Average Error: 29.5 → 0.0
Time: 2.3m
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007721035358892515:\\ \;\;\;\;\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\\ \mathbf{elif}\;x \le 0.00689805232726187:\\ \;\;\;\;(\left(\frac{-1}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((\frac{2}{15} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{(\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}}\right) \cdot \left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}}\right) + -1)_*} \cdot \log \left(e^{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}}\right)\\ \end{array}\]
double f(double x, double __attribute__((unused)) y) {
        double r14858427 = 2.0;
        double r14858428 = 1.0;
        double r14858429 = -2.0;
        double r14858430 = x;
        double r14858431 = r14858429 * r14858430;
        double r14858432 = exp(r14858431);
        double r14858433 = r14858428 + r14858432;
        double r14858434 = r14858427 / r14858433;
        double r14858435 = r14858434 - r14858428;
        return r14858435;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r14858436 = x;
        double r14858437 = -0.007721035358892515;
        bool r14858438 = r14858436 <= r14858437;
        double r14858439 = 2.0;
        double r14858440 = -2.0;
        double r14858441 = r14858440 * r14858436;
        double r14858442 = exp(r14858441);
        double r14858443 = 1.0;
        double r14858444 = r14858442 + r14858443;
        double r14858445 = r14858439 / r14858444;
        double r14858446 = r14858445 - r14858443;
        double r14858447 = exp(r14858446);
        double r14858448 = log(r14858447);
        double r14858449 = 0.00689805232726187;
        bool r14858450 = r14858436 <= r14858449;
        double r14858451 = -0.3333333333333333;
        double r14858452 = r14858451 * r14858436;
        double r14858453 = r14858436 * r14858436;
        double r14858454 = 0.13333333333333333;
        double r14858455 = 5.0;
        double r14858456 = pow(r14858436, r14858455);
        double r14858457 = fma(r14858454, r14858456, r14858436);
        double r14858458 = fma(r14858452, r14858453, r14858457);
        double r14858459 = sqrt(r14858445);
        double r14858460 = -1.0;
        double r14858461 = fma(r14858459, r14858459, r14858460);
        double r14858462 = cbrt(r14858461);
        double r14858463 = cbrt(r14858446);
        double r14858464 = r14858463 * r14858463;
        double r14858465 = exp(r14858464);
        double r14858466 = log(r14858465);
        double r14858467 = r14858462 * r14858466;
        double r14858468 = r14858450 ? r14858458 : r14858467;
        double r14858469 = r14858438 ? r14858448 : r14858468;
        return r14858469;
}


\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.007721035358892515:\\
\;\;\;\;\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\\

\mathbf{elif}\;x \le 0.00689805232726187:\\
\;\;\;\;(\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}:\\
\;\;\;\;\sqrt[3]{(\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}}\right) \cdot \left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}}\right) + -1)_*} \cdot \log \left(e^{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}}\right)\\

\end{array}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if x < -0.007721035358892515

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    if -0.007721035358892515 < x < 0.00689805232726187

    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

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

    if 0.00689805232726187 < x

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt0.0

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

    6. Applied exp-prod0.0

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

    7. Applied log-pow0.0

      \[\leadsto \color{blue}{\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \log \left(e^{\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt0.0

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

    10. Applied fma-neg0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007721035358892515:\\ \;\;\;\;\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1} - 1}\right)\\ \mathbf{elif}\;x \le 0.00689805232726187:\\ \;\;\;\;(\left(\frac{-1}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((\frac{2}{15} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{(\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}}\right) \cdot \left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}}\right) + -1)_*} \cdot \log \left(e^{\sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1} \cdot \sqrt[3]{\frac{2}{e^{-2 \cdot x} + 1} - 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))