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

Average Error: 6.0 → 2.4

Time: 2.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -4.3132305778902384 \cdot 10^{-122}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.7182579779497088 \cdot 10^{-201}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 3.6065555586875704 \cdot 10^{+160}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) z))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -4.3132305778902384e-122)
   (* (* x y) (/ 1.0 z))
   (if (<= (* x y) 1.7182579779497088e-201)
     (* y (/ x z))
     (if (<= (* x y) 3.6065555586875704e+160)
       (* (* x y) (/ 1.0 z))
       (* x (/ y z))))))

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)) <= -4.3132305778902384e-122)) {
		VAR = ((double) (((double) (x * y)) * (1.0 / z)));
	} else {
		double VAR_1;
		if ((((double) (x * y)) <= 1.7182579779497088e-201)) {
			VAR_1 = ((double) (y * (x / z)));
		} else {
			double VAR_2;
			if ((((double) (x * y)) <= 3.6065555586875704e+160)) {
				VAR_2 = ((double) (((double) (x * y)) * (1.0 / z)));
			} else {
				VAR_2 = ((double) (x * (y / z)));
			}
			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	2.4

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < -4.31323057789023836e-122 or 1.7182579779497088e-201 < (* x y) < 3.6065555586875704e160

   1. Initial program 2.9

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

   3. Applied div-inv 3.0

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

   if -4.31323057789023836e-122 < (* x y) < 1.7182579779497088e-201

   1. Initial program 8.1

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

   3. Applied add-cube-cbrt 8.6

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

   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 8.1

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

   6. Simplified 1.4

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

   if 3.6065555586875704e160 < (* x y)

   1. Initial program 20.7

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

   2. Simplified 2.2

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4.3132305778902384 \cdot 10^{-122}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.7182579779497088 \cdot 10^{-201}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 3.6065555586875704 \cdot 10^{+160}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(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))