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

Average Error: 6.1 → 2.7

Time: 1.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \le -3.95253 \cdot 10^{-323}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 1.25608554375454006 \cdot 10^{-255}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 9.4322673509474408 \cdot 10^{246}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

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) (((double) (x * y)) / z)) <= -3.95252516673e-323)) {
		VAR = ((double) (((double) (x * y)) / z));
	} else {
		double VAR_1;
		if ((((double) (((double) (x * y)) / z)) <= 1.25608554375454e-255)) {
			VAR_1 = ((double) (((double) (x / z)) * y));
		} else {
			double VAR_2;
			if ((((double) (((double) (x * y)) / z)) <= 9.43226735094744e+246)) {
				VAR_2 = ((double) (((double) (x * y)) / z));
			} else {
				VAR_2 = ((double) (x / ((double) (z / y))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	6.3
Herbie	2.7

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

Derivation

1. Split input into 3 regimes

2. if (/ (* x y) z) < -3.95252516673e-323 or 1.25608554375454e-255 < (/ (* x y) z) < 9.43226735094744e+246

   1. Initial program 2.6

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

   if -3.95252516673e-323 < (/ (* x y) z) < 1.25608554375454e-255

   1. Initial program 10.3

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

   3. Applied associate-/l* 1.6

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

   5. Applied associate-/r/ 1.8

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

   if 9.43226735094744e+246 < (/ (* x y) z)

   1. Initial program 33.8

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

   3. Applied associate-/l* 8.5

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

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \le -3.95253 \cdot 10^{-323}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 1.25608554375454006 \cdot 10^{-255}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 9.4322673509474408 \cdot 10^{246}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 
(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))