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

Average Error: 6.2 → 2.4

Time: 4.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \le -8.0087809409445095 \cdot 10^{-299}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 5.495922425828504 \cdot 10^{-288}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 7.8039931848164805 \cdot 10^{301}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

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) (((double) (x * y)) / z)) <= -8.00878094094451e-299)) {
		VAR = ((double) (((double) (x * y)) / z));
	} else {
		double VAR_1;
		if ((((double) (((double) (x * y)) / z)) <= 5.495922425828504e-288)) {
			VAR_1 = ((double) (((double) (x / z)) * y));
		} else {
			double VAR_2;
			if ((((double) (((double) (x * y)) / z)) <= 7.8039931848164805e+301)) {
				VAR_2 = ((double) (((double) (x * y)) * ((double) (1.0 / z))));
			} else {
				VAR_2 = ((double) (x / ((double) (z / y))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.2
Target	6.3
Herbie	2.4

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

Derivation

1. Split input into 4 regimes

2. if (/ (* x y) z) < -8.0087809409445095e-299

   1. Initial program 5.0

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

   if -8.0087809409445095e-299 < (/ (* x y) z) < 5.495922425828504e-288

   1. Initial program 9.5

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

   3. Applied clear-num 10.6

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

   5. Applied associate-/r* 2.2

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{z}{x}}{y}}}\]
   6. Using strategy rm

   7. Applied div-inv 2.2

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

   8. Applied add-cube-cbrt 2.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{z}{x} \cdot \frac{1}{y}}\]

   9. Applied times-frac 1.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{z}{x}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{y}}}\]

   10. Simplified 1.2

       \[\leadsto \color{blue}{\frac{x}{z}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{y}}\]

   11. Simplified 1.2

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

   if 5.495922425828504e-288 < (/ (* x y) z) < 7.8039931848164805e301

   1. Initial program 0.4

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

   3. Applied div-inv 0.5

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

   if 7.8039931848164805e301 < (/ (* x y) z)

   1. Initial program 59.9

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

   3. Applied associate-/l* 2.2

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \le -8.0087809409445095 \cdot 10^{-299}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 5.495922425828504 \cdot 10^{-288}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 7.8039931848164805 \cdot 10^{301}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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))