Average Error: 29.2 → 0.0
Time: 3.9m
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.006778640856078821:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.007603939260661412:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{-1}{3} \cdot x\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\frac{2}{15}, \left({x}^{5}\right), x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right), \left(\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)\right), -1\right)}{\left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \left(\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) + \left(1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)\right)}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.006778640856078821:\\
\;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\

\mathbf{elif}\;x \le 0.007603939260661412:\\
\;\;\;\;\mathsf{fma}\left(\left(\frac{-1}{3} \cdot x\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\frac{2}{15}, \left({x}^{5}\right), x\right)\right)\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r23339881 = 2.0;
        double r23339882 = 1.0;
        double r23339883 = -2.0;
        double r23339884 = x;
        double r23339885 = r23339883 * r23339884;
        double r23339886 = exp(r23339885);
        double r23339887 = r23339882 + r23339886;
        double r23339888 = r23339881 / r23339887;
        double r23339889 = r23339888 - r23339882;
        return r23339889;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r23339890 = x;
        double r23339891 = -0.006778640856078821;
        bool r23339892 = r23339890 <= r23339891;
        double r23339893 = 2.0;
        double r23339894 = 1.0;
        double r23339895 = -2.0;
        double r23339896 = r23339895 * r23339890;
        double r23339897 = exp(r23339896);
        double r23339898 = r23339894 + r23339897;
        double r23339899 = r23339893 / r23339898;
        double r23339900 = r23339899 - r23339894;
        double r23339901 = 0.007603939260661412;
        bool r23339902 = r23339890 <= r23339901;
        double r23339903 = -0.3333333333333333;
        double r23339904 = r23339903 * r23339890;
        double r23339905 = r23339890 * r23339890;
        double r23339906 = 0.13333333333333333;
        double r23339907 = 5.0;
        double r23339908 = pow(r23339890, r23339907);
        double r23339909 = fma(r23339906, r23339908, r23339890);
        double r23339910 = fma(r23339904, r23339905, r23339909);
        double r23339911 = r23339899 * r23339899;
        double r23339912 = r23339911 * r23339911;
        double r23339913 = -1.0;
        double r23339914 = fma(r23339911, r23339912, r23339913);
        double r23339915 = r23339899 + r23339894;
        double r23339916 = r23339894 + r23339911;
        double r23339917 = r23339912 + r23339916;
        double r23339918 = r23339915 * r23339917;
        double r23339919 = r23339914 / r23339918;
        double r23339920 = r23339902 ? r23339910 : r23339919;
        double r23339921 = r23339892 ? r23339900 : r23339920;
        return r23339921;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if x < -0.006778640856078821

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

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

    if -0.006778640856078821 < x < 0.007603939260661412

    1. Initial program 58.9

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot \frac{-1}{3}\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\frac{2}{15}, \left({x}^{5}\right), x\right)\right)\right)}\]

    if 0.007603939260661412 < x

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    5. Applied flip3--0.0

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

    6. Applied associate-/l/0.0

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

    7. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.006778640856078821:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.007603939260661412:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{-1}{3} \cdot x\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\frac{2}{15}, \left({x}^{5}\right), x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right), \left(\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)\right), -1\right)}{\left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \left(\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) + \left(1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))