Average Error: 29.3 → 0.2
Time: 4.6s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -7.14473254542662855 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{1 + e^{-2 \cdot x}}}, \frac{2}{\sqrt{1 + e^{-2 \cdot x}}}, -1\right)\\ \mathbf{elif}\;-2 \cdot x \le 1.26447032404754653 \cdot 10^{-10}:\\ \;\;\;\;1 \cdot x - \left(\mathsf{fma}\left({x}^{3}, 0.33333333333333337, 1\right) - 0.5 \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}, \frac{\sqrt[3]{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}, -1\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -7.14473254542662855 \cdot 10^{-4}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{1 + e^{-2 \cdot x}}}, \frac{2}{\sqrt{1 + e^{-2 \cdot x}}}, -1\right)\\

\mathbf{elif}\;-2 \cdot x \le 1.26447032404754653 \cdot 10^{-10}:\\
\;\;\;\;1 \cdot x - \left(\mathsf{fma}\left({x}^{3}, 0.33333333333333337, 1\right) - 0.5 \cdot 2\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r80848 = 2.0;
        double r80849 = 1.0;
        double r80850 = -2.0;
        double r80851 = x;
        double r80852 = r80850 * r80851;
        double r80853 = exp(r80852);
        double r80854 = r80849 + r80853;
        double r80855 = r80848 / r80854;
        double r80856 = r80855 - r80849;
        return r80856;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r80857 = -2.0;
        double r80858 = x;
        double r80859 = r80857 * r80858;
        double r80860 = -0.0007144732545426629;
        bool r80861 = r80859 <= r80860;
        double r80862 = 1.0;
        double r80863 = 1.0;
        double r80864 = exp(r80859);
        double r80865 = r80863 + r80864;
        double r80866 = sqrt(r80865);
        double r80867 = r80862 / r80866;
        double r80868 = 2.0;
        double r80869 = r80868 / r80866;
        double r80870 = -r80863;
        double r80871 = fma(r80867, r80869, r80870);
        double r80872 = 1.2644703240475465e-10;
        bool r80873 = r80859 <= r80872;
        double r80874 = r80863 * r80858;
        double r80875 = 3.0;
        double r80876 = pow(r80858, r80875);
        double r80877 = 0.33333333333333337;
        double r80878 = fma(r80876, r80877, r80863);
        double r80879 = 0.5;
        double r80880 = r80879 * r80868;
        double r80881 = r80878 - r80880;
        double r80882 = r80874 - r80881;
        double r80883 = cbrt(r80868);
        double r80884 = r80883 * r80883;
        double r80885 = cbrt(r80865);
        double r80886 = r80885 * r80885;
        double r80887 = r80884 / r80886;
        double r80888 = r80883 / r80885;
        double r80889 = fma(r80887, r80888, r80870);
        double r80890 = r80873 ? r80882 : r80889;
        double r80891 = r80861 ? r80871 : r80890;
        return r80891;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if (* -2.0 x) < -0.0007144732545426629

    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 *-un-lft-identity0.1

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

    5. Applied times-frac0.1

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

    6. Applied fma-neg0.1

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

    if -0.0007144732545426629 < (* -2.0 x) < 1.2644703240475465e-10

    1. Initial program 59.7

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

    3. Applied add-sqr-sqrt60.7

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

    4. Applied *-un-lft-identity60.7

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

    5. Applied times-frac60.7

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

    6. Applied fma-neg60.7

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

    7. Taylor expanded around 0 60.7

      \[\leadsto \color{blue}{\left(2 \cdot {\left(\sqrt{0.5}\right)}^{2} + 1 \cdot x\right) - \left(1 + 0.33333333333333337 \cdot {x}^{3}\right)}\]

    8. Simplified0.1

      \[\leadsto \color{blue}{1 \cdot x - \left(\mathsf{fma}\left({x}^{3}, 0.33333333333333337, 1\right) - 0.5 \cdot 2\right)}\]

    if 1.2644703240475465e-10 < (* -2.0 x)

    1. Initial program 0.4

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

    3. Applied add-cube-cbrt0.5

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

    4. Applied add-cube-cbrt0.5

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

    5. Applied times-frac0.5

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

    6. Applied fma-neg0.5

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -7.14473254542662855 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{1 + e^{-2 \cdot x}}}, \frac{2}{\sqrt{1 + e^{-2 \cdot x}}}, -1\right)\\ \mathbf{elif}\;-2 \cdot x \le 1.26447032404754653 \cdot 10^{-10}:\\ \;\;\;\;1 \cdot x - \left(\mathsf{fma}\left({x}^{3}, 0.33333333333333337, 1\right) - 0.5 \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}, \frac{\sqrt[3]{2}}{\sqrt[3]{1 + e^{-2 \cdot x}}}, -1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))