Average Error: 29.0 → 0.1
Time: 17.5s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -15.66122047665012395611938700312748551369:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right)\right)}\\ \mathbf{elif}\;-2 \cdot x \le 5.846791957771713948576082497954331529399 \cdot 10^{-6}:\\ \;\;\;\;\left(1 - 0.3333333333333333703407674875052180141211 \cdot \left(x \cdot x\right)\right) \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right)\right)}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -15.66122047665012395611938700312748551369:\\
\;\;\;\;\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right)\right)}\\

\mathbf{elif}\;-2 \cdot x \le 5.846791957771713948576082497954331529399 \cdot 10^{-6}:\\
\;\;\;\;\left(1 - 0.3333333333333333703407674875052180141211 \cdot \left(x \cdot x\right)\right) \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r1738243 = 2.0;
        double r1738244 = 1.0;
        double r1738245 = -2.0;
        double r1738246 = x;
        double r1738247 = r1738245 * r1738246;
        double r1738248 = exp(r1738247);
        double r1738249 = r1738244 + r1738248;
        double r1738250 = r1738243 / r1738249;
        double r1738251 = r1738250 - r1738244;
        return r1738251;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r1738252 = -2.0;
        double r1738253 = x;
        double r1738254 = r1738252 * r1738253;
        double r1738255 = -15.661220476650124;
        bool r1738256 = r1738254 <= r1738255;
        double r1738257 = 2.0;
        double r1738258 = exp(r1738254);
        double r1738259 = 1.0;
        double r1738260 = r1738258 + r1738259;
        double r1738261 = r1738257 / r1738260;
        double r1738262 = r1738261 - r1738259;
        double r1738263 = r1738262 * r1738262;
        double r1738264 = r1738262 * r1738263;
        double r1738265 = cbrt(r1738264);
        double r1738266 = 5.846791957771714e-06;
        bool r1738267 = r1738254 <= r1738266;
        double r1738268 = 0.33333333333333337;
        double r1738269 = r1738253 * r1738253;
        double r1738270 = r1738268 * r1738269;
        double r1738271 = r1738259 - r1738270;
        double r1738272 = r1738271 * r1738253;
        double r1738273 = 5.551115123125783e-17;
        double r1738274 = r1738269 * r1738273;
        double r1738275 = r1738269 * r1738274;
        double r1738276 = r1738272 - r1738275;
        double r1738277 = r1738267 ? r1738276 : r1738265;
        double r1738278 = r1738256 ? r1738265 : r1738277;
        return r1738278;
}

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 (* -2.0 x) < -15.661220476650124 or 5.846791957771714e-06 < (* -2.0 x)

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    if -15.661220476650124 < (* -2.0 x) < 5.846791957771714e-06

    1. Initial program 58.9

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{1 \cdot x - \left(0.3333333333333333703407674875052180141211 \cdot {x}^{3} + 5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4}\right)}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{x \cdot \left(1 - 0.3333333333333333703407674875052180141211 \cdot \left(x \cdot x\right)\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -15.66122047665012395611938700312748551369:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right)\right)}\\ \mathbf{elif}\;-2 \cdot x \le 5.846791957771713948576082497954331529399 \cdot 10^{-6}:\\ \;\;\;\;\left(1 - 0.3333333333333333703407674875052180141211 \cdot \left(x \cdot x\right)\right) \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} - 1\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))