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

Average Error: 6.2 → 0.3

Time: 2.7s

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 r580749 = x;
        double r580750 = y;
        double r580751 = r580749 * r580750;
        double r580752 = z;
        double r580753 = r580751 / r580752;
        return r580753;
}

↓

double f(double x, double y, double z) {
        double r580754 = x;
        double r580755 = y;
        double r580756 = r580754 * r580755;
        double r580757 = -inf.0;
        bool r580758 = r580756 <= r580757;
        double r580759 = 1.0;
        double r580760 = z;
        double r580761 = r580760 / r580755;
        double r580762 = r580761 / r580754;
        double r580763 = r580759 / r580762;
        double r580764 = -2.510490940855015e-305;
        bool r580765 = r580756 <= r580764;
        double r580766 = r580756 / r580760;
        double r580767 = 2.3541375655288775e-189;
        bool r580768 = r580756 <= r580767;
        double r580769 = r580754 / r580761;
        double r580770 = 3.5319662921484826e+186;
        bool r580771 = r580756 <= r580770;
        double r580772 = r580771 ? r580766 : r580769;
        double r580773 = r580768 ? r580769 : r580772;
        double r580774 = r580765 ? r580766 : r580773;
        double r580775 = r580758 ? r580763 : r580774;
        return r580775;
}

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))