Logistic function from Lakshay Garg

Average Error: 28.9 → 0.2

Time: 3.9s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -29.4323329991655491:\\ \;\;\;\;\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1}}\right) - 1\\ \mathbf{elif}\;-2 \cdot x \le 3.8844474649051155 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right) - 1\\ \end{array}\]

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) <= -29.43233299916555)) {
		VAR = (log(exp((2.0 / (exp((-2.0 * x)) + 1.0)))) - 1.0);
	} else {
		double VAR_1;
		if (((-2.0 * x) <= 3.8844474649051155e-06)) {
			VAR_1 = fma(1.0, x, -fma(5.551115123125783e-17, pow(x, 4.0), (0.33333333333333337 * pow(x, 3.0))));
		} else {
			VAR_1 = (expm1(log1p((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) < -29.43233299916555

   1. Initial program 0.0

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

   3. Applied add-log-exp 0.0

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

   4. Simplified 0.0

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

   if -29.43233299916555 < (* -2.0 x) < 3.8844474649051155e-06

   1. Initial program 58.8

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

   2. Taylor expanded around 0 0.3

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

   3. Simplified 0.3

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

   if 3.8844474649051155e-06 < (* -2.0 x)

   1. Initial program 0.1

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

   3. Applied expm1-log1p-u 0.1

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)} - 1\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -29.4323329991655491:\\ \;\;\;\;\log \left(e^{\frac{2}{e^{-2 \cdot x} + 1}}\right) - 1\\ \mathbf{elif}\;-2 \cdot x \le 3.8844474649051155 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(1, x, -\mathsf{fma}\left(5.55112 \cdot 10^{-17}, {x}^{4}, 0.33333333333333337 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right) - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))