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

Average Error: 6.3 → 1.3

Time: 2.9s

Precision: binary64

[x y]: =sort([x y])

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -1.4056475562933142 \cdot 10^{-63}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 7.3976120488473 \cdot 10^{-313}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 7.413653572078785 \cdot 10^{+103}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) (- INFINITY))
   (* x (/ y z))
   (if (<= (* x y) -1.4056475562933142e-63)
     (/ (* x y) z)
     (if (<= (* x y) 7.3976120488473e-313)
       (* x (/ y z))
       (if (<= (* x y) 7.413653572078785e+103) (/ (* 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) <= -((double) INFINITY)) {
		tmp = x * (y / z);
	} else if ((x * y) <= -1.4056475562933142e-63) {
		tmp = (x * y) / z;
	} else if ((x * y) <= 7.3976120488473e-313) {
		tmp = x * (y / z);
	} else if ((x * y) <= 7.413653572078785e+103) {
		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	6.2
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -inf.0 or -1.4056475562933142e-63 < (*.f64 x y) < 7.3976120488473e-313

   1. Initial program 12.7

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

   3. Applied *-un-lft-identity_binary64_20538 12.7

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

   4. Applied times-frac_binary64_20544 2.4

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

   5. Simplified 2.4

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

   if -inf.0 < (*.f64 x y) < -1.4056475562933142e-63 or 7.3976120488473e-313 < (*.f64 x y) < 7.4136535720787846e103

   1. Initial program 0.2

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

   if 7.4136535720787846e103 < (*.f64 x y)

   1. Initial program 14.2

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

   3. Applied associate-/l*_binary64_20483 4.0

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

   5. Applied associate-/r/_binary64_20484 3.2

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -1.4056475562933142 \cdot 10^{-63}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 7.3976120488473 \cdot 10^{-313}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 7.413653572078785 \cdot 10^{+103}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(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))