Logistic function from Lakshay Garg

Average Error: 29.2 → 0.1

Time: 5.1s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -12.7972291612644522:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}, 1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \le 1.50750368274967835 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]

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) <= -12.797229161264452)) {
		VAR = fma((2.0 / (pow(1.0, 3.0) + pow(exp((-2.0 * x)), 3.0))), ((1.0 * 1.0) + ((exp((-2.0 * x)) * exp((-2.0 * x))) - (1.0 * exp((-2.0 * x))))), -1.0);
	} else {
		double VAR_1;
		if (((-2.0 * x) <= 1.5075036827496784e-05)) {
			VAR_1 = fma(1.0, x, -fma(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) < -12.797229161264452

   1. Initial program 0.0

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

   3. Applied flip3-+ 0.0

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

   4. Applied associate-/r/ 0.0

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

   5. Applied fma-neg 0.0

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

   if -12.797229161264452 < (* -2.0 x) < 1.5075036827496784e-05

   1. Initial program 59.1

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

   2. Taylor expanded around 0 0.1

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

   3. Simplified 0.1

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

   if 1.5075036827496784e-05 < (* -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

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

   4. Applied associate-/r* 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -12.7972291612644522:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}, 1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \le 1.50750368274967835 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))