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

Average Error: 6.1 → 1.5

Time: 4.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le 2.02229087115117 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right)\\ \mathbf{elif}\;x \cdot y \le 1.72227241281623525 \cdot 10^{268}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

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)) <= 2.0222908711512e-311)) {
		VAR = ((double) (((double) (((double) (((double) cbrt(x)) * ((double) cbrt(x)))) / ((double) cbrt(z)))) * ((double) (((double) (((double) cbrt(x)) / ((double) cbrt(z)))) * ((double) (y / ((double) cbrt(z))))))));
	} else {
		double VAR_1;
		if ((((double) (x * y)) <= 1.7222724128162353e+268)) {
			VAR_1 = ((double) (((double) (x * y)) / z));
		} else {
			VAR_1 = ((double) (x * ((double) (y / z))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	6.3
Herbie	1.5

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < 2.0222908711512e-311

   1. Initial program 7.5

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

   3. Applied add-cube-cbrt 8.2

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

   4. Applied times-frac 4.9

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 5.1

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

   7. Applied times-frac 5.1

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

   8. Applied associate-*l* 2.5

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

   if 2.0222908711512e-311 < (* x y) < 1.7222724128162353e+268

   1. Initial program 0.2

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

   if 1.7222724128162353e+268 < (* x y)

   1. Initial program 45.1

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

   3. Applied *-un-lft-identity 45.1

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

   4. Applied times-frac 0.3

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

   5. Simplified 0.3

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le 2.02229087115117 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right)\\ \mathbf{elif}\;x \cdot y \le 1.72227241281623525 \cdot 10^{268}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020129 
(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))