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

Average Error: 6.5 → 1.6

Time: 11.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.6207759867400651 \cdot 10^{260}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -5.6900917550780378 \cdot 10^{-61}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le -0.0:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 4.3629336611497376 \cdot 10^{104}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r778448 = x;
        double r778449 = y;
        double r778450 = r778448 * r778449;
        double r778451 = z;
        double r778452 = r778450 / r778451;
        return r778452;
}

↓

double f(double x, double y, double z) {
        double r778453 = x;
        double r778454 = y;
        double r778455 = r778453 * r778454;
        double r778456 = -1.6207759867400651e+260;
        bool r778457 = r778455 <= r778456;
        double r778458 = z;
        double r778459 = r778454 / r778458;
        double r778460 = r778453 * r778459;
        double r778461 = -5.690091755078038e-61;
        bool r778462 = r778455 <= r778461;
        double r778463 = r778455 / r778458;
        double r778464 = -0.0;
        bool r778465 = r778455 <= r778464;
        double r778466 = 4.3629336611497376e+104;
        bool r778467 = r778455 <= r778466;
        double r778468 = r778458 / r778454;
        double r778469 = r778453 / r778468;
        double r778470 = r778467 ? r778463 : r778469;
        double r778471 = r778465 ? r778460 : r778470;
        double r778472 = r778462 ? r778463 : r778471;
        double r778473 = r778457 ? r778460 : r778472;
        return r778473;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.5
Target	6.4
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.70421306606504721 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* x y) < -1.6207759867400651e+260 or -5.690091755078038e-61 < (* x y) < -0.0

   1. Initial program 13.1

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

   3. Applied *-un-lft-identity 13.1

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

   4. Applied times-frac 2.7

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

   5. Simplified 2.7

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

   if -1.6207759867400651e+260 < (* x y) < -5.690091755078038e-61 or -0.0 < (* x y) < 4.3629336611497376e+104

   1. Initial program 0.4

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

   if 4.3629336611497376e+104 < (* x y)

   1. Initial program 14.3

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

   3. Applied associate-/l* 3.9

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.6207759867400651 \cdot 10^{260}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le -5.6900917550780378 \cdot 10^{-61}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le -0.0:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \le 4.3629336611497376 \cdot 10^{104}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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))