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

Average Error: 6.3 → 1.8

Time: 3.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le 3.4794371166802771 \cdot 10^{-260}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\\ \mathbf{elif}\;x \cdot y \le 1.24317524122921827 \cdot 10^{103}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

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) (x * y)) <= 3.479437116680277e-260)) {
		VAR = ((double) (((double) (x * ((double) (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) / ((double) (((double) cbrt(z)) * ((double) cbrt(z)))))))) * ((double) (((double) cbrt(y)) / ((double) cbrt(z))))));
	} else {
		double VAR_1;
		if ((((double) (x * y)) <= 1.2431752412292183e+103)) {
			VAR_1 = ((double) (1.0 / ((double) (z / ((double) (x * y))))));
		} else {
			VAR_1 = ((double) (x / ((double) (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	6.0
Herbie	1.8

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < 3.479437116680277e-260

   1. Initial program 7.5

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

   3. Applied *-un-lft-identity 7.5

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

   4. Applied times-frac 5.1

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

   5. Simplified 5.1

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

   7. Applied add-cube-cbrt 5.9

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

   8. Applied add-cube-cbrt 6.0

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

   9. Applied times-frac 6.0

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

   10. Applied associate-*r* 1.9

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

   if 3.479437116680277e-260 < (* x y) < 1.2431752412292183e+103

   1. Initial program 0.2

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

   3. Applied clear-num 0.8

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

   if 1.2431752412292183e+103 < (* x y)

   1. Initial program 14.3

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

   3. Applied associate-/l* 3.5

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le 3.4794371166802771 \cdot 10^{-260}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\\ \mathbf{elif}\;x \cdot y \le 1.24317524122921827 \cdot 10^{103}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 
(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))