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

Average Error: 6.2 → 0.3

Time: 2.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq -2.8178694528456616 \cdot 10^{-218}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 3.8548113336237 \cdot 10^{-317}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.9992129103769233 \cdot 10^{+239}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) (- INFINITY))
   (/ x (/ z y))
   (if (<= (* x y) -2.8178694528456616e-218)
     (/ (* x y) z)
     (if (<= (* x y) 3.8548113336237e-317)
       (* x (/ y z))
       (if (<= (* x y) 1.9992129103769233e+239)
         (/ (* x y) z)
         (/ x (/ z y)))))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if ((((double) (x * y)) <= ((double) -(((double) INFINITY))))) {
		tmp = (x / (z / y));
	} else {
		double tmp_1;
		if ((((double) (x * y)) <= -2.8178694528456616e-218)) {
			tmp_1 = (((double) (x * y)) / z);
		} else {
			double tmp_2;
			if ((((double) (x * y)) <= 3.8548113336237e-317)) {
				tmp_2 = ((double) (x * (y / z)));
			} else {
				double tmp_3;
				if ((((double) (x * y)) <= 1.9992129103769233e+239)) {
					tmp_3 = (((double) (x * y)) / z);
				} else {
					tmp_3 = (x / (z / y));
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.2
Target	6.3
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < -inf.0 or 1.9992129103769233e239 < (* x y)

   1. Initial program Error: 43.5 bits

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

   3. Applied associate-/l* Error: 0.6 bits

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

   if -inf.0 < (* x y) < -2.8178694528456616e-218 or 3.85481133e-317 < (* x y) < 1.9992129103769233e239

   1. Initial program Error: 0.3 bits

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

   if -2.8178694528456616e-218 < (* x y) < 3.85481133e-317

   1. Initial program Error: 14.7 bits

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

   2. Simplified Error: 0.4 bits

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

4. Final simplification Error: 0.3 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq -2.8178694528456616 \cdot 10^{-218}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 3.8548113336237 \cdot 10^{-317}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.9992129103769233 \cdot 10^{+239}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))