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

Average Error: 6.5 → 1.6

Time: 9.8s

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 r396755 = x;
        double r396756 = y;
        double r396757 = r396755 * r396756;
        double r396758 = z;
        double r396759 = r396757 / r396758;
        return r396759;
}

↓

double f(double x, double y, double z) {
        double r396760 = x;
        double r396761 = y;
        double r396762 = r396760 * r396761;
        double r396763 = -1.6207759867400651e+260;
        bool r396764 = r396762 <= r396763;
        double r396765 = -r396761;
        double r396766 = z;
        double r396767 = -r396766;
        double r396768 = r396765 / r396767;
        double r396769 = r396760 * r396768;
        double r396770 = -5.690091755078038e-61;
        bool r396771 = r396762 <= r396770;
        double r396772 = r396762 / r396766;
        double r396773 = -0.0;
        bool r396774 = r396762 <= r396773;
        double r396775 = 4.3629336611497376e+104;
        bool r396776 = r396762 <= r396775;
        double r396777 = r396766 / r396761;
        double r396778 = r396760 / r396777;
        double r396779 = r396776 ? r396772 : r396778;
        double r396780 = r396774 ? r396769 : r396779;
        double r396781 = r396771 ? r396772 : r396780;
        double r396782 = r396764 ? r396769 : r396781;
        return r396782;
}

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 clear-num 13.4

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x \cdot y}}}\]
   4. Using strategy rm

   5. Applied frac-2neg 13.4

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

   6. Simplified 13.4

      \[\leadsto \frac{\color{blue}{-1}}{-\frac{z}{x \cdot y}}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 13.4

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

   9. Applied times-frac 3.3

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

   10. Applied distribute-rgt-neg-in 3.3

       \[\leadsto \frac{-1}{\color{blue}{\frac{1}{x} \cdot \left(-\frac{z}{y}\right)}}\]

   11. Applied *-un-lft-identity 3.3

       \[\leadsto \frac{\color{blue}{1 \cdot -1}}{\frac{1}{x} \cdot \left(-\frac{z}{y}\right)}\]

   12. Applied times-frac 3.0

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

   13. Simplified 3.0

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

   14. Simplified 2.7

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

   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 
(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))