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

Average Error: 6.2 → 0.3

Time: 2.8s

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 r685618 = x;
        double r685619 = y;
        double r685620 = r685618 * r685619;
        double r685621 = z;
        double r685622 = r685620 / r685621;
        return r685622;
}

↓

double f(double x, double y, double z) {
        double r685623 = x;
        double r685624 = y;
        double r685625 = r685623 * r685624;
        double r685626 = -inf.0;
        bool r685627 = r685625 <= r685626;
        double r685628 = 1.0;
        double r685629 = z;
        double r685630 = r685629 / r685624;
        double r685631 = r685630 / r685623;
        double r685632 = r685628 / r685631;
        double r685633 = -2.510490940855015e-305;
        bool r685634 = r685625 <= r685633;
        double r685635 = r685625 / r685629;
        double r685636 = 2.3541375655288775e-189;
        bool r685637 = r685625 <= r685636;
        double r685638 = r685623 / r685630;
        double r685639 = 3.5319662921484826e+186;
        bool r685640 = r685625 <= r685639;
        double r685641 = r685640 ? r685635 : r685638;
        double r685642 = r685637 ? r685638 : r685641;
        double r685643 = r685634 ? r685635 : r685642;
        double r685644 = r685627 ? r685632 : r685643;
        return r685644;
}

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 +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))