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

Average Error: 6.4 → 0.5

Time: 2.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.3223259751325834 \cdot 10^{+220}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -5.368407950621961 \cdot 10^{-169}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5.4838713756939765 \cdot 10^{-233} \lor \neg \left(x \cdot y \leq 1.413306160411407 \cdot 10^{+170}\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if ((((double) (x * y)) <= -1.3223259751325834e+220)) {
		VAR = ((double) (y * (x / z)));
	} else {
		double VAR_1;
		if ((((double) (x * y)) <= -5.368407950621961e-169)) {
			VAR_1 = (((double) (x * y)) / z);
		} else {
			double VAR_2;
			if (((((double) (x * y)) <= 5.4838713756939765e-233) || !(((double) (x * y)) <= 1.413306160411407e+170))) {
				VAR_2 = ((double) (x * (y / z)));
			} else {
				VAR_2 = ((double) (((double) (x * y)) * (1.0 / z)));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.4
Target	6.0
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (* x y) < -1.32232597513258336e220

   1. Initial program 31.4

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

   2. Simplified 0.8

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

   4. Applied add-cube-cbrt 1.9

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

   5. Applied associate-*l* 1.9

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

   6. Simplified 1.5

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

   7. Taylor expanded around 0 31.4

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

   8. Simplified 0.8

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

   if -1.32232597513258336e220 < (* x y) < -5.368407950621961e-169

   1. Initial program 0.2

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

   if -5.368407950621961e-169 < (* x y) < 5.4838713756939765e-233 or 1.413306160411407e170 < (* x y)

   1. Initial program 13.1

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

   2. Simplified 0.8

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

   if 5.4838713756939765e-233 < (* x y) < 1.413306160411407e170

   1. Initial program 0.2

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

   2. Simplified 9.3

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

   4. Applied div-inv 9.4

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{1}{z}\right)}\]

   5. Applied associate-*r* 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.3223259751325834 \cdot 10^{+220}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -5.368407950621961 \cdot 10^{-169}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 5.4838713756939765 \cdot 10^{-233} \lor \neg \left(x \cdot y \leq 1.413306160411407 \cdot 10^{+170}\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))