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

Average Error: 5.9 → 0.7

Time: 3.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} = -inf.0:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le -6.808467034095899 \cdot 10^{-309}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 1.933499529089061 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 6.41496115728645654 \cdot 10^{271}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

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)) <= -inf.0)) {
		VAR = ((double) (x * ((double) (y / z))));
	} else {
		double VAR_1;
		if ((((double) (((double) (x * y)) / z)) <= -6.8084670340959e-309)) {
			VAR_1 = ((double) (((double) (x * y)) / z));
		} else {
			double VAR_2;
			if ((((double) (((double) (x * y)) / z)) <= 1.93349952908906e-310)) {
				VAR_2 = ((double) (x / ((double) (z / y))));
			} else {
				double VAR_3;
				if ((((double) (((double) (x * y)) / z)) <= 6.4149611572864565e+271)) {
					VAR_3 = ((double) (((double) (x * y)) / z));
				} else {
					VAR_3 = ((double) (x * ((double) (y / z))));
				}
				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	5.9
Target	5.9
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (/ (* x y) z) < -inf.0 or 6.41496115728645654e271 < (/ (* x y) z)

   1. Initial program 49.7

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

   3. Applied *-un-lft-identity 49.7

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

   4. Applied times-frac 4.7

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

   5. Simplified 4.7

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

   if -inf.0 < (/ (* x y) z) < -6.808467034095899e-309 or 1.933499529089061e-310 < (/ (* x y) z) < 6.41496115728645654e271

   1. Initial program 0.5

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

   if -6.808467034095899e-309 < (/ (* x y) z) < 1.933499529089061e-310

   1. Initial program 10.4

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

   3. Applied associate-/l* 0.4

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} = -inf.0:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le -6.808467034095899 \cdot 10^{-309}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 1.933499529089061 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 6.41496115728645654 \cdot 10^{271}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020168 
(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))