Average Error: 29.7 → 0.9
Time: 52.9s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1.7297142739317407 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right) - 1}{\left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right) - \sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}} \cdot \left(\left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right) \cdot \left(\sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}}\right)\right)} \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - \left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right)\right)\\ \mathbf{elif}\;-2 \cdot x \le 1.5895218096725198 \cdot 10^{-06}:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} - \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right) - 1}{\left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right) - \sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}} \cdot \left(\left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right) \cdot \left(\sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}}\right)\right)} \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - \left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right)\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -1.7297142739317407 \cdot 10^{+18}:\\
\;\;\;\;\frac{\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right) - 1}{\left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right) - \sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}} \cdot \left(\left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right) \cdot \left(\sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}}\right)\right)} \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - \left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right)\right)\\

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

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{e^{-2 \cdot x} + 1} \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right) - 1}{\left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right) - \sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}} \cdot \left(\left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right) \cdot \left(\sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}} \cdot \sqrt[3]{1 + \frac{2}{e^{-2 \cdot x} + 1}}\right)\right)} \cdot \left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - \left(1 + \frac{2}{e^{-2 \cdot x} + 1}\right)\right)\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r2402575 = 2.0;
        double r2402576 = 1.0;
        double r2402577 = -2.0;
        double r2402578 = x;
        double r2402579 = r2402577 * r2402578;
        double r2402580 = exp(r2402579);
        double r2402581 = r2402576 + r2402580;
        double r2402582 = r2402575 / r2402581;
        double r2402583 = r2402582 - r2402576;
        return r2402583;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r2402584 = -2.0;
        double r2402585 = x;
        double r2402586 = r2402584 * r2402585;
        double r2402587 = -1.7297142739317407e+18;
        bool r2402588 = r2402586 <= r2402587;
        double r2402589 = 2.0;
        double r2402590 = exp(r2402586);
        double r2402591 = 1.0;
        double r2402592 = r2402590 + r2402591;
        double r2402593 = r2402589 / r2402592;
        double r2402594 = r2402593 * r2402593;
        double r2402595 = r2402593 * r2402594;
        double r2402596 = r2402595 - r2402591;
        double r2402597 = r2402594 * r2402594;
        double r2402598 = r2402591 + r2402593;
        double r2402599 = cbrt(r2402598);
        double r2402600 = r2402599 * r2402599;
        double r2402601 = r2402598 * r2402600;
        double r2402602 = r2402599 * r2402601;
        double r2402603 = r2402597 - r2402602;
        double r2402604 = r2402596 / r2402603;
        double r2402605 = r2402594 - r2402598;
        double r2402606 = r2402604 * r2402605;
        double r2402607 = 1.5895218096725198e-06;
        bool r2402608 = r2402586 <= r2402607;
        double r2402609 = 0.13333333333333333;
        double r2402610 = 5.0;
        double r2402611 = pow(r2402585, r2402610);
        double r2402612 = r2402609 * r2402611;
        double r2402613 = r2402585 * r2402585;
        double r2402614 = 0.3333333333333333;
        double r2402615 = r2402613 * r2402614;
        double r2402616 = r2402615 * r2402585;
        double r2402617 = r2402612 - r2402616;
        double r2402618 = r2402617 + r2402585;
        double r2402619 = r2402608 ? r2402618 : r2402606;
        double r2402620 = r2402588 ? r2402606 : r2402619;
        return r2402620;
}

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 2 regimes

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

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied associate-/r*0.1

      \[\leadsto \color{blue}{\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}}} - 1\]
    5. Using strategy rm

    6. Applied flip3--0.1

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

    7. Simplified0.1

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

    8. Simplified0.1

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

    10. Applied flip-+0.1

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

    11. Applied associate-/r/0.1

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

    13. Applied add-cube-cbrt0.1

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

    14. Applied associate-*r*0.1

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

    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

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

  4. Final simplification0.9

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

Reproduce

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