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

Average Error: 6.2 → 0.3

Time: 2.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y = -\infty:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{y}}{x}}\\ \mathbf{elif}\;x \cdot y \le -2.51049094085501509 \cdot 10^{-305}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 2.3541375655288775 \cdot 10^{-189}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 3.53196629214848261 \cdot 10^{186}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r772444 = x;
        double r772445 = y;
        double r772446 = r772444 * r772445;
        double r772447 = z;
        double r772448 = r772446 / r772447;
        return r772448;
}

↓

double f(double x, double y, double z) {
        double r772449 = x;
        double r772450 = y;
        double r772451 = r772449 * r772450;
        double r772452 = -inf.0;
        bool r772453 = r772451 <= r772452;
        double r772454 = 1.0;
        double r772455 = z;
        double r772456 = r772455 / r772450;
        double r772457 = r772456 / r772449;
        double r772458 = r772454 / r772457;
        double r772459 = -2.510490940855015e-305;
        bool r772460 = r772451 <= r772459;
        double r772461 = r772451 / r772455;
        double r772462 = 2.3541375655288775e-189;
        bool r772463 = r772451 <= r772462;
        double r772464 = r772449 / r772456;
        double r772465 = 3.5319662921484826e+186;
        bool r772466 = r772451 <= r772465;
        double r772467 = r772466 ? r772461 : r772464;
        double r772468 = r772463 ? r772464 : r772467;
        double r772469 = r772460 ? r772461 : r772468;
        double r772470 = r772453 ? r772458 : r772469;
        return r772470;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.2
Target	6.3
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-/l* 0.3

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

   5. Applied clear-num 0.4

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

   if -inf.0 < (* x y) < -2.510490940855015e-305 or 2.3541375655288775e-189 < (* x y) < 3.5319662921484826e+186

   1. Initial program 0.2

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

   if -2.510490940855015e-305 < (* x y) < 2.3541375655288775e-189 or 3.5319662921484826e+186 < (* x y)

   1. Initial program 15.4

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

   3. Applied associate-/l* 0.6

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y = -\infty:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{y}}{x}}\\ \mathbf{elif}\;x \cdot y \le -2.51049094085501509 \cdot 10^{-305}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 2.3541375655288775 \cdot 10^{-189}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 3.53196629214848261 \cdot 10^{186}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))