Average Error: 30.2 → 0.1
Time: 15.1s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -4.521691874239059:\\ \;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 9.96322552535665 \cdot 10^{-05}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{3} + \left({x}^{5} \cdot \frac{2}{15} + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -4.521691874239059:\\
\;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\

\mathbf{elif}\;-2 \cdot x \le 9.96322552535665 \cdot 10^{-05}:\\
\;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{3} + \left({x}^{5} \cdot \frac{2}{15} + x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r1750093 = 2.0;
        double r1750094 = 1.0;
        double r1750095 = -2.0;
        double r1750096 = x;
        double r1750097 = r1750095 * r1750096;
        double r1750098 = exp(r1750097);
        double r1750099 = r1750094 + r1750098;
        double r1750100 = r1750093 / r1750099;
        double r1750101 = r1750100 - r1750094;
        return r1750101;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r1750102 = -2.0;
        double r1750103 = x;
        double r1750104 = r1750102 * r1750103;
        double r1750105 = -4.521691874239059;
        bool r1750106 = r1750104 <= r1750105;
        double r1750107 = 2.0;
        double r1750108 = exp(r1750104);
        double r1750109 = 1.0;
        double r1750110 = r1750108 + r1750109;
        double r1750111 = sqrt(r1750110);
        double r1750112 = r1750107 / r1750111;
        double r1750113 = r1750112 / r1750111;
        double r1750114 = r1750113 - r1750109;
        double r1750115 = 9.96322552535665e-05;
        bool r1750116 = r1750104 <= r1750115;
        double r1750117 = r1750103 * r1750103;
        double r1750118 = r1750117 * r1750103;
        double r1750119 = -0.3333333333333333;
        double r1750120 = r1750118 * r1750119;
        double r1750121 = 5.0;
        double r1750122 = pow(r1750103, r1750121);
        double r1750123 = 0.13333333333333333;
        double r1750124 = r1750122 * r1750123;
        double r1750125 = r1750124 + r1750103;
        double r1750126 = r1750120 + r1750125;
        double r1750127 = r1750116 ? r1750126 : r1750114;
        double r1750128 = r1750106 ? r1750114 : r1750127;
        return r1750128;
}

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) < -4.521691874239059 or 9.96322552535665e-05 < (* -2 x)

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied associate-/r*0.0

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

    if -4.521691874239059 < (* -2 x) < 9.96322552535665e-05

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -4.521691874239059:\\ \;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 9.96322552535665 \cdot 10^{-05}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{3} + \left({x}^{5} \cdot \frac{2}{15} + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))