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

Average Error: 5.8 → 1.9

Time: 3.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -3.1553374956545372 \cdot 10^{-173}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 3.30239338386023779 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le 3.6874275339874201 \cdot 10^{173}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if ((((double) (x * y)) <= -3.1553374956545372e-173)) {
		VAR = ((double) (((double) (x * y)) / z));
	} else {
		double VAR_1;
		if ((((double) (x * y)) <= 3.302393383860238e-200)) {
			VAR_1 = ((double) (((double) (x / z)) * y));
		} else {
			double VAR_2;
			if ((((double) (x * y)) <= 3.68742753398742e+173)) {
				VAR_2 = ((double) (((double) (x * y)) / z));
			} else {
				VAR_2 = ((double) (x * ((double) (y / z))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	6.3
Herbie	1.9

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < -3.1553374956545372e-173 or 3.30239338386023779e-200 < (* x y) < 3.6874275339874201e173

   1. Initial program 2.4

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

   if -3.1553374956545372e-173 < (* x y) < 3.30239338386023779e-200

   1. Initial program 9.6

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

   3. Applied associate-/l* 1.1

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]
   4. Using strategy rm

   5. Applied associate-/r/ 0.7

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

   if 3.6874275339874201e173 < (* x y)

   1. Initial program 19.3

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

   3. Applied *-un-lft-identity 19.3

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 \cdot z}}\]

   4. Applied times-frac 2.2

      \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y}{z}}\]

   5. Simplified 2.2

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -3.1553374956545372 \cdot 10^{-173}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 3.30239338386023779 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le 3.6874275339874201 \cdot 10^{173}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020164 
(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))