Logistic function from Lakshay Garg

Average Error: 29.8 → 0.1

Time: 3.3s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.022522952500362264:\\ \;\;\;\;{e}^{\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right)}\\ \mathbf{elif}\;-2 \cdot x \le 5.27732541853695279 \cdot 10^{-8}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]

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) <= -0.022522952500362264)) {
		VAR = pow(((double) M_E), log(((2.0 / (1.0 + exp((-2.0 * x)))) - 1.0)));
	} else {
		double VAR_1;
		if (((-2.0 * x) <= 5.277325418536953e-08)) {
			VAR_1 = ((1.0 * x) - ((5.551115123125783e-17 * pow(x, 4.0)) + (0.33333333333333337 * pow(x, 3.0))));
		} else {
			VAR_1 = (((2.0 / sqrt((1.0 + exp((-2.0 * x))))) / sqrt((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) < -0.022522952500362264

   1. Initial program 0.0

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

   3. Applied add-exp-log 0.0

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

   5. Applied pow1 0.0

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

   6. Applied log-pow 0.0

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

   7. Applied exp-prod 0.0

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

   8. Simplified 0.0

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

   if -0.022522952500362264 < (* -2.0 x) < 5.277325418536953e-08

   1. Initial program 59.4

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

   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 5.277325418536953e-08 < (* -2.0 x)

   1. Initial program 0.2

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

   3. Applied add-sqr-sqrt 0.3

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

   4. Applied associate-/r* 0.3

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.022522952500362264:\\ \;\;\;\;{e}^{\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right)}\\ \mathbf{elif}\;-2 \cdot x \le 5.27732541853695279 \cdot 10^{-8}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))