Logistic function from Lakshay Garg

Average Error: 29.7 → 0.3

Time: 5.0s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -246925.840479128616:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {\left(1 \cdot 1\right)}^{3}}{\left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{4} + \left(\left(2 \cdot 2\right) \cdot \frac{1}{{\left(e^{-2 \cdot x} + 1\right)}^{2}}\right) \cdot \left(1 \cdot 1\right)\right) + {1}^{4}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)}\\ \mathbf{elif}\;-2 \cdot x \le 1.62448898541434414 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {\left(1 \cdot 1\right)}^{3}}{\left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{4} + \left(\left(2 \cdot 2\right) \cdot \frac{1}{{\left(e^{-2 \cdot x} + 1\right)}^{2}}\right) \cdot \left(1 \cdot 1\right)\right) + {1}^{4}\right) \cdot \mathsf{fma}\left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}}, \frac{\sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}}, 1\right)}\\ \end{array}\]

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)) <= -246925.84047912862)) {
		VAR = ((double) (((double) (((double) pow(((double) (((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) * ((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))), 3.0)) - ((double) pow(((double) (1.0 * 1.0)), 3.0)))) / ((double) (((double) (((double) (((double) pow(((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))), 4.0)) + ((double) (((double) (((double) (2.0 * 2.0)) * ((double) (1.0 / ((double) pow(((double) (((double) exp(((double) (-2.0 * x)))) + 1.0)), 2.0)))))) * ((double) (1.0 * 1.0)))))) + ((double) pow(1.0, 4.0)))) * ((double) (((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) + 1.0))))));
	} else {
		double VAR_1;
		if ((((double) (-2.0 * x)) <= 0.00016244889854143441)) {
			VAR_1 = ((double) fma(1.0, x, ((double) -(((double) fma(5.551115123125783e-17, ((double) pow(x, 4.0)), ((double) (0.33333333333333337 * ((double) pow(x, 3.0))))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) pow(((double) (((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) * ((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))), 3.0)) - ((double) pow(((double) (1.0 * 1.0)), 3.0)))) / ((double) (((double) (((double) (((double) pow(((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))), 4.0)) + ((double) (((double) (((double) (2.0 * 2.0)) * ((double) (1.0 / ((double) pow(((double) (((double) exp(((double) (-2.0 * x)))) + 1.0)), 2.0)))))) * ((double) (1.0 * 1.0)))))) + ((double) pow(1.0, 4.0)))) * ((double) fma(((double) (((double) (((double) cbrt(2.0)) * ((double) cbrt(2.0)))) / ((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))), ((double) (((double) cbrt(2.0)) / ((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) < -246925.84047912862

   1. Initial program 0

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

   3. Applied flip-- 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 flip3-- 0

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

   6. Applied associate-/l/ 0

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

   7. Simplified 0

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

   if -246925.84047912862 < (* -2.0 x) < 0.00016244889854143441

   1. Initial program 58.5

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

   2. Taylor expanded around 0 0.6

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

   3. Simplified 0.6

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

   if 0.00016244889854143441 < (* -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 flip3-- 0.1

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

   6. Applied associate-/l/ 0.1

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

   7. Simplified 0.1

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

   9. Applied add-sqr-sqrt 0.1

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

   10. Applied add-cube-cbrt 0.1

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

   11. Applied times-frac 0.1

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

   12. Applied fma-def 0.1

       \[\leadsto \frac{{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {\left(1 \cdot 1\right)}^{3}}{\left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{4} + \left(\left(2 \cdot 2\right) \cdot \frac{1}{{\left(e^{-2 \cdot x} + 1\right)}^{2}}\right) \cdot \left(1 \cdot 1\right)\right) + {1}^{4}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}}, \frac{\sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}}, 1\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -246925.840479128616:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {\left(1 \cdot 1\right)}^{3}}{\left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{4} + \left(\left(2 \cdot 2\right) \cdot \frac{1}{{\left(e^{-2 \cdot x} + 1\right)}^{2}}\right) \cdot \left(1 \cdot 1\right)\right) + {1}^{4}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)}\\ \mathbf{elif}\;-2 \cdot x \le 1.62448898541434414 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {\left(1 \cdot 1\right)}^{3}}{\left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{4} + \left(\left(2 \cdot 2\right) \cdot \frac{1}{{\left(e^{-2 \cdot x} + 1\right)}^{2}}\right) \cdot \left(1 \cdot 1\right)\right) + {1}^{4}\right) \cdot \mathsf{fma}\left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}}, \frac{\sqrt[3]{2}}{\sqrt{1 + e^{-2 \cdot x}}}, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))