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

Average Error: 6.5 → 0.8

Time: 2.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -2.2129676655986834 \cdot 10^{108}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \le -9.1676459104440448 \cdot 10^{-181}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 7.03767637656364 \cdot 10^{-242}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \le 1.60210316985900323 \cdot 10^{190}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

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)) <= -2.2129676655986834e+108)) {
		VAR = ((double) (y / ((double) (z / x))));
	} else {
		double VAR_1;
		if ((((double) (x * y)) <= -9.167645910444045e-181)) {
			VAR_1 = ((double) (((double) (x * y)) / z));
		} else {
			double VAR_2;
			if ((((double) (x * y)) <= 7.037676376563638e-242)) {
				VAR_2 = ((double) (y / ((double) (z / x))));
			} else {
				double VAR_3;
				if ((((double) (x * y)) <= 1.6021031698590032e+190)) {
					VAR_3 = ((double) (((double) (x * y)) / z));
				} else {
					VAR_3 = ((double) (x * ((double) (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	6.5
Target	6.3
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < -2.2129676655986834e+108 or -9.167645910444045e-181 < (* x y) < 7.037676376563638e-242

   1. Initial program 12.4

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

   3. Applied clear-num 12.6

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x \cdot y}}}\]
   4. Using strategy rm

   5. Applied associate-/r* 2.0

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

   6. Taylor expanded around 0 12.4

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

   7. Simplified 1.5

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

   if -2.2129676655986834e+108 < (* x y) < -9.167645910444045e-181 or 7.037676376563638e-242 < (* x y) < 1.6021031698590032e+190

   1. Initial program 0.2

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

   if 1.6021031698590032e+190 < (* x y)

   1. Initial program 25.5

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

   3. Applied *-un-lft-identity 25.5

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

   4. Applied times-frac 1.7

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

   5. Simplified 1.7

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -2.2129676655986834 \cdot 10^{108}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \le -9.1676459104440448 \cdot 10^{-181}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 7.03767637656364 \cdot 10^{-242}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \le 1.60210316985900323 \cdot 10^{190}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020126 
(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))