Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, B

Average Error: 25.7 → 0.5

Time: 3.2s

Precision: 64

Math

\[x \cdot \sqrt{y \cdot y - z \cdot z}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -7.9985446061757004 \cdot 10^{-268}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

C

double code(double x, double y, double z) {
	return (x * sqrt(((y * y) - (z * z))));
}

↓

double code(double x, double y, double z) {
	double temp;
	if ((y <= -7.9985446061757e-268)) {
		temp = (-1.0 * (x * y));
	} else {
		temp = (x * y);
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	25.7
Target	0.5
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y \lt 2.58160964882516951 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -7.9985446061757e-268

   1. Initial program 25.6

      \[x \cdot \sqrt{y \cdot y - z \cdot z}\]

   2. Taylor expanded around -inf 0.5

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)}\]

   if -7.9985446061757e-268 < y

   1. Initial program 25.8

      \[x \cdot \sqrt{y \cdot y - z \cdot z}\]

   2. Taylor expanded around inf 0.6

      \[\leadsto x \cdot \color{blue}{y}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.9985446061757004 \cdot 10^{-268}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, B"
  :precision binary64

  :herbie-target
  (if (< y 2.5816096488251695e-278) (- (* x y)) (* x (* (sqrt (+ y z)) (sqrt (- y z)))))

  (* x (sqrt (- (* y y) (* z z)))))