Logistic function from Lakshay Garg

Average Error: 29.7 → 0.1

Time: 5.0s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -12.329649134843827:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 5.30640056390034621 \cdot 10^{-6}:\\ \;\;\;\;1 \cdot x - \left(4.996 \cdot 10^{-16} \cdot {x}^{4} + 0.33333333333333348 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}{\left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \left(\frac{1}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \left(\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) + 1 \cdot 1\right)}\\ \end{array}\]

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 temp;
	if (((-2.0 * x) <= -12.329649134843827)) {
		temp = ((((2.0 / (1.0 + exp((-2.0 * x)))) * (2.0 / (1.0 + exp((-2.0 * x))))) - (1.0 * 1.0)) / ((2.0 / (1.0 + exp((-2.0 * x)))) + 1.0));
	} else {
		double temp_1;
		if (((-2.0 * x) <= 5.306400563900346e-06)) {
			temp_1 = ((1.0 * x) - ((4.996003610813204e-16 * pow(x, 4.0)) + (0.3333333333333335 * pow(x, 3.0))));
		} else {
			temp_1 = (((((2.0 / (1.0 + exp((-2.0 * x)))) * (2.0 / (1.0 + exp((-2.0 * x))))) * ((2.0 / (1.0 + exp((-2.0 * x)))) * (2.0 / (1.0 + exp((-2.0 * x)))))) - ((1.0 * 1.0) * (1.0 * 1.0))) / (((2.0 / (1.0 + exp((-2.0 * x)))) + 1.0) * (((1.0 / (cbrt((1.0 + exp((-2.0 * x)))) * cbrt((1.0 + exp((-2.0 * x)))))) * ((2.0 / cbrt((1.0 + exp((-2.0 * x))))) * (2.0 / (1.0 + exp((-2.0 * x)))))) + (1.0 * 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) < -12.329649134843827

   1. Initial program 0.0

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

   3. Applied flip-- 0.0

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

   if -12.329649134843827 < (* -2.0 x) < 5.306400563900346e-06

   1. Initial program 59.0

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

   3. Applied flip-- 59.0

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

   4. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{1 \cdot x - \left(4.996 \cdot 10^{-16} \cdot {x}^{4} + 0.33333333333333348 \cdot {x}^{3}\right)}\]

   if 5.306400563900346e-06 < (* -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 flip-- 0.1

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

   6. Applied associate-/l/ 0.1

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

   8. Applied add-cube-cbrt 0.1

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

   9. Applied *-un-lft-identity 0.1

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

   10. Applied times-frac 0.1

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

   11. Applied associate-*l* 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -12.329649134843827:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 5.30640056390034621 \cdot 10^{-6}:\\ \;\;\;\;1 \cdot x - \left(4.996 \cdot 10^{-16} \cdot {x}^{4} + 0.33333333333333348 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) - \left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)}{\left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \left(\frac{1}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \left(\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) + 1 \cdot 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))