Logistic function from Lakshay Garg

Average Error: 29.5 → 0.1

Time: 4.8s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1.1154491279793497:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{\frac{64}{{\left(e^{-2 \cdot x} + 1\right)}^{6}}}} + 1\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 1.7050588330471929 \cdot 10^{-4}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} + 1\right) \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}^{3}}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]

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 temp;
	if (((-2.0 * x) <= -1.1154491279793497)) {
		temp = (((cbrt(sqrt((64.0 / pow((exp((-2.0 * x)) + 1.0), 6.0)))) + 1.0) * (cbrt(sqrt(pow((2.0 / (1.0 + exp((-2.0 * x)))), 6.0))) - 1.0)) / ((2.0 / (1.0 + exp((-2.0 * x)))) + 1.0));
	} else {
		double temp_1;
		if (((-2.0 * x) <= 0.0001705058833047193)) {
			temp_1 = ((1.0 * x) - ((5.551115123125783e-17 * pow(x, 4.0)) + (0.33333333333333337 * pow(x, 3.0))));
		} else {
			temp_1 = (((cbrt(sqrt(pow((2.0 / (1.0 + exp((-2.0 * x)))), 6.0))) + 1.0) * cbrt(pow((cbrt(sqrt(pow((2.0 / (1.0 + exp((-2.0 * x)))), 6.0))) - 1.0), 3.0))) / ((2.0 / (1.0 + exp((-2.0 * x)))) + 1.0));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 3 regimes

2. if (* -2.0 x) < -1.1154491279793497

   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 add-cbrt-cube 0.0

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

   6. Applied add-cbrt-cube 0.0

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

   7. Applied cbrt-undiv 0.0

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

   8. Applied add-cbrt-cube 0.0

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

   9. Applied add-cbrt-cube 0.0

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

   10. Applied cbrt-undiv 0.0

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

   11. Applied cbrt-unprod 0.0

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

   12. Simplified 0.0

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

   14. Applied add-sqr-sqrt 0.0

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

   15. Applied cbrt-prod 0.0

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

   16. Applied difference-of-squares 0.0

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

   17. Taylor expanded around inf 0.0

       \[\leadsto \frac{\left(\sqrt[3]{\sqrt{\color{blue}{\frac{64}{{\left(e^{-2 \cdot x} + 1\right)}^{6}}}}} + 1\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   if -1.1154491279793497 < (* -2.0 x) < 0.0001705058833047193

   1. Initial program 59.0

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

   2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)}\]

   if 0.0001705058833047193 < (* -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 add-cbrt-cube 0.1

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

   6. Applied add-cbrt-cube 0.1

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

   7. Applied cbrt-undiv 0.1

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

   8. Applied add-cbrt-cube 0.1

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

   9. Applied add-cbrt-cube 0.1

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

   10. Applied cbrt-undiv 0.1

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

   11. Applied cbrt-unprod 0.1

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

   12. Simplified 0.1

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

   14. Applied add-sqr-sqrt 0.1

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

   15. Applied cbrt-prod 0.1

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

   16. Applied difference-of-squares 0.1

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

   18. Applied add-cbrt-cube 0.1

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

   19. Simplified 0.1

       \[\leadsto \frac{\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} + 1\right) \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}^{3}}}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1.1154491279793497:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{\frac{64}{{\left(e^{-2 \cdot x} + 1\right)}^{6}}}} + 1\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 1.7050588330471929 \cdot 10^{-4}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} + 1\right) \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}^{3}}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))