Average Error: 29.2 → 0.3
Time: 6.7s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -10.55391129503279401546933513600379228592 \lor \neg \left(-2 \cdot x \le 3.456201489394080829830849858040359851019 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\frac{2}{1 + e^{-2 \cdot x}} \cdot \left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) + 1 \cdot 1}} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4} + 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -10.55391129503279401546933513600379228592 \lor \neg \left(-2 \cdot x \le 3.456201489394080829830849858040359851019 \cdot 10^{-10}\right):\\
\;\;\;\;\left(\sqrt[3]{\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\frac{2}{1 + e^{-2 \cdot x}} \cdot \left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) + 1 \cdot 1}} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4} + 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r66012 = 2.0;
        double r66013 = 1.0;
        double r66014 = -2.0;
        double r66015 = x;
        double r66016 = r66014 * r66015;
        double r66017 = exp(r66016);
        double r66018 = r66013 + r66017;
        double r66019 = r66012 / r66018;
        double r66020 = r66019 - r66013;
        return r66020;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r66021 = -2.0;
        double r66022 = x;
        double r66023 = r66021 * r66022;
        double r66024 = -10.553911295032794;
        bool r66025 = r66023 <= r66024;
        double r66026 = 3.456201489394081e-10;
        bool r66027 = r66023 <= r66026;
        double r66028 = !r66027;
        bool r66029 = r66025 || r66028;
        double r66030 = 2.0;
        double r66031 = 1.0;
        double r66032 = exp(r66023);
        double r66033 = r66031 + r66032;
        double r66034 = r66030 / r66033;
        double r66035 = 3.0;
        double r66036 = pow(r66034, r66035);
        double r66037 = pow(r66031, r66035);
        double r66038 = r66036 - r66037;
        double r66039 = r66031 + r66034;
        double r66040 = r66034 * r66039;
        double r66041 = r66031 * r66031;
        double r66042 = r66040 + r66041;
        double r66043 = r66038 / r66042;
        double r66044 = cbrt(r66043);
        double r66045 = r66034 - r66031;
        double r66046 = cbrt(r66045);
        double r66047 = r66044 * r66046;
        double r66048 = r66047 * r66046;
        double r66049 = r66031 * r66022;
        double r66050 = 5.551115123125783e-17;
        double r66051 = 4.0;
        double r66052 = pow(r66022, r66051);
        double r66053 = r66050 * r66052;
        double r66054 = 0.33333333333333337;
        double r66055 = pow(r66022, r66035);
        double r66056 = r66054 * r66055;
        double r66057 = r66053 + r66056;
        double r66058 = r66049 - r66057;
        double r66059 = r66029 ? r66048 : r66058;
        return r66059;
}

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 (* -2.0 x) < -10.553911295032794 or 3.456201489394081e-10 < (* -2.0 x)

    1. Initial program 0.2

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

    3. Applied add-cube-cbrt0.2

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

    5. Applied flip3--0.2

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

    6. Simplified0.2

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

    if -10.553911295032794 < (* -2.0 x) < 3.456201489394081e-10

    1. Initial program 59.0

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

    2. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4} + 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -10.55391129503279401546933513600379228592 \lor \neg \left(-2 \cdot x \le 3.456201489394080829830849858040359851019 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\frac{2}{1 + e^{-2 \cdot x}} \cdot \left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) + 1 \cdot 1}} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4} + 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))