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

Average Error: 6.0 → 0.7

Time: 2.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.7044901820336683 \cdot 10^{+112}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq -1.5672693784664743 \cdot 10^{-263}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 4.935100862797952 \cdot 10^{-136}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 5.612311914307996 \cdot 10^{+255}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if ((((double) (x * y)) <= -2.7044901820336683e+112)) {
		VAR = (x / (z / y));
	} else {
		double VAR_1;
		if ((((double) (x * y)) <= -1.5672693784664743e-263)) {
			VAR_1 = (((double) (x * y)) / z);
		} else {
			double VAR_2;
			if ((((double) (x * y)) <= 4.935100862797952e-136)) {
				VAR_2 = ((double) (y * (x / z)));
			} else {
				double VAR_3;
				if ((((double) (x * y)) <= 5.612311914307996e+255)) {
					VAR_3 = ((double) (((double) (x * y)) * (1.0 / z)));
				} else {
					VAR_3 = (x / (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	6.0
Target	6.2
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (* x y) < -2.704490182033668e112 or 5.61231191430799618e255 < (* x y)

   1. Initial program 21.7

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

   3. Applied associate-/l* 2.3

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

   if -2.704490182033668e112 < (* x y) < -1.5672693784664743e-263

   1. Initial program 0.2

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

   if -1.5672693784664743e-263 < (* x y) < 4.93510086279795205e-136

   1. Initial program 9.6

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

   3. Applied clear-num 9.9

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

   5. Applied associate-/r* 1.4

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

   7. Applied div-inv 1.5

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

   8. Applied add-sqr-sqrt 1.5

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

   9. Applied times-frac 1.0

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

   10. Simplified 0.9

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

   11. Simplified 0.8

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

   if 4.93510086279795205e-136 < (* x y) < 5.61231191430799618e255

   1. Initial program 0.3

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

   3. Applied div-inv 0.3

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.7044901820336683 \cdot 10^{+112}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq -1.5672693784664743 \cdot 10^{-263}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 4.935100862797952 \cdot 10^{-136}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 5.612311914307996 \cdot 10^{+255}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))