Average Error: 29.2 → 0.0
Time: 26.2s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007359924193442822:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.007101371656256956:\\ \;\;\;\;\left(\left(\frac{-1}{3} \cdot \left(x \cdot x\right)\right) \cdot x + x\right) + {x}^{5} \cdot \frac{2}{15}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.007359924193442822:\\
\;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\

\mathbf{elif}\;x \le 0.007101371656256956:\\
\;\;\;\;\left(\left(\frac{-1}{3} \cdot \left(x \cdot x\right)\right) \cdot x + x\right) + {x}^{5} \cdot \frac{2}{15}\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r3154915 = 2.0;
        double r3154916 = 1.0;
        double r3154917 = -2.0;
        double r3154918 = x;
        double r3154919 = r3154917 * r3154918;
        double r3154920 = exp(r3154919);
        double r3154921 = r3154916 + r3154920;
        double r3154922 = r3154915 / r3154921;
        double r3154923 = r3154922 - r3154916;
        return r3154923;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r3154924 = x;
        double r3154925 = -0.007359924193442822;
        bool r3154926 = r3154924 <= r3154925;
        double r3154927 = 2.0;
        double r3154928 = 1.0;
        double r3154929 = -2.0;
        double r3154930 = r3154929 * r3154924;
        double r3154931 = exp(r3154930);
        double r3154932 = r3154928 + r3154931;
        double r3154933 = r3154927 / r3154932;
        double r3154934 = r3154933 - r3154928;
        double r3154935 = 0.007101371656256956;
        bool r3154936 = r3154924 <= r3154935;
        double r3154937 = -0.3333333333333333;
        double r3154938 = r3154924 * r3154924;
        double r3154939 = r3154937 * r3154938;
        double r3154940 = r3154939 * r3154924;
        double r3154941 = r3154940 + r3154924;
        double r3154942 = 5.0;
        double r3154943 = pow(r3154924, r3154942);
        double r3154944 = 0.13333333333333333;
        double r3154945 = r3154943 * r3154944;
        double r3154946 = r3154941 + r3154945;
        double r3154947 = r3154936 ? r3154946 : r3154934;
        double r3154948 = r3154926 ? r3154934 : r3154947;
        return r3154948;
}

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 x < -0.007359924193442822 or 0.007101371656256956 < x

    1. Initial program 0.0

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

    2. Taylor expanded around -inf 0.0

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

    if -0.007359924193442822 < x < 0.007101371656256956

    1. Initial program 59.1

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

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007359924193442822:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.007101371656256956:\\ \;\;\;\;\left(\left(\frac{-1}{3} \cdot \left(x \cdot x\right)\right) \cdot x + x\right) + {x}^{5} \cdot \frac{2}{15}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))