Logistic function from Lakshay Garg

Average Error: 29.1 → 0.5

Time: 35.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.012573984788303481 \lor \neg \left(-2 \cdot x \leq 3.709510049264789 \cdot 10^{-21}\right):\\ \;\;\;\;\left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)\right) \cdot \left(\frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3}}{\left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right) \cdot \left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{12} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right)} - \frac{1}{\left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right) \cdot \left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{12} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \end{array}\]

FPCore

(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.012573984788303481)
         (not (<= (* -2.0 x) 3.709510049264789e-21)))
   (*
    (*
     (-
      (pow (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 6.0)
      (+ 1.0 (pow (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 3.0)))
     (-
      1.0
      (/ (+ 2.0 (/ 4.0 (+ 1.0 (exp (* -2.0 x))))) (+ 1.0 (exp (* -2.0 x))))))
    (-
     (/
      (pow (pow (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 3.0) 3.0)
      (*
       (-
        1.0
        (*
         (/ (+ 2.0 (/ 4.0 (+ 1.0 (exp (* -2.0 x))))) (+ 1.0 (exp (* -2.0 x))))
         (/
          (+ 2.0 (/ 4.0 (+ 1.0 (exp (* -2.0 x)))))
          (+ 1.0 (exp (* -2.0 x))))))
       (-
        (pow (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 12.0)
        (*
         (+ 1.0 (pow (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 3.0))
         (+ 1.0 (pow (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 3.0))))))
     (/
      1.0
      (*
       (-
        1.0
        (*
         (/ (+ 2.0 (/ 4.0 (+ 1.0 (exp (* -2.0 x))))) (+ 1.0 (exp (* -2.0 x))))
         (/
          (+ 2.0 (/ 4.0 (+ 1.0 (exp (* -2.0 x)))))
          (+ 1.0 (exp (* -2.0 x))))))
       (-
        (pow (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 12.0)
        (*
         (+ 1.0 (pow (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 3.0))
         (+ 1.0 (pow (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 3.0))))))))
   (-
    (+ x (* 0.13333333333333333 (pow x 5.0)))
    (* 0.3333333333333333 (pow x 3.0)))))

C

double code(double x, double y) {
	return (2.0 / (1.0 + exp(-2.0 * x))) - 1.0;
}

↓

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.012573984788303481) || !((-2.0 * x) <= 3.709510049264789e-21)) {
		tmp = ((pow((2.0 / (1.0 + exp(-2.0 * x))), 6.0) - (1.0 + pow((2.0 / (1.0 + exp(-2.0 * x))), 3.0))) * (1.0 - ((2.0 + (4.0 / (1.0 + exp(-2.0 * x)))) / (1.0 + exp(-2.0 * x))))) * ((pow(pow((2.0 / (1.0 + exp(-2.0 * x))), 3.0), 3.0) / ((1.0 - (((2.0 + (4.0 / (1.0 + exp(-2.0 * x)))) / (1.0 + exp(-2.0 * x))) * ((2.0 + (4.0 / (1.0 + exp(-2.0 * x)))) / (1.0 + exp(-2.0 * x))))) * (pow((2.0 / (1.0 + exp(-2.0 * x))), 12.0) - ((1.0 + pow((2.0 / (1.0 + exp(-2.0 * x))), 3.0)) * (1.0 + pow((2.0 / (1.0 + exp(-2.0 * x))), 3.0)))))) - (1.0 / ((1.0 - (((2.0 + (4.0 / (1.0 + exp(-2.0 * x)))) / (1.0 + exp(-2.0 * x))) * ((2.0 + (4.0 / (1.0 + exp(-2.0 * x)))) / (1.0 + exp(-2.0 * x))))) * (pow((2.0 / (1.0 + exp(-2.0 * x))), 12.0) - ((1.0 + pow((2.0 / (1.0 + exp(-2.0 * x))), 3.0)) * (1.0 + pow((2.0 / (1.0 + exp(-2.0 * x))), 3.0)))))));
	} else {
		tmp = (x + (0.13333333333333333 * pow(x, 5.0))) - (0.3333333333333333 * pow(x, 3.0));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -0.012573984788303481 or 3.709510049264789e-21 < (*.f64 -2 x)

   1. Initial program 1.0

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

   3. Applied flip3--_binary64_418 1.0

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

   4. Simplified 1.0

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

   5. Simplified 1.0

      \[\leadsto \frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - 1}{\color{blue}{1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}}\]
   6. Using strategy rm

   7. Applied flip3--_binary64_418 1.0

      \[\leadsto \frac{\color{blue}{\frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3} - {1}^{3}}{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} \cdot {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} + \left(1 \cdot 1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} \cdot 1\right)}}}{1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}\]

   8. Applied associate-/l/_binary64_363 1.0

      \[\leadsto \color{blue}{\frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3} - {1}^{3}}{\left(1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right) \cdot \left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} \cdot {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} + \left(1 \cdot 1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} \cdot 1\right)\right)}}\]

   9. Simplified 1.0

      \[\leadsto \frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3} - {1}^{3}}{\color{blue}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} + \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)}}\]
   10. Using strategy rm

   11. Applied div-sub_binary64_419 1.0

       \[\leadsto \color{blue}{\frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3}}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} + \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)} - \frac{{1}^{3}}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} + \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)}}\]

   12. Simplified 1.0

       \[\leadsto \frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3}}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} + \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)} - \color{blue}{\frac{1}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} + \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)}}\]
   13. Using strategy rm

   14. Applied flip-+_binary64_388 1.0

       \[\leadsto \frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3}}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} + \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)} - \frac{1}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} + \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \color{blue}{\frac{1 \cdot 1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}{1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}}}\]

   15. Applied flip-+_binary64_388 1.0

       \[\leadsto \frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3}}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} + \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)} - \frac{1}{\color{blue}{\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} \cdot {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}} \cdot \frac{1 \cdot 1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}{1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}}\]

   16. Applied frac-times_binary64_424 1.0

       \[\leadsto \frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3}}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} + \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)} - \frac{1}{\color{blue}{\frac{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} \cdot {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 \cdot 1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)}}}\]

   17. Applied associate-/r/_binary64_362 1.0

       \[\leadsto \frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3}}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} + \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 + \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)} - \color{blue}{\frac{1}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} \cdot {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 \cdot 1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)} \cdot \left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)\right)}\]

   18. Applied flip-+_binary64_388 1.0

       \[\leadsto \frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3}}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} + \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \color{blue}{\frac{1 \cdot 1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}{1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}}} - \frac{1}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} \cdot {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 \cdot 1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)} \cdot \left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)\right)\]

   19. Applied flip-+_binary64_388 1.0

       \[\leadsto \frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3}}{\color{blue}{\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} \cdot {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}} \cdot \frac{1 \cdot 1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}{1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}}} - \frac{1}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} \cdot {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 \cdot 1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)} \cdot \left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)\right)\]

   20. Applied frac-times_binary64_424 1.0

       \[\leadsto \frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3}}{\color{blue}{\frac{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} \cdot {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 \cdot 1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)}}} - \frac{1}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} \cdot {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 \cdot 1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)} \cdot \left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)\right)\]

   21. Applied associate-/r/_binary64_362 1.0

       \[\leadsto \color{blue}{\frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3}}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} \cdot {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 \cdot 1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)} \cdot \left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)\right)} - \frac{1}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} \cdot {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 \cdot 1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)} \cdot \left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)\right)\]

   22. Applied distribute-rgt-out--_binary64_370 1.0

       \[\leadsto \color{blue}{\left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)\right) \cdot \left(\frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3}}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} \cdot {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 \cdot 1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)} - \frac{1}{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} \cdot {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 \cdot 1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)}\right)}\]

   23. Simplified 1.0

       \[\leadsto \left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)\right) \cdot \color{blue}{\left(\frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3}}{\left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right) \cdot \left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{12} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right)} - \frac{1}{\left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right) \cdot \left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{12} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right)}\right)}\]

   if -0.012573984788303481 < (*.f64 -2 x) < 3.709510049264789e-21

   1. Initial program 59.7

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

   2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.012573984788303481 \lor \neg \left(-2 \cdot x \leq 3.709510049264789 \cdot 10^{-21}\right):\\ \;\;\;\;\left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right) \cdot \left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right)\right) \cdot \left(\frac{{\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}^{3}}{\left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right) \cdot \left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{12} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right)} - \frac{1}{\left(1 - \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}} \cdot \frac{2 + \frac{4}{1 + e^{-2 \cdot x}}}{1 + e^{-2 \cdot x}}\right) \cdot \left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{12} - \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right) \cdot \left(1 + {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020303 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))