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

Average Error: 6.3 → 1.0

Time: 3.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.0224873737954097 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -3.6601650605813124 \cdot 10^{-178}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.2067845442957687 \cdot 10^{-174}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 4.487592179574517 \cdot 10^{+209}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -2.0224873737954097e+104)
   (* y (/ x z))
   (if (<= (* x y) -3.6601650605813124e-178)
     (/ (* x y) z)
     (if (<= (* x y) 1.2067845442957687e-174)
       (/ x (/ z y))
       (if (<= (* x y) 4.487592179574517e+209) (/ (* 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) <= -2.0224873737954097e+104) {
		tmp = y * (x / z);
	} else if ((x * y) <= -3.6601650605813124e-178) {
		tmp = (x * y) / z;
	} else if ((x * y) <= 1.2067845442957687e-174) {
		tmp = x / (z / y);
	} else if ((x * y) <= 4.487592179574517e+209) {
		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	6.3
Target	6.4
Herbie	1.0

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.0224873737954097e104 or 4.4875921795745169e209 < (*.f64 x y)

   1. Initial program 18.5

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

   3. Applied associate-/l*_binary64_22529 3.7

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

   5. Applied associate-/r/_binary64_22530 3.2

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

   if -2.0224873737954097e104 < (*.f64 x y) < -3.66016506058131239e-178 or 1.20678454429576868e-174 < (*.f64 x y) < 4.4875921795745169e209

   1. Initial program 0.2

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

   if -3.66016506058131239e-178 < (*.f64 x y) < 1.20678454429576868e-174

   1. Initial program 9.2

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

   3. Applied associate-/l*_binary64_22529 1.0

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.0224873737954097 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -3.6601650605813124 \cdot 10^{-178}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.2067845442957687 \cdot 10^{-174}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 4.487592179574517 \cdot 10^{+209}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020353 
(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))