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

Average Error: 6.1 → 0.9

Time: 2.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.05754127488798247 \cdot 10^{197}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le -2.2367107212282394 \cdot 10^{-172}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.011892524644406 \cdot 10^{-194}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 9.3742492436314431 \cdot 10^{91}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]

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) (x * y)) <= -1.0575412748879825e+197)) {
		VAR = ((double) (x / ((double) (z / y))));
	} else {
		double VAR_1;
		if ((((double) (x * y)) <= -2.2367107212282394e-172)) {
			VAR_1 = ((double) (((double) (x * y)) / z));
		} else {
			double VAR_2;
			if ((((double) (x * y)) <= 1.011892524644406e-194)) {
				VAR_2 = ((double) (x * ((double) (y / z))));
			} else {
				double VAR_3;
				if ((((double) (x * y)) <= 9.374249243631443e+91)) {
					VAR_3 = ((double) (((double) (x * y)) / z));
				} else {
					VAR_3 = ((double) (y * ((double) (x / 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	6.1
Target	6.1
Herbie	0.9

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

Derivation

1. Split input into 4 regimes

2. if (* x y) < -1.05754127488798247e197

   1. Initial program 24.3

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

   3. Applied associate-/l* 0.9

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

   if -1.05754127488798247e197 < (* x y) < -2.2367107212282394e-172 or 1.011892524644406e-194 < (* x y) < 9.3742492436314431e91

   1. Initial program 0.2

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

   if -2.2367107212282394e-172 < (* x y) < 1.011892524644406e-194

   1. Initial program 9.6

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

   2. Simplified 0.9

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

   if 9.3742492436314431e91 < (* x y)

   1. Initial program 13.1

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

   3. Applied add-cube-cbrt 13.9

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

   4. Applied times-frac 3.7

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}\]

   5. Taylor expanded around 0 13.1

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

   6. Simplified 4.2

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.05754127488798247 \cdot 10^{197}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le -2.2367107212282394 \cdot 10^{-172}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.011892524644406 \cdot 10^{-194}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 9.3742492436314431 \cdot 10^{91}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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))