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

Average Error: 6.4 → 1.2

Time: 4.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} = -\infty:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le -1.6297677216893168 \cdot 10^{-306}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double temp;
	if ((((x * y) / z) <= -inf.0)) {
		temp = ((x / z) * y);
	} else {
		double temp_1;
		if ((((x * y) / z) <= -1.6297677216893168e-306)) {
			temp_1 = ((x * y) / z);
		} else {
			temp_1 = (((cbrt(y) * cbrt(y)) / (cbrt(z) * cbrt(z))) * (x / (cbrt(z) / cbrt(y))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.4
Target	6.3
Herbie	1.2

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

Derivation

1. Split input into 3 regimes

2. if (/ (* x y) z) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-/l* 0.3

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

   5. Applied associate-/r/ 0.2

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

   if -inf.0 < (/ (* x y) z) < -1.6297677216893168e-306

   1. Initial program 0.6

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

   if -1.6297677216893168e-306 < (/ (* x y) z)

   1. Initial program 7.4

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

   3. Applied associate-/l* 5.2

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

   5. Applied add-cube-cbrt 5.8

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

   6. Applied add-cube-cbrt 6.0

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

   7. Applied times-frac 6.0

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

   8. Applied *-un-lft-identity 6.0

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

   9. Applied times-frac 1.7

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

   10. Simplified 1.6

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} = -\infty:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le -1.6297677216893168 \cdot 10^{-306}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(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))