Average Error: 29.7 → 0.9
Time: 22.6s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1.7297142739317407 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;-2 \cdot x \le 1.5895218096725198 \cdot 10^{-06}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{3}, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -1.7297142739317407 \cdot 10^{+18}:\\
\;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\

\mathbf{elif}\;-2 \cdot x \le 1.5895218096725198 \cdot 10^{-06}:\\
\;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{3}, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r1702089 = 2.0;
        double r1702090 = 1.0;
        double r1702091 = -2.0;
        double r1702092 = x;
        double r1702093 = r1702091 * r1702092;
        double r1702094 = exp(r1702093);
        double r1702095 = r1702090 + r1702094;
        double r1702096 = r1702089 / r1702095;
        double r1702097 = r1702096 - r1702090;
        return r1702097;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r1702098 = -2.0;
        double r1702099 = x;
        double r1702100 = r1702098 * r1702099;
        double r1702101 = -1.7297142739317407e+18;
        bool r1702102 = r1702100 <= r1702101;
        double r1702103 = 2.0;
        double r1702104 = exp(r1702100);
        double r1702105 = 1.0;
        double r1702106 = r1702104 + r1702105;
        double r1702107 = r1702103 / r1702106;
        double r1702108 = r1702107 - r1702105;
        double r1702109 = 1.5895218096725198e-06;
        bool r1702110 = r1702100 <= r1702109;
        double r1702111 = r1702099 * r1702099;
        double r1702112 = r1702099 * r1702111;
        double r1702113 = -0.3333333333333333;
        double r1702114 = 5.0;
        double r1702115 = pow(r1702099, r1702114);
        double r1702116 = 0.13333333333333333;
        double r1702117 = fma(r1702115, r1702116, r1702099);
        double r1702118 = fma(r1702112, r1702113, r1702117);
        double r1702119 = r1702110 ? r1702118 : r1702108;
        double r1702120 = r1702102 ? r1702108 : r1702119;
        return r1702120;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if (* -2 x) < -1.7297142739317407e+18 or 1.5895218096725198e-06 < (* -2 x)

    1. Initial program 0.1

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

    if -1.7297142739317407e+18 < (* -2 x) < 1.5895218096725198e-06

    1. Initial program 57.5

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

    2. Taylor expanded around 0 1.7

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

    3. Simplified1.7

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1.7297142739317407 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;-2 \cdot x \le 1.5895218096725198 \cdot 10^{-06}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{3}, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))