Logistic function from Lakshay Garg

Average Error: 29.6 → 0.5

Time: 4.5s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.01756904497623861:\\ \;\;\;\;\frac{\left(-1 \cdot 1\right) + \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot 2}{1 + e^{-2 \cdot x}}}{\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 2.50044575718993887 \cdot 10^{-18}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\left(-1 \cdot 1\right) + \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot 2}{1 + e^{-2 \cdot x}}} \cdot \sqrt[3]{\left(-1 \cdot 1\right) + \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot 2}{1 + e^{-2 \cdot x}}}\right) \cdot \sqrt[3]{\left(-1 \cdot 1\right) + \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot 2}{1 + e^{-2 \cdot x}}}}{\frac{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} + 1}\\ \end{array}\]

C

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)) <= -0.01756904497623861)) {
		VAR = ((double) (((double) (((double) -(((double) (1.0 * 1.0)))) + ((double) (((double) (((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) * 2.0)) / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))) / ((double) (((double) (((double) (2.0 / ((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))) / ((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))) + 1.0))));
	} else {
		double VAR_1;
		if ((((double) (-2.0 * x)) <= 2.500445757189939e-18)) {
			VAR_1 = ((double) (((double) (1.0 * x)) - ((double) (((double) (5.551115123125783e-17 * ((double) pow(x, 4.0)))) + ((double) (0.33333333333333337 * ((double) pow(x, 3.0))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) cbrt(((double) (((double) -(((double) (1.0 * 1.0)))) + ((double) (((double) (((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) * 2.0)) / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))))) * ((double) cbrt(((double) (((double) -(((double) (1.0 * 1.0)))) + ((double) (((double) (((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) * 2.0)) / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))))))) * ((double) cbrt(((double) (((double) -(((double) (1.0 * 1.0)))) + ((double) (((double) (((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) * 2.0)) / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))))))) / ((double) (((double) (((double) (((double) (((double) cbrt(2.0)) * ((double) cbrt(2.0)))) * ((double) (((double) cbrt(2.0)) / ((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))))) / ((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))) + 1.0))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 3 regimes

2. if (* -2.0 x) < -0.01756904497623861

   1. Initial program 0.0

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

   3. Applied add-sqr-sqrt 0.0

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

   4. Applied associate-/r* 0.0

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

   6. Applied flip-- 0.0

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

   7. Simplified 0.0

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

   if -0.01756904497623861 < (* -2.0 x) < 2.500445757189939e-18

   1. Initial program 59.8

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

   2. Taylor expanded around 0 0.0

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

   if 2.500445757189939e-18 < (* -2.0 x)

   1. Initial program 1.7

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

   3. Applied add-sqr-sqrt 1.7

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

   4. Applied associate-/r* 1.7

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

   6. Applied flip-- 1.7

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

   7. Simplified 1.8

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

   9. Applied add-cube-cbrt 1.8

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

   11. Applied *-un-lft-identity 1.8

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

   12. Applied sqrt-prod 1.8

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

   13. Applied add-cube-cbrt 1.8

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

   14. Applied times-frac 1.8

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

   15. Simplified 1.8

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.01756904497623861:\\ \;\;\;\;\frac{\left(-1 \cdot 1\right) + \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot 2}{1 + e^{-2 \cdot x}}}{\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 2.50044575718993887 \cdot 10^{-18}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\left(-1 \cdot 1\right) + \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot 2}{1 + e^{-2 \cdot x}}} \cdot \sqrt[3]{\left(-1 \cdot 1\right) + \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot 2}{1 + e^{-2 \cdot x}}}\right) \cdot \sqrt[3]{\left(-1 \cdot 1\right) + \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot 2}{1 + e^{-2 \cdot x}}}}{\frac{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \frac{\sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))