Average Error: 29.2 → 0.5
Time: 19.1s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1147158.01509999507106840610504150390625:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 9.326698951531978754675718365670100684639 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.3333333333333333703407674875052180141211\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right) \cdot \left(\sqrt{1} + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -1147158.01509999507106840610504150390625:\\
\;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\

\mathbf{elif}\;-2 \cdot x \le 9.326698951531978754675718365670100684639 \cdot 10^{-11}:\\
\;\;\;\;x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.3333333333333333703407674875052180141211\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r2312979 = 2.0;
        double r2312980 = 1.0;
        double r2312981 = -2.0;
        double r2312982 = x;
        double r2312983 = r2312981 * r2312982;
        double r2312984 = exp(r2312983);
        double r2312985 = r2312980 + r2312984;
        double r2312986 = r2312979 / r2312985;
        double r2312987 = r2312986 - r2312980;
        return r2312987;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r2312988 = -2.0;
        double r2312989 = x;
        double r2312990 = r2312988 * r2312989;
        double r2312991 = -1147158.015099995;
        bool r2312992 = r2312990 <= r2312991;
        double r2312993 = 2.0;
        double r2312994 = 1.0;
        double r2312995 = exp(r2312990);
        double r2312996 = r2312994 + r2312995;
        double r2312997 = r2312993 / r2312996;
        double r2312998 = r2312997 - r2312994;
        double r2312999 = 9.326698951531979e-11;
        bool r2313000 = r2312990 <= r2312999;
        double r2313001 = r2312989 * r2312989;
        double r2313002 = 0.33333333333333337;
        double r2313003 = r2313001 * r2313002;
        double r2313004 = r2312994 - r2313003;
        double r2313005 = r2312989 * r2313004;
        double r2313006 = 5.551115123125783e-17;
        double r2313007 = r2313001 * r2313006;
        double r2313008 = r2313001 * r2313007;
        double r2313009 = r2313005 - r2313008;
        double r2313010 = sqrt(r2312997);
        double r2313011 = sqrt(r2312994);
        double r2313012 = r2313010 - r2313011;
        double r2313013 = r2313011 + r2313010;
        double r2313014 = r2313012 * r2313013;
        double r2313015 = r2313000 ? r2313009 : r2313014;
        double r2313016 = r2312992 ? r2312998 : r2313015;
        return r2313016;
}

Error

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Results

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Derivation

  1. Split input into 3 regimes

  2. if (* -2.0 x) < -1147158.015099995

    1. Initial program 0

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

    3. Applied add-sqr-sqrt0

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

    4. Applied add-sqr-sqrt1.6

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

    5. Applied difference-of-squares1.0

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

    7. Applied *-un-lft-identity1.0

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

    8. Applied associate-*l*1.0

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

    9. Simplified0

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

    if -1147158.015099995 < (* -2.0 x) < 9.326698951531979e-11

    1. Initial program 58.5

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

    2. Taylor expanded around 0 0.8

      \[\leadsto \color{blue}{1 \cdot x - \left(0.3333333333333333703407674875052180141211 \cdot {x}^{3} + 5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4}\right)}\]

    3. Simplified0.8

      \[\leadsto \color{blue}{x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.3333333333333333703407674875052180141211\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)}\]

    if 9.326698951531979e-11 < (* -2.0 x)

    1. Initial program 0.4

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

    3. Applied add-sqr-sqrt0.4

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

    4. Applied add-sqr-sqrt0.4

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

    5. Applied difference-of-squares0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1147158.01509999507106840610504150390625:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 9.326698951531978754675718365670100684639 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.3333333333333333703407674875052180141211\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right) \cdot \left(\sqrt{1} + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))