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

Average Error: 6.3 → 0.6

Time: 2.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.4024571977887486 \cdot 10^{+230}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -5.9589954090699305 \cdot 10^{-145} \lor \neg \left(x \cdot y \leq -0\right) \land x \cdot y \leq 1.7296203136662388 \cdot 10^{+181}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

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)) <= -2.4024571977887486e+230)) {
		VAR = ((double) (y * (x / z)));
	} else {
		double VAR_1;
		if (((((double) (x * y)) <= -5.9589954090699305e-145) || (!(((double) (x * y)) <= -0.0) && (((double) (x * y)) <= 1.7296203136662388e+181)))) {
			VAR_1 = ((double) (((double) (x * y)) * (1.0 / z)));
		} else {
			VAR_1 = ((double) (x * (y / z)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	6.2
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < -2.40245719778874865e230

   1. Initial program 33.3

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

   3. Applied add-cube-cbrt 33.8

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

   4. Applied times-frac 1.4

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}\]

   5. Taylor expanded around 0 33.3

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

   6. Simplified 0.6

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

   if -2.40245719778874865e230 < (* x y) < -5.9589954090699305e-145 or -0.0 < (* x y) < 1.72962031366623875e181

   1. Initial program 0.4

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

   3. Applied div-inv 0.5

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}}\]

   if -5.9589954090699305e-145 < (* x y) < -0.0 or 1.72962031366623875e181 < (* x y)

   1. Initial program 14.2

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

   2. Simplified 1.0

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.4024571977887486 \cdot 10^{+230}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -5.9589954090699305 \cdot 10^{-145} \lor \neg \left(x \cdot y \leq -0\right) \land x \cdot y \leq 1.7296203136662388 \cdot 10^{+181}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(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))