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

Average Error: 6.3 → 2.6

Time: 5.5s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le 7.14147297711 \cdot 10^{-314}:\\ \;\;\;\;\left(\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{z}}}\\ \mathbf{elif}\;x \cdot y \le 4.09863767648200739 \cdot 10^{175}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if ((((double) (x * y)) <= 7.1414729771084e-314)) {
		VAR = ((double) (((double) ((x / ((double) (((double) cbrt(z)) * ((double) cbrt(z))))) * (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) / ((double) cbrt(((double) (((double) cbrt(z)) * ((double) cbrt(z))))))))) * (((double) cbrt(y)) / ((double) cbrt(((double) cbrt(z)))))));
	} else {
		double VAR_1;
		if ((((double) (x * y)) <= 4.0986376764820074e+175)) {
			VAR_1 = (((double) (x * y)) / z);
		} else {
			VAR_1 = (x / (z / y));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	6.3
Herbie	2.6

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < 7.14147297711e-314

   1. Initial program 8.2

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

   3. Applied add-cube-cbrt 8.9

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

   4. Applied times-frac 5.1

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

   6. Applied add-cube-cbrt 5.2

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

   7. Applied cbrt-prod 5.2

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

   8. Applied add-cube-cbrt 5.4

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

   9. Applied times-frac 5.4

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

   10. Applied associate-*r* 4.2

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

   if 7.14147297711e-314 < (* x y) < 4.09863767648200739e175

   1. Initial program 0.2

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

   if 4.09863767648200739e175 < (* x y)

   1. Initial program 21.6

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

   3. Applied associate-/l* 1.5

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

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le 7.14147297711 \cdot 10^{-314}:\\ \;\;\;\;\left(\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{z}}}\\ \mathbf{elif}\;x \cdot y \le 4.09863767648200739 \cdot 10^{175}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))