Average Error: 29.2 → 0.0
Time: 54.6s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007359924193442822:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.007101371656256956:\\ \;\;\;\;(\left(\frac{-1}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((\frac{2}{15} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]
\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(\frac{-1}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((\frac{2}{15} \cdot \left({x}^{5}\right) + x)_*\right))_*\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r5199715 = 2.0;
        double r5199716 = 1.0;
        double r5199717 = -2.0;
        double r5199718 = x;
        double r5199719 = r5199717 * r5199718;
        double r5199720 = exp(r5199719);
        double r5199721 = r5199716 + r5199720;
        double r5199722 = r5199715 / r5199721;
        double r5199723 = r5199722 - r5199716;
        return r5199723;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r5199724 = x;
        double r5199725 = -0.007359924193442822;
        bool r5199726 = r5199724 <= r5199725;
        double r5199727 = 2.0;
        double r5199728 = 1.0;
        double r5199729 = -2.0;
        double r5199730 = r5199729 * r5199724;
        double r5199731 = exp(r5199730);
        double r5199732 = r5199728 + r5199731;
        double r5199733 = r5199727 / r5199732;
        double r5199734 = r5199733 - r5199728;
        double r5199735 = 0.007101371656256956;
        bool r5199736 = r5199724 <= r5199735;
        double r5199737 = -0.3333333333333333;
        double r5199738 = r5199737 * r5199724;
        double r5199739 = r5199724 * r5199724;
        double r5199740 = 0.13333333333333333;
        double r5199741 = 5.0;
        double r5199742 = pow(r5199724, r5199741);
        double r5199743 = fma(r5199740, r5199742, r5199724);
        double r5199744 = fma(r5199738, r5199739, r5199743);
        double r5199745 = r5199736 ? r5199744 : r5199734;
        double r5199746 = r5199726 ? r5199734 : r5199745;
        return r5199746;
}

Error

Bits error versus x

Bits error versus y

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 -inf 59.1

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

    3. Taylor expanded around 0 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

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