Logistic function from Lakshay Garg

Average Error: 29.5 → 0.4

Time: 5.7s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.0107417321816840241:\\ \;\;\;\;e^{\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\\ \mathbf{elif}\;-2 \cdot x \le 1.14760911155323326 \cdot 10^{-17}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{\frac{1}{e^{-2 \cdot x} + 1}} + \sqrt{1}\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}} - \sqrt{1}\right)\\ \end{array}\]

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 VAR;
	if (((-2.0 * x) <= -0.010741732181684024)) {
		VAR = exp(log(((2.0 / (1.0 + exp((-2.0 * x)))) - 1.0)));
	} else {
		double VAR_1;
		if (((-2.0 * x) <= 1.1476091115532333e-17)) {
			VAR_1 = ((1.0 * x) - ((5.551115123125783e-17 * pow(x, 4.0)) + (0.33333333333333337 * pow(x, 3.0))));
		} else {
			VAR_1 = (((sqrt(2.0) * sqrt((1.0 / (exp((-2.0 * x)) + 1.0)))) + sqrt(1.0)) * (cbrt(sqrt(pow((2.0 / (1.0 + exp((-2.0 * x)))), 3.0))) - sqrt(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) < -0.010741732181684024

   1. Initial program 0.0

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

   3. Applied add-exp-log 0.0

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

   if -0.010741732181684024 < (* -2.0 x) < 1.1476091115532333e-17

   1. Initial program 59.7

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

   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 1.1476091115532333e-17 < (* -2.0 x)

   1. Initial program 1.2

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

   3. Applied add-cbrt-cube 1.3

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

   4. Applied add-cbrt-cube 1.3

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

   5. Applied cbrt-undiv 1.3

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

   6. Simplified 1.3

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

   8. Applied add-sqr-sqrt 1.3

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

   9. Applied add-sqr-sqrt 1.3

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

   10. Applied cbrt-prod 1.3

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

   11. Applied difference-of-squares 1.3

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

   12. Taylor expanded around inf 1.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.0107417321816840241:\\ \;\;\;\;e^{\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\\ \mathbf{elif}\;-2 \cdot x \le 1.14760911155323326 \cdot 10^{-17}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{\frac{1}{e^{-2 \cdot x} + 1}} + \sqrt{1}\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}} - \sqrt{1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))