Average Error: 29.7 → 0.1
Time: 4.5s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -12.329649134843827:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 5.30640056390034621 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(4.996 \cdot 10^{-16}, {x}^{4}, 0.33333333333333348 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}{\mathsf{fma}\left(\frac{2}{1 + e^{-2 \cdot x}}, \frac{2}{1 + e^{-2 \cdot x}}, 1 \cdot 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -12.329649134843827:\\
\;\;\;\;\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}\\

\mathbf{elif}\;-2 \cdot x \le 5.30640056390034621 \cdot 10^{-6}:\\
\;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(4.996 \cdot 10^{-16}, {x}^{4}, 0.33333333333333348 \cdot {x}^{3}\right)\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r57760 = 2.0;
        double r57761 = 1.0;
        double r57762 = -2.0;
        double r57763 = x;
        double r57764 = r57762 * r57763;
        double r57765 = exp(r57764);
        double r57766 = r57761 + r57765;
        double r57767 = r57760 / r57766;
        double r57768 = r57767 - r57761;
        return r57768;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r57769 = -2.0;
        double r57770 = x;
        double r57771 = r57769 * r57770;
        double r57772 = -12.329649134843827;
        bool r57773 = r57771 <= r57772;
        double r57774 = 2.0;
        double r57775 = 1.0;
        double r57776 = exp(r57771);
        double r57777 = r57775 + r57776;
        double r57778 = r57774 / r57777;
        double r57779 = r57778 * r57778;
        double r57780 = r57775 * r57775;
        double r57781 = r57779 - r57780;
        double r57782 = r57778 + r57775;
        double r57783 = r57781 / r57782;
        double r57784 = 5.306400563900346e-06;
        bool r57785 = r57771 <= r57784;
        double r57786 = 4.996003610813204e-16;
        double r57787 = 4.0;
        double r57788 = pow(r57770, r57787);
        double r57789 = 0.3333333333333335;
        double r57790 = 3.0;
        double r57791 = pow(r57770, r57790);
        double r57792 = r57789 * r57791;
        double r57793 = fma(r57786, r57788, r57792);
        double r57794 = -r57793;
        double r57795 = fma(r57775, r57770, r57794);
        double r57796 = r57779 * r57779;
        double r57797 = r57780 * r57780;
        double r57798 = r57796 - r57797;
        double r57799 = fma(r57778, r57778, r57780);
        double r57800 = r57799 * r57782;
        double r57801 = r57798 / r57800;
        double r57802 = r57785 ? r57795 : r57801;
        double r57803 = r57773 ? r57783 : r57802;
        return r57803;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if (* -2.0 x) < -12.329649134843827

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    if -12.329649134843827 < (* -2.0 x) < 5.306400563900346e-06

    1. Initial program 59.0

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

    3. Applied flip--59.0

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

    4. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{1 \cdot x - \left(4.996 \cdot 10^{-16} \cdot {x}^{4} + 0.33333333333333348 \cdot {x}^{3}\right)}\]

    5. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, x, -\mathsf{fma}\left(4.996 \cdot 10^{-16}, {x}^{4}, 0.33333333333333348 \cdot {x}^{3}\right)\right)}\]

    if 5.306400563900346e-06 < (* -2.0 x)

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    5. Applied flip--0.1

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

    6. Applied associate-/l/0.1

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

    7. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -12.329649134843827:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 5.30640056390034621 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(4.996 \cdot 10^{-16}, {x}^{4}, 0.33333333333333348 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}{\mathsf{fma}\left(\frac{2}{1 + e^{-2 \cdot x}}, \frac{2}{1 + e^{-2 \cdot x}}, 1 \cdot 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))