Logistic function from Lakshay Garg

Average Error: 29.0 → 0.2

Time: 1.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2549.842835266359 \lor \neg \left(-2 \cdot x \leq 1.0636739907298338 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -2549.842835266359)
         (not (<= (* -2.0 x) 1.0636739907298338e-5)))
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (fma (pow x 3.0) -0.3333333333333333 x)))

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 tmp;
	if (((-2.0 * x) <= -2549.842835266359) || !((-2.0 * x) <= 1.0636739907298338e-5)) {
		tmp = (2.0 / (1.0 + exp(-2.0 * x))) - 1.0;
	} else {
		tmp = fma(pow(x, 3.0), -0.3333333333333333, x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -2549.84283526635909 or 1.0636739907298338e-5 < (*.f64 -2 x)

   1. Initial program 0.0

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

   if -2549.84283526635909 < (*.f64 -2 x) < 1.0636739907298338e-5

   1. Initial program 58.6

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

   2. Taylor expanded in x around 0 0.4

      \[\leadsto \color{blue}{x - 0.3333333333333333 \cdot {x}^{3}} \]

   3. Simplified 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2549.842835266359 \lor \neg \left(-2 \cdot x \leq 1.0636739907298338 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))