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

Average Error: 6.0 → 0.3

Time: 3.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.747793645483722 \cdot 10^{244}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y}}\\ \mathbf{elif}\;x \cdot y \le -7.5284985913962166 \cdot 10^{-228}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 5.91997159885211997 \cdot 10^{-264}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 3.61414911308014051 \cdot 10^{279}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if ((((double) (x * y)) <= -1.747793645483722e+244)) {
		VAR = ((double) (1.0 / ((double) (((double) (z / x)) / y))));
	} else {
		double VAR_1;
		if ((((double) (x * y)) <= -7.528498591396217e-228)) {
			VAR_1 = ((double) (((double) (x * y)) / z));
		} else {
			double VAR_2;
			if ((((double) (x * y)) <= 5.91997159885212e-264)) {
				VAR_2 = ((double) (x / ((double) (z / y))));
			} else {
				double VAR_3;
				if ((((double) (x * y)) <= 3.6141491130801405e+279)) {
					VAR_3 = ((double) (((double) (x * y)) / z));
				} else {
					VAR_3 = ((double) (x / ((double) (z / y))));
				}
				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	6.0
Target	6.2
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < -1.747793645483722e+244

   1. Initial program 35.2

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

   3. Applied clear-num 35.2

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x \cdot y}}}\]
   4. Using strategy rm

   5. Applied associate-/r* 0.7

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

   if -1.747793645483722e+244 < (* x y) < -7.528498591396217e-228 or 5.91997159885212e-264 < (* x y) < 3.6141491130801405e+279

   1. Initial program 0.2

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

   if -7.528498591396217e-228 < (* x y) < 5.91997159885212e-264 or 3.6141491130801405e+279 < (* x y)

   1. Initial program 16.7

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

   3. Applied associate-/l* 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.747793645483722 \cdot 10^{244}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y}}\\ \mathbf{elif}\;x \cdot y \le -7.5284985913962166 \cdot 10^{-228}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 5.91997159885211997 \cdot 10^{-264}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 3.61414911308014051 \cdot 10^{279}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(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))