Average Error: 29.2 → 0.5
Time: 18.0s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1147158.01509999507106840610504150390625:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 9.326698951531978754675718365670100684639 \cdot 10^{-11}:\\ \;\;\;\;x \cdot 1 - \left(x \cdot x\right) \cdot \left(0.3333333333333333703407674875052180141211 \cdot x + \left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right) \cdot \left(\sqrt{1} + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right)\\ \end{array}\]
\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 1 - \left(x \cdot x\right) \cdot \left(0.3333333333333333703407674875052180141211 \cdot x + \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 r2431153 = 2.0;
        double r2431154 = 1.0;
        double r2431155 = -2.0;
        double r2431156 = x;
        double r2431157 = r2431155 * r2431156;
        double r2431158 = exp(r2431157);
        double r2431159 = r2431154 + r2431158;
        double r2431160 = r2431153 / r2431159;
        double r2431161 = r2431160 - r2431154;
        return r2431161;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r2431162 = -2.0;
        double r2431163 = x;
        double r2431164 = r2431162 * r2431163;
        double r2431165 = -1147158.015099995;
        bool r2431166 = r2431164 <= r2431165;
        double r2431167 = 2.0;
        double r2431168 = 1.0;
        double r2431169 = exp(r2431164);
        double r2431170 = r2431168 + r2431169;
        double r2431171 = r2431167 / r2431170;
        double r2431172 = r2431171 - r2431168;
        double r2431173 = 9.326698951531979e-11;
        bool r2431174 = r2431164 <= r2431173;
        double r2431175 = r2431163 * r2431168;
        double r2431176 = r2431163 * r2431163;
        double r2431177 = 0.33333333333333337;
        double r2431178 = r2431177 * r2431163;
        double r2431179 = 5.551115123125783e-17;
        double r2431180 = r2431176 * r2431179;
        double r2431181 = r2431178 + r2431180;
        double r2431182 = r2431176 * r2431181;
        double r2431183 = r2431175 - r2431182;
        double r2431184 = sqrt(r2431171);
        double r2431185 = sqrt(r2431168);
        double r2431186 = r2431184 - r2431185;
        double r2431187 = r2431185 + r2431184;
        double r2431188 = r2431186 * r2431187;
        double r2431189 = r2431174 ? r2431183 : r2431188;
        double r2431190 = r2431166 ? r2431172 : r2431189;
        return r2431190;
}

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 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

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

    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

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

Reproduce

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