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

Average Error: 6.1 → 1.1

Time: 2.9s

Precision: binary64

\[[x, y]=\mathsf{sort}([x, y])\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq -2.9952545328734584 \cdot 10^{-293}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 5.188044960065 \cdot 10^{-299}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 3.103356213688716 \cdot 10^{+251}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* x y) z) (- INFINITY))
   (/ x (/ z y))
   (if (<= (/ (* x y) z) -2.9952545328734584e-293)
     (/ (* x y) z)
     (if (<= (/ (* x y) z) 5.188044960065e-299)
       (* x (/ y z))
       (if (<= (/ (* x y) z) 3.103356213688716e+251)
         (* (* x y) (/ 1.0 z))
         (/ x (/ z y)))))))

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) / z) <= -((double) INFINITY)) {
		tmp = x / (z / y);
	} else if (((x * y) / z) <= -2.9952545328734584e-293) {
		tmp = (x * y) / z;
	} else if (((x * y) / z) <= 5.188044960065e-299) {
		tmp = x * (y / z);
	} else if (((x * y) / z) <= 3.103356213688716e+251) {
		tmp = (x * y) * (1.0 / z);
	} else {
		tmp = x / (z / y);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	6.5
Herbie	1.1

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x y) z) < -inf.0 or 3.1033562136887158e251 < (/.f64 (*.f64 x y) z)

   1. Initial program 42.9

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

   3. Applied associate-/l*_binary64_15709 5.9

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

   if -inf.0 < (/.f64 (*.f64 x y) z) < -2.99525453287345835e-293

   1. Initial program 0.5

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

   if -2.99525453287345835e-293 < (/.f64 (*.f64 x y) z) < 5.1880449600649995e-299

   1. Initial program 10.2

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

   3. Applied *-un-lft-identity_binary64_15764 10.2

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

   4. Applied times-frac_binary64_15770 1.1

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

   if 5.1880449600649995e-299 < (/.f64 (*.f64 x y) z) < 3.1033562136887158e251

   1. Initial program 0.5

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

   3. Applied div-inv_binary64_15761 0.6

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{z}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq -2.9952545328734584 \cdot 10^{-293}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 5.188044960065 \cdot 10^{-299}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 3.103356213688716 \cdot 10^{+251}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021059 
(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))