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

Average Error: 6.1 → 1.4

Time: 5.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \le -2.69026972134117347 \cdot 10^{284}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le -2.91064644301977049 \cdot 10^{-253}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 5.34918334653899883 \cdot 10^{-270}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 5.89657563244339658 \cdot 10^{300}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

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) (((double) (x * y)) / z)) <= -2.6902697213411735e+284)) {
		VAR = ((double) (x / ((double) (z / y))));
	} else {
		double VAR_1;
		if ((((double) (((double) (x * y)) / z)) <= -2.9106464430197705e-253)) {
			VAR_1 = ((double) (1.0 / ((double) (z / ((double) (x * y))))));
		} else {
			double VAR_2;
			if ((((double) (((double) (x * y)) / z)) <= 5.349183346538999e-270)) {
				VAR_2 = ((double) (x * ((double) (y / z))));
			} else {
				double VAR_3;
				if ((((double) (((double) (x * y)) / z)) <= 5.896575632443397e+300)) {
					VAR_3 = ((double) (1.0 / ((double) (z / ((double) (x * y))))));
				} else {
					VAR_3 = ((double) (x / ((double) (z / 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	6.1
Target	6.1
Herbie	1.4

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

Derivation

1. Split input into 3 regimes

2. if (/ (* x y) z) < -2.6902697213411735e+284 or 5.896575632443397e+300 < (/ (* x y) z)

   1. Initial program 52.1

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

   3. Applied associate-/l* 4.0

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

   if -2.6902697213411735e+284 < (/ (* x y) z) < -2.9106464430197705e-253 or 5.349183346538999e-270 < (/ (* x y) z) < 5.896575632443397e+300

   1. Initial program 0.5

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

   3. Applied clear-num 0.6

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

   if -2.9106464430197705e-253 < (/ (* x y) z) < 5.349183346538999e-270

   1. Initial program 9.0

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

   3. Applied *-un-lft-identity 9.0

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

   4. Applied times-frac 2.6

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

   5. Simplified 2.6

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \le -2.69026972134117347 \cdot 10^{284}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le -2.91064644301977049 \cdot 10^{-253}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 5.34918334653899883 \cdot 10^{-270}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \le 5.89657563244339658 \cdot 10^{300}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(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))