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

Average Error: 6.3 → 0.8

Time: 1.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -7.725807749934587 \cdot 10^{96}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le -3.0807367441454416 \cdot 10^{-282}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le -0.0:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 5.31151602867187701 \cdot 10^{255}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{\frac{z}{y}}}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double temp;
	if (((x * y) <= -7.725807749934587e+96)) {
		temp = ((x / z) * y);
	} else {
		double temp_1;
		if (((x * y) <= -3.0807367441454416e-282)) {
			temp_1 = ((x * y) / z);
		} else {
			double temp_2;
			if (((x * y) <= -0.0)) {
				temp_2 = (x * (y / z));
			} else {
				double temp_3;
				if (((x * y) <= 5.311516028671877e+255)) {
					temp_3 = ((x * y) / z);
				} else {
					temp_3 = (1.0 / (1.0 / (x / (z / y))));
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	6.4
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 4 regimes

2. if (* x y) < -7.725807749934587e+96

   1. Initial program 14.6

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

   3. Applied associate-/l* 3.3

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

   5. Applied associate-/r/ 4.1

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

   if -7.725807749934587e+96 < (* x y) < -3.0807367441454416e-282 or -0.0 < (* x y) < 5.311516028671877e+255

   1. Initial program 0.4

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

   if -3.0807367441454416e-282 < (* x y) < -0.0

   1. Initial program 17.6

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

   3. Applied *-un-lft-identity 17.6

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

   4. Applied times-frac 0.1

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

   5. Simplified 0.1

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

   if 5.311516028671877e+255 < (* x y)

   1. Initial program 40.0

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

   3. Applied associate-/l* 0.3

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

   5. Applied clear-num 0.3

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

   7. Applied *-un-lft-identity 0.3

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

   8. Applied *-un-lft-identity 0.3

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

   9. Applied times-frac 0.3

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

   10. Applied associate-/l* 0.4

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -7.725807749934587 \cdot 10^{96}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le -3.0807367441454416 \cdot 10^{-282}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le -0.0:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 5.31151602867187701 \cdot 10^{255}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{\frac{z}{y}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))