Logistic function from Lakshay Garg

Average Error: 30.0 → 0.4

Time: 3.4s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -714.4055638101147:\\ \;\;\;\;\frac{\left(-{1}^{3}\right) + {\left(\left(e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right) + 1 \cdot 1\right) \cdot \frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}\right)}^{3}}{\left(\left(\frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}} \cdot \frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}\right) \cdot \left(\left(e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right) + 1 \cdot 1\right) \cdot \left(e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right) + 1 \cdot 1\right)\right) + 1 \cdot \left(\left(e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right) + 1 \cdot 1\right) \cdot \frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}\right)\right) + 1 \cdot 1}\\ \mathbf{elif}\;-2 \cdot x \le 1.06724691143434056 \cdot 10^{-10}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]

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) <= -714.4055638101147)) {
		VAR = ((-pow(1.0, 3.0) + pow((((exp((-2.0 * x)) * (exp((-2.0 * x)) - 1.0)) + (1.0 * 1.0)) * (2.0 / (pow(1.0, 3.0) + pow(exp((-2.0 * x)), 3.0)))), 3.0)) / (((((2.0 / (pow(1.0, 3.0) + pow(exp((-2.0 * x)), 3.0))) * (2.0 / (pow(1.0, 3.0) + pow(exp((-2.0 * x)), 3.0)))) * (((exp((-2.0 * x)) * (exp((-2.0 * x)) - 1.0)) + (1.0 * 1.0)) * ((exp((-2.0 * x)) * (exp((-2.0 * x)) - 1.0)) + (1.0 * 1.0)))) + (1.0 * (((exp((-2.0 * x)) * (exp((-2.0 * x)) - 1.0)) + (1.0 * 1.0)) * (2.0 / (pow(1.0, 3.0) + pow(exp((-2.0 * x)), 3.0)))))) + (1.0 * 1.0)));
	} else {
		double VAR_1;
		if (((-2.0 * x) <= 1.0672469114343406e-10)) {
			VAR_1 = ((1.0 * x) - ((5.551115123125783e-17 * pow(x, 4.0)) + (0.33333333333333337 * pow(x, 3.0))));
		} else {
			VAR_1 = ((((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));
		}
		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) < -714.4055638101147

   1. Initial program 0

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

   3. Applied flip3-+ 0

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

   4. Applied associate-/r/ 0

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

   6. Applied flip3-- 0

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

   7. Simplified 0

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

   8. Simplified 0

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

   if -714.4055638101147 < (* -2.0 x) < 1.0672469114343406e-10

   1. Initial program 59.1

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

   2. Taylor expanded around 0 0.5

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

   if 1.0672469114343406e-10 < (* -2.0 x)

   1. Initial program 0.5

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

   3. Applied flip-- 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -714.4055638101147:\\ \;\;\;\;\frac{\left(-{1}^{3}\right) + {\left(\left(e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right) + 1 \cdot 1\right) \cdot \frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}\right)}^{3}}{\left(\left(\frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}} \cdot \frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}\right) \cdot \left(\left(e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right) + 1 \cdot 1\right) \cdot \left(e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right) + 1 \cdot 1\right)\right) + 1 \cdot \left(\left(e^{-2 \cdot x} \cdot \left(e^{-2 \cdot x} - 1\right) + 1 \cdot 1\right) \cdot \frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}\right)\right) + 1 \cdot 1}\\ \mathbf{elif}\;-2 \cdot x \le 1.06724691143434056 \cdot 10^{-10}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))