Logistic function from Lakshay Garg

Average Error: 29.3 → 0.4

Time: 4.4s

Precision: binary64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -14820470.435843565:\\ \;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 4.51305363822253686 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{\sqrt{1 + e^{-2 \cdot x}}} \cdot \sqrt[3]{\sqrt{1 + e^{-2 \cdot x}}}}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)\\ \end{array}\]

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)) <= -14820470.435843565)) {
		VAR = ((double) (((double) (((double) (2.0 / ((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))) / ((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))) - 1.0));
	} else {
		double VAR_1;
		if ((((double) (-2.0 * x)) <= 4.513053638222537e-05)) {
			VAR_1 = ((double) (((double) (1.0 * x)) - ((double) (((double) (5.551115123125783e-17 * ((double) pow(x, 4.0)))) + ((double) (0.33333333333333337 * ((double) pow(x, 3.0))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) sqrt(((double) (((double) (2.0 / ((double) (((double) cbrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))) * ((double) cbrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))))) / ((double) (((double) cbrt(((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))) * ((double) cbrt(((double) sqrt(((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))))))))) + ((double) sqrt(1.0)))) * ((double) (((double) sqrt(((double) (2.0 / ((double) (1.0 + ((double) exp(((double) (-2.0 * x)))))))))) - ((double) 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) < -14820470.435843565

   1. Initial program 0

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

   3. Applied add-sqr-sqrt 0

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

   4. Applied associate-/r* 0

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

   if -14820470.435843565 < (* -2.0 x) < 4.51305363822253686e-5

   1. Initial program 58.4

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

   2. Taylor expanded around 0 0.7

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

   if 4.51305363822253686e-5 < (* -2.0 x)

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt 0.1

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

   4. Applied add-sqr-sqrt 0.1

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

   5. Applied difference-of-squares 0.1

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

   7. Applied add-cube-cbrt 0.1

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

   8. Applied associate-/r* 0.1

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

   10. Applied add-sqr-sqrt 0.1

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

   11. Applied cbrt-prod 0.1

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -14820470.435843565:\\ \;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 4.51305363822253686 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{2}{\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}}}{\sqrt[3]{\sqrt{1 + e^{-2 \cdot x}}} \cdot \sqrt[3]{\sqrt{1 + e^{-2 \cdot x}}}}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020164 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))