;
        double r920491 = pow(r920473, r920490);
        double r920492 = 0.13333333333333333;
        double r920493 = r920491 * r920492;
        double r920494 = r920493 + r920473;
        double r920495 = r920473 * r920473;
        double r920496 = r920473 * r920495;
        double r920497 = -0.3333333333333333;
        double r920498 = r920496 * r920497;
        double r920499 = r920494 + r920498;
        double r920500 = r920489 ? r920499 : r920487;
        double r920501 = r920476 ? r920487 : r920500;
        return r920501;
}

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))