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

Average Error: 6.3 → 0.6

Time: 1.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.26037315653932019 \cdot 10^{194}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le -5.02402572648450271 \cdot 10^{-243}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.2224174804579308 \cdot 10^{-187}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \le 1.468043545250476 \cdot 10^{146}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{1}{x}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if (((x * y) <= -1.2603731565393202e+194)) {
		VAR = (x / (z / y));
	} else {
		double VAR_1;
		if (((x * y) <= -5.024025726484503e-243)) {
			VAR_1 = ((x * y) / z);
		} else {
			double VAR_2;
			if (((x * y) <= 1.2224174804579308e-187)) {
				VAR_2 = (y / (z / x));
			} else {
				double VAR_3;
				if (((x * y) <= 1.468043545250476e+146)) {
					VAR_3 = ((x * y) / z);
				} else {
					VAR_3 = ((y / z) / (1.0 / x));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	6.3
Herbie	0.6

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

Derivation

1. Split input into 4 regimes

2. if (* x y) < -1.2603731565393202e+194

   1. Initial program 27.3

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

   3. Applied associate-/l* 1.4

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

   if -1.2603731565393202e+194 < (* x y) < -5.024025726484503e-243 or 1.2224174804579308e-187 < (* x y) < 1.468043545250476e+146

   1. Initial program 0.2

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

   if -5.024025726484503e-243 < (* x y) < 1.2224174804579308e-187

   1. Initial program 11.1

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

   3. Applied *-commutative 11.1

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

   4. Applied associate-/l* 0.6

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

   if 1.468043545250476e+146 < (* x y)

   1. Initial program 18.0

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

   3. Applied *-commutative 18.0

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

   4. Applied associate-/l* 2.8

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

   6. Applied div-inv 2.9

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

   7. Applied associate-/r* 2.6

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{\frac{1}{x}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.26037315653932019 \cdot 10^{194}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le -5.02402572648450271 \cdot 10^{-243}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.2224174804579308 \cdot 10^{-187}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \le 1.468043545250476 \cdot 10^{146}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{1}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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))