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

Average Error: 6.5 → 1.2

Time: 2.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \le -3.93530838127716306 \cdot 10^{278} \lor \neg \left(\frac{x \cdot y}{z} \le -9.1878331426997 \cdot 10^{-312} \lor \neg \left(\frac{x \cdot y}{z} \le 8.6535795721578118 \cdot 10^{-242} \lor \neg \left(\frac{x \cdot y}{z} \le 1.2254646193436702 \cdot 10^{295}\right)\right)\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if (((((x * y) / z) <= -3.935308381277163e+278) || !((((x * y) / z) <= -9.1878331426997e-312) || !((((x * y) / z) <= 8.653579572157812e-242) || !(((x * y) / z) <= 1.2254646193436702e+295))))) {
		VAR = (x * (y / z));
	} else {
		VAR = ((x * y) / z);
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.5
Target	6.2
Herbie	1.2

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

Derivation

1. Split input into 2 regimes

2. if (/ (* x y) z) < -3.935308381277163e+278 or -9.1878331426997e-312 < (/ (* x y) z) < 8.653579572157812e-242 or 1.2254646193436702e+295 < (/ (* x y) z)

   1. Initial program 18.7

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

   3. Applied *-un-lft-identity 18.7

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

   4. Applied times-frac 2.7

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

   5. Simplified 2.7

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

   if -3.935308381277163e+278 < (/ (* x y) z) < -9.1878331426997e-312 or 8.653579572157812e-242 < (/ (* x y) z) < 1.2254646193436702e+295

   1. Initial program 0.5

      \[\frac{x \cdot y}{z}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \le -3.93530838127716306 \cdot 10^{278} \lor \neg \left(\frac{x \cdot y}{z} \le -9.1878331426997 \cdot 10^{-312} \lor \neg \left(\frac{x \cdot y}{z} \le 8.6535795721578118 \cdot 10^{-242} \lor \neg \left(\frac{x \cdot y}{z} \le 1.2254646193436702 \cdot 10^{295}\right)\right)\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 
(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))