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

Average Error: 6.1 → 0.4

Time: 2.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.10921153159549348 \cdot 10^{268} \lor \neg \left(x \cdot y \le -4.26705345296296743 \cdot 10^{-203} \lor \neg \left(x \cdot y \le 3.8494643137002 \cdot 10^{-312} \lor \neg \left(x \cdot y \le 3.19422899203229664 \cdot 10^{178}\right)\right)\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

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)) <= -1.1092115315954935e+268) || !((((double) (x * y)) <= -4.2670534529629674e-203) || !((((double) (x * y)) <= 3.8494643137002e-312) || !(((double) (x * y)) <= 3.1942289920322966e+178))))) {
		VAR = ((double) (((double) (x / z)) * y));
	} else {
		VAR = ((double) (((double) (x * y)) / z));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	6.1
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (* x y) < -1.1092115315954935e+268 or -4.2670534529629674e-203 < (* x y) < 3.8494643137002e-312 or 3.1942289920322966e+178 < (* x y)

   1. Initial program 19.1

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

   3. Applied associate-/l* 0.7

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

   5. Applied associate-/r/ 0.6

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

   if -1.1092115315954935e+268 < (* x y) < -4.2670534529629674e-203 or 3.8494643137002e-312 < (* x y) < 3.1942289920322966e+178

   1. Initial program 0.2

      \[\frac{x \cdot y}{z}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.10921153159549348 \cdot 10^{268} \lor \neg \left(x \cdot y \le -4.26705345296296743 \cdot 10^{-203} \lor \neg \left(x \cdot y \le 3.8494643137002 \cdot 10^{-312} \lor \neg \left(x \cdot y \le 3.19422899203229664 \cdot 10^{178}\right)\right)\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020124 
(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))