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

Average Error: 6.3 → 0.7

Time: 1.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -3.51744488014109726 \cdot 10^{187}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le -3.5445091415403063 \cdot 10^{-180}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \le 1.57690935885075214 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le 4.9136453820221977 \cdot 10^{217}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{y}}\\ \end{array}\]

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)) <= -3.517444880141097e+187)) {
		VAR = ((double) (((double) (x / z)) * y));
	} else {
		double VAR_1;
		if ((((double) (x * y)) <= -3.5445091415403063e-180)) {
			VAR_1 = ((double) (((double) (x * y)) * ((double) (1.0 / z))));
		} else {
			double VAR_2;
			if ((((double) (x * y)) <= 1.5769093588507521e-150)) {
				VAR_2 = ((double) (((double) (x / z)) * y));
			} else {
				double VAR_3;
				if ((((double) (x * y)) <= 4.913645382022198e+217)) {
					VAR_3 = ((double) (((double) (x * y)) / z));
				} else {
					VAR_3 = ((double) (((double) (x / z)) / ((double) (1.0 / y))));
				}
				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	6.4
Herbie	0.7

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

Derivation

1. Split input into 4 regimes

2. if (* x y) < -3.517444880141097e+187 or -3.5445091415403063e-180 < (* x y) < 1.5769093588507521e-150

   1. Initial program 11.9

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

   3. Applied associate-/l* 1.0

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

   5. Applied associate-/r/ 1.1

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

   if -3.517444880141097e+187 < (* x y) < -3.5445091415403063e-180

   1. Initial program 0.3

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

   3. Applied div-inv 0.3

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

   if 1.5769093588507521e-150 < (* x y) < 4.913645382022198e+217

   1. Initial program 0.2

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

   if 4.913645382022198e+217 < (* x y)

   1. Initial program 30.7

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

   3. Applied associate-/l* 1.1

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

   5. Applied div-inv 1.1

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

   6. Applied associate-/r* 1.5

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{1}{y}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -3.51744488014109726 \cdot 10^{187}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le -3.5445091415403063 \cdot 10^{-180}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \le 1.57690935885075214 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le 4.9136453820221977 \cdot 10^{217}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020126 
(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))