Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A

Average Error: 5.9 → 0.3

Time: 1.3s

Precision: 64

Math

\[\frac{x \cdot y}{z}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -3.2208625570158223 \cdot 10^{267}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le -1.09971896141137388 \cdot 10^{-228}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 5.0223574014577535 \cdot 10^{-307}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \le 2.87915916409507102 \cdot 10^{171}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if (((x * y) <= -3.2208625570158223e+267)) {
		VAR = (x / (z / y));
	} else {
		double VAR_1;
		if (((x * y) <= -1.0997189614113739e-228)) {
			VAR_1 = ((x * y) / z);
		} else {
			double VAR_2;
			if (((x * y) <= 5.0223574014577535e-307)) {
				VAR_2 = (y / (z / x));
			} else {
				double VAR_3;
				if (((x * y) <= 2.879159164095071e+171)) {
					VAR_3 = ((x * y) / z);
				} else {
					VAR_3 = (x * (y / z));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.9
Target	6.2
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.70421306606504721 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (* x y) < -3.2208625570158223e+267

   1. Initial program 41.3

      \[\frac{x \cdot y}{z}\]
   2. Using strategy rm

   3. Applied associate-/l* 0.3

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]

   if -3.2208625570158223e+267 < (* x y) < -1.0997189614113739e-228 or 5.0223574014577535e-307 < (* x y) < 2.879159164095071e+171

   1. Initial program 0.2

      \[\frac{x \cdot y}{z}\]

   if -1.0997189614113739e-228 < (* x y) < 5.0223574014577535e-307

   1. Initial program 14.1

      \[\frac{x \cdot y}{z}\]
   2. Using strategy rm

   3. Applied *-commutative 14.1

      \[\leadsto \frac{\color{blue}{y \cdot x}}{z}\]

   4. Applied associate-/l* 0.2

      \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}}\]

   if 2.879159164095071e+171 < (* x y)

   1. Initial program 21.5

      \[\frac{x \cdot y}{z}\]
   2. Using strategy rm

   3. Applied div-inv 21.5

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}}\]
   4. Using strategy rm

   5. Applied associate-*l* 1.7

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

   6. Simplified 1.6

      \[\leadsto x \cdot \color{blue}{\frac{y}{z}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -3.2208625570158223 \cdot 10^{267}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le -1.09971896141137388 \cdot 10^{-228}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 5.0223574014577535 \cdot 10^{-307}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \le 2.87915916409507102 \cdot 10^{171}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -4.262230790519429e-138) (/ (* x y) z) (if (< z 1.7042130660650472e-164) (/ x (/ z y)) (* (/ x z) y)))

  (/ (* x y) z))