Average Error: 29.3 → 1.6
Time: 5.6s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -2.66125213953473545 \cdot 10^{30}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 2.69769158723092915 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}} + \sqrt{1}\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}} - \sqrt{1}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -2.66125213953473545 \cdot 10^{30}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}} - 1\\

\mathbf{elif}\;-2 \cdot x \le 2.69769158723092915 \cdot 10^{-5}:\\
\;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r52010 = 2.0;
        double r52011 = 1.0;
        double r52012 = -2.0;
        double r52013 = x;
        double r52014 = r52012 * r52013;
        double r52015 = exp(r52014);
        double r52016 = r52011 + r52015;
        double r52017 = r52010 / r52016;
        double r52018 = r52017 - r52011;
        return r52018;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r52019 = -2.0;
        double r52020 = x;
        double r52021 = r52019 * r52020;
        double r52022 = -2.6612521395347354e+30;
        bool r52023 = r52021 <= r52022;
        double r52024 = 2.0;
        double r52025 = 1.0;
        double r52026 = exp(r52021);
        double r52027 = r52025 + r52026;
        double r52028 = r52024 / r52027;
        double r52029 = 3.0;
        double r52030 = pow(r52028, r52029);
        double r52031 = cbrt(r52030);
        double r52032 = r52031 - r52025;
        double r52033 = 2.697691587230929e-05;
        bool r52034 = r52021 <= r52033;
        double r52035 = r52025 * r52020;
        double r52036 = 5.551115123125783e-17;
        double r52037 = 4.0;
        double r52038 = pow(r52020, r52037);
        double r52039 = r52036 * r52038;
        double r52040 = 0.33333333333333337;
        double r52041 = pow(r52020, r52029);
        double r52042 = r52040 * r52041;
        double r52043 = r52039 + r52042;
        double r52044 = r52035 - r52043;
        double r52045 = sqrt(r52030);
        double r52046 = cbrt(r52045);
        double r52047 = sqrt(r52025);
        double r52048 = r52046 + r52047;
        double r52049 = r52046 - r52047;
        double r52050 = r52048 * r52049;
        double r52051 = r52034 ? r52044 : r52050;
        double r52052 = r52023 ? r52032 : r52051;
        return r52052;
}

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) < -2.6612521395347354e+30

    1. Initial program 0

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

    3. Applied add-cbrt-cube0

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

    4. Applied add-cbrt-cube0

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

    5. Applied cbrt-undiv0

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

    6. Simplified0

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

    if -2.6612521395347354e+30 < (* -2.0 x) < 2.697691587230929e-05

    1. Initial program 56.5

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

    2. Taylor expanded around 0 3.0

      \[\leadsto \color{blue}{1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)}\]

    if 2.697691587230929e-05 < (* -2.0 x)

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    4. Applied add-cbrt-cube0.1

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

    5. Applied cbrt-undiv0.1

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

    6. Simplified0.1

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

    8. Applied add-sqr-sqrt0.1

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

    9. Applied add-sqr-sqrt0.1

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

    10. Applied cbrt-prod0.1

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

    11. Applied difference-of-squares0.1

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -2.66125213953473545 \cdot 10^{30}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 2.69769158723092915 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}} + \sqrt{1}\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}} - \sqrt{1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))