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

Average Error: 6.4 → 0.4

Time: 2.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.1864768529540578 \cdot 10^{+247}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -6.403791384877416 \cdot 10^{-306} \lor \neg \left(x \cdot y \leq 6.414678060669069 \cdot 10^{-224}\right) \land x \cdot y \leq 1.9567238211991542 \cdot 10^{+162}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -1.1864768529540578e+247)
   (* x (/ y z))
   (if (or (<= (* x y) -6.403791384877416e-306)
           (and (not (<= (* x y) 6.414678060669069e-224))
                (<= (* x y) 1.9567238211991542e+162)))
     (/ (* x y) z)
     (* y (/ x z)))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -1.1864768529540578e+247) {
		tmp = x * (y / z);
	} else if (((x * y) <= -6.403791384877416e-306) || (!((x * y) <= 6.414678060669069e-224) && ((x * y) <= 1.9567238211991542e+162))) {
		tmp = (x * y) / z;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.4
Target	6.2
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.1864768529540578e247

   1. Initial program 39.3

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

   3. Applied *-un-lft-identity_binary64 39.3

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

   4. Applied times-frac_binary64 0.3

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

   5. Simplified 0.3

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

   if -1.1864768529540578e247 < (*.f64 x y) < -6.40379138487741635e-306 or 6.41467806066906876e-224 < (*.f64 x y) < 1.9567238211991542e162

   1. Initial program 0.2

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

   if -6.40379138487741635e-306 < (*.f64 x y) < 6.41467806066906876e-224 or 1.9567238211991542e162 < (*.f64 x y)

   1. Initial program 15.4

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

   3. Applied add-cube-cbrt_binary64 15.8

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

   4. Applied times-frac_binary64 1.0

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

   5. Taylor expanded around 0 15.4

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

   6. Simplified 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.1864768529540578 \cdot 10^{+247}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -6.403791384877416 \cdot 10^{-306} \lor \neg \left(x \cdot y \leq 6.414678060669069 \cdot 10^{-224}\right) \land x \cdot y \leq 1.9567238211991542 \cdot 10^{+162}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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))