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

Average Error: 6.2 → 0.3

Time: 1.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -4.3017054784378062 \cdot 10^{233}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le -5.8770184402377573 \cdot 10^{-273}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 7.90505 \cdot 10^{-323}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 2.68053028644473876 \cdot 10^{263}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y}}\\ \end{array}\]

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)) <= -4.301705478437806e+233)) {
		VAR = ((double) (x / ((double) (z / y))));
	} else {
		double VAR_1;
		if ((((double) (x * y)) <= -5.877018440237757e-273)) {
			VAR_1 = ((double) (((double) (x * y)) / z));
		} else {
			double VAR_2;
			if ((((double) (x * y)) <= 7.9050503334599e-323)) {
				VAR_2 = ((double) (x * ((double) (y / z))));
			} else {
				double VAR_3;
				if ((((double) (x * y)) <= 2.680530286444739e+263)) {
					VAR_3 = ((double) (((double) (x * y)) / z));
				} else {
					VAR_3 = ((double) (1.0 / ((double) (((double) (z / x)) / 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	6.2
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 4 regimes

2. if (* x y) < -4.301705478437806e+233

   1. Initial program 32.2

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

   3. Applied associate-/l* 0.9

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

   if -4.301705478437806e+233 < (* x y) < -5.877018440237757e-273 or 7.9050503334599e-323 < (* x y) < 2.680530286444739e+263

   1. Initial program 0.3

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

   if -5.877018440237757e-273 < (* x y) < 7.9050503334599e-323

   1. Initial program 16.8

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

   3. Applied *-un-lft-identity 16.8

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

   4. Applied times-frac 0.1

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

   5. Simplified 0.1

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

   if 2.680530286444739e+263 < (* x y)

   1. Initial program 42.0

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

   3. Applied clear-num 42.1

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

   5. Applied associate-/r* 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -4.3017054784378062 \cdot 10^{233}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le -5.8770184402377573 \cdot 10^{-273}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 7.90505 \cdot 10^{-323}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 2.68053028644473876 \cdot 10^{263}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(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))