Average Error: 29.0 → 0.0
Time: 4.1s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.063635990551007771 \lor \neg \left(-2 \cdot x \le 2.3421360244665261 \cdot 10^{-4}\right):\\ \;\;\;\;\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right) + \frac{\frac{\sqrt{2}}{\sqrt{1 + e^{-2 \cdot x}}} - \sqrt{1}}{2} \cdot \left(\frac{\sqrt{2}}{\sqrt{1 + e^{-2 \cdot x}}} + \sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.063635990551007771 \lor \neg \left(-2 \cdot x \le 2.3421360244665261 \cdot 10^{-4}\right):\\
\;\;\;\;\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right) + \frac{\frac{\sqrt{2}}{\sqrt{1 + e^{-2 \cdot x}}} - \sqrt{1}}{2} \cdot \left(\frac{\sqrt{2}}{\sqrt{1 + e^{-2 \cdot x}}} + \sqrt{1}\right)\\

\mathbf{else}:\\
\;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\

\end{array}
double code(double x, double y) {
	return ((double) (((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) - 1.0));
}
↓
double code(double x, double y) {
	double VAR;
	if (((((double) (-2.0 * x)) <= -0.06363599055100777) || !(((double) (-2.0 * x)) <= 0.0002342136024466526))) {
		VAR = ((double) (((double) log(((double) sqrt(((double) exp(((double) (((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) - 1.0)))))))) + ((double) (((double) (((double) (((double) (((double) sqrt(2.0)) / ((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))) - ((double) sqrt(1.0)))) / 2.0)) * ((double) (((double) (((double) sqrt(2.0)) / ((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))) + ((double) sqrt(1.0))))))));
	} else {
		VAR = ((double) (((double) (1.0 * x)) - ((double) (((double) (5.551115123125783e-17 * ((double) pow(x, 4.0)))) + ((double) (0.33333333333333337 * ((double) pow(x, 3.0))))))));
	}
	return VAR;
}

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) < -0.063635990551007771 or 2.3421360244665261e-4 < (* -2.0 x)

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    4. Applied add-log-exp0.0

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

    5. Applied diff-log0.0

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

    6. Simplified0.0

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

    8. Applied add-sqr-sqrt0.0

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

    9. Applied log-prod0.0

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

    11. Applied add-sqr-sqrt0.0

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

    12. Applied add-sqr-sqrt0.0

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

    13. Applied add-sqr-sqrt0.0

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

    14. Applied times-frac0.0

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

    15. Applied difference-of-squares0.0

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

    16. Applied exp-prod0.0

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

    17. Applied sqrt-pow10.0

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

    18. Applied log-pow0.0

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

    19. Simplified0.0

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

    if -0.063635990551007771 < (* -2.0 x) < 2.3421360244665261e-4

    1. Initial program 59.0

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.063635990551007771 \lor \neg \left(-2 \cdot x \le 2.3421360244665261 \cdot 10^{-4}\right):\\ \;\;\;\;\log \left(\sqrt{e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right) + \frac{\frac{\sqrt{2}}{\sqrt{1 + e^{-2 \cdot x}}} - \sqrt{1}}{2} \cdot \left(\frac{\sqrt{2}}{\sqrt{1 + e^{-2 \cdot x}}} + \sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020174 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))