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

Average Error: 6.3 → 0.5

Time: 3.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -2.946526206817621 \cdot 10^{283}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -4.1421979300826392 \cdot 10^{-146}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \le 8.693023314594202 \cdot 10^{-292}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 2.3461036638854905 \cdot 10^{236}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

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)) <= -2.9465262068176215e+283)) {
		VAR = ((double) (x * ((double) (y / z))));
	} else {
		double VAR_1;
		if ((((double) (x * y)) <= -4.142197930082639e-146)) {
			VAR_1 = ((double) (((double) (x * y)) * ((double) (1.0 / z))));
		} else {
			double VAR_2;
			if ((((double) (x * y)) <= 8.693023314594202e-292)) {
				VAR_2 = ((double) (x / ((double) (z / y))));
			} else {
				double VAR_3;
				if ((((double) (x * y)) <= 2.3461036638854905e+236)) {
					VAR_3 = ((double) (((double) (x * y)) * ((double) (1.0 / z))));
				} else {
					VAR_3 = ((double) (x * ((double) (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	6.3
Target	6.2
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < -2.946526206817621e283 or 2.3461036638854905e236 < (* x y)

   1. Initial program 40.8

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

   3. Applied *-un-lft-identity 40.8

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

   4. Applied times-frac 0.5

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

   5. Simplified 0.5

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

   if -2.946526206817621e283 < (* x y) < -4.1421979300826392e-146 or 8.693023314594202e-292 < (* x y) < 2.3461036638854905e236

   1. Initial program 0.2

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

   3. Applied div-inv 0.3

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

   if -4.1421979300826392e-146 < (* x y) < 8.693023314594202e-292

   1. Initial program 11.0

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

   3. Applied associate-/l* 0.9

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -2.946526206817621 \cdot 10^{283}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -4.1421979300826392 \cdot 10^{-146}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \le 8.693023314594202 \cdot 10^{-292}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 2.3461036638854905 \cdot 10^{236}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(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))