Average Error: 29.5 → 0.5
Time: 5.5s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -209.68920730807355:\\ \;\;\;\;\frac{\log \left({\left(e^{1 + \frac{2}{{\left(e^{-2 \cdot x}\right)}^{3} + {1}^{3}} \cdot \left(1 \cdot 1 + e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right)\right)}\right)}^{\left(\frac{2}{{\left(e^{-2 \cdot x}\right)}^{3} + {1}^{3}} \cdot \left(1 \cdot 1 + e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right)\right) - 1\right)}\right)}{1 + \frac{2}{e^{-2 \cdot x} + 1}}\\ \mathbf{elif}\;-2 \cdot x \leq 1.2623611574830718 \cdot 10^{-16}:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1 \cdot 1}{1 + \frac{2}{e^{-2 \cdot x} + 1}}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -209.68920730807355:\\
\;\;\;\;\frac{\log \left({\left(e^{1 + \frac{2}{{\left(e^{-2 \cdot x}\right)}^{3} + {1}^{3}} \cdot \left(1 \cdot 1 + e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right)\right)}\right)}^{\left(\frac{2}{{\left(e^{-2 \cdot x}\right)}^{3} + {1}^{3}} \cdot \left(1 \cdot 1 + e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right)\right) - 1\right)}\right)}{1 + \frac{2}{e^{-2 \cdot x} + 1}}\\

\mathbf{elif}\;-2 \cdot x \leq 1.2623611574830718 \cdot 10^{-16}:\\
\;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1 \cdot 1}{1 + \frac{2}{e^{-2 \cdot x} + 1}}\\

\end{array}
(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))
↓
(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -209.68920730807355)
   (/
    (log
     (pow
      (exp
       (+
        1.0
        (*
         (/ 2.0 (+ (pow (exp (* -2.0 x)) 3.0) (pow 1.0 3.0)))
         (+ (* 1.0 1.0) (* (exp (* -2.0 x)) (- (exp (* -2.0 x)) 1.0))))))
      (-
       (*
        (/ 2.0 (+ (pow (exp (* -2.0 x)) 3.0) (pow 1.0 3.0)))
        (+ (* 1.0 1.0) (* (exp (* -2.0 x)) (- (exp (* -2.0 x)) 1.0))))
       1.0)))
    (+ 1.0 (/ 2.0 (+ (exp (* -2.0 x)) 1.0))))
   (if (<= (* -2.0 x) 1.2623611574830718e-16)
     (-
      (* x 1.0)
      (* (pow x 3.0) (+ (* x 5.551115123125783e-17) 0.33333333333333337)))
     (/
      (-
       (* (/ 2.0 (+ (exp (* -2.0 x)) 1.0)) (/ 2.0 (+ (exp (* -2.0 x)) 1.0)))
       (* 1.0 1.0))
      (+ 1.0 (/ 2.0 (+ (exp (* -2.0 x)) 1.0)))))))
double code(double x, double y) {
	return ((double) ((2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x))))))) - 1.0));
}
↓
double code(double x, double y) {
	double tmp;
	if ((((double) (-2.0 * x)) <= -209.68920730807355)) {
		tmp = (((double) log(((double) pow(((double) exp(((double) (1.0 + ((double) ((2.0 / ((double) (((double) pow(((double) exp(((double) (-2.0 * x)))), 3.0)) + ((double) pow(1.0, 3.0))))) * ((double) (((double) (1.0 * 1.0)) + ((double) (((double) exp(((double) (-2.0 * x)))) * ((double) (((double) exp(((double) (-2.0 * x)))) - 1.0)))))))))))), ((double) (((double) ((2.0 / ((double) (((double) pow(((double) exp(((double) (-2.0 * x)))), 3.0)) + ((double) pow(1.0, 3.0))))) * ((double) (((double) (1.0 * 1.0)) + ((double) (((double) exp(((double) (-2.0 * x)))) * ((double) (((double) exp(((double) (-2.0 * x)))) - 1.0)))))))) - 1.0)))))) / ((double) (1.0 + (2.0 / ((double) (((double) exp(((double) (-2.0 * x)))) + 1.0))))));
	} else {
		double tmp_1;
		if ((((double) (-2.0 * x)) <= 1.2623611574830718e-16)) {
			tmp_1 = ((double) (((double) (x * 1.0)) - ((double) (((double) pow(x, 3.0)) * ((double) (((double) (x * 5.551115123125783e-17)) + 0.33333333333333337))))));
		} else {
			tmp_1 = (((double) (((double) ((2.0 / ((double) (((double) exp(((double) (-2.0 * x)))) + 1.0))) * (2.0 / ((double) (((double) exp(((double) (-2.0 * x)))) + 1.0))))) - ((double) (1.0 * 1.0)))) / ((double) (1.0 + (2.0 / ((double) (((double) exp(((double) (-2.0 * x)))) + 1.0))))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

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 3 regimes

  2. if (* -2.0 x) < -209.68920730807355

    1. Initial program Error: 0 bits

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

    3. Applied flip--Error: 0 bits

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

    4. SimplifiedError: 0 bits

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

    6. Applied flip3-+Error: 0 bits

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

    7. Applied associate-/r/Error: 0 bits

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

    8. Applied flip3-+Error: 0 bits

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

    9. Applied associate-/r/Error: 0 bits

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

    10. Applied swap-sqrError: 0 bits

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

    11. SimplifiedError: 0 bits

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

    13. Applied add-log-expError: 0 bits

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

    14. Applied add-log-expError: 0 bits

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

    15. Applied diff-logError: 0 bits

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

    16. SimplifiedError: 0 bits

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

    if -209.68920730807355 < (* -2.0 x) < 1.26236115748307184e-16

    1. Initial program Error: 59.2 bits

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

    2. Taylor expanded around 0 Error: 0.4 bits

      \[\leadsto \color{blue}{1 \cdot x - \left(5.551115123125783 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)}\]

    3. SimplifiedError: 0.4 bits

      \[\leadsto \color{blue}{1 \cdot x - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)}\]

    if 1.26236115748307184e-16 < (* -2.0 x)

    1. Initial program Error: 1.2 bits

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

    3. Applied flip--Error: 1.2 bits

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

    4. SimplifiedError: 1.2 bits

      \[\leadsto \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\color{blue}{1 + \frac{2}{1 + e^{-2 \cdot x}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplificationError: 0.5 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -209.68920730807355:\\ \;\;\;\;\frac{\log \left({\left(e^{1 + \frac{2}{{\left(e^{-2 \cdot x}\right)}^{3} + {1}^{3}} \cdot \left(1 \cdot 1 + e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right)\right)}\right)}^{\left(\frac{2}{{\left(e^{-2 \cdot x}\right)}^{3} + {1}^{3}} \cdot \left(1 \cdot 1 + e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right)\right) - 1\right)}\right)}{1 + \frac{2}{e^{-2 \cdot x} + 1}}\\ \mathbf{elif}\;-2 \cdot x \leq 1.2623611574830718 \cdot 10^{-16}:\\ \;\;\;\;x \cdot 1 - {x}^{3} \cdot \left(x \cdot 5.551115123125783 \cdot 10^{-17} + 0.33333333333333337\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1} - 1 \cdot 1}{1 + \frac{2}{e^{-2 \cdot x} + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))