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

Average Error: 6.4 → 0.9

Time: 1.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.38237468439397263 \cdot 10^{254}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{y}}\\ \mathbf{elif}\;x \cdot y \le -1.23451703565074252 \cdot 10^{-158}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 4.9613720119372489 \cdot 10^{-181}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le 1.2503160602535021 \cdot 10^{109}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

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) <= -1.3823746843939726e+254)) {
		VAR = ((x / z) / (1.0 / y));
	} else {
		double VAR_1;
		if (((x * y) <= -1.2345170356507425e-158)) {
			VAR_1 = ((x * y) / z);
		} else {
			double VAR_2;
			if (((x * y) <= 4.961372011937249e-181)) {
				VAR_2 = ((x / z) * y);
			} else {
				double VAR_3;
				if (((x * y) <= 1.250316060253502e+109)) {
					VAR_3 = ((x * y) / z);
				} else {
					VAR_3 = (x * (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	6.4
Target	6.3
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) < -1.3823746843939726e+254

   1. Initial program 40.5

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

   3. Applied associate-/l* 0.2

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

   5. Applied div-inv 0.3

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

   6. Applied associate-/r* 0.6

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

   if -1.3823746843939726e+254 < (* x y) < -1.2345170356507425e-158 or 4.961372011937249e-181 < (* x y) < 1.250316060253502e+109

   1. Initial program 0.2

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

   if -1.2345170356507425e-158 < (* x y) < 4.961372011937249e-181

   1. Initial program 9.5

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

   3. Applied associate-/l* 1.2

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

   5. Applied associate-/r/ 1.0

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

   if 1.250316060253502e+109 < (* x y)

   1. Initial program 15.0

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

   3. Applied *-un-lft-identity 15.0

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

   4. Applied times-frac 3.5

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

   5. Simplified 3.5

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.38237468439397263 \cdot 10^{254}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{1}{y}}\\ \mathbf{elif}\;x \cdot y \le -1.23451703565074252 \cdot 10^{-158}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 4.9613720119372489 \cdot 10^{-181}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le 1.2503160602535021 \cdot 10^{109}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 +o rules:numerics
(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))