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

Average Error: 6.5 → 2.1

Time: 10.8s

Precision: 64

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

Math

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

↓

\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\]

C

double f(double x, double y, double z) {
        double r672425 = x;
        double r672426 = y;
        double r672427 = r672425 * r672426;
        double r672428 = z;
        double r672429 = r672427 / r672428;
        return r672429;
}

↓

double f(double x, double y, double z) {
        double r672430 = x;
        double r672431 = cbrt(r672430);
        double r672432 = r672431 * r672431;
        double r672433 = z;
        double r672434 = cbrt(r672433);
        double r672435 = r672434 * r672434;
        double r672436 = y;
        double r672437 = cbrt(r672436);
        double r672438 = r672437 * r672437;
        double r672439 = r672435 / r672438;
        double r672440 = r672432 / r672439;
        double r672441 = r672434 / r672437;
        double r672442 = r672431 / r672441;
        double r672443 = r672440 * r672442;
        return r672443;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.5
Target	6.0
Herbie	2.1

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

Derivation

1. Initial program 6.5

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

3. Applied associate-/l* 6.0

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

5. Applied add-cube-cbrt 6.8

   \[\leadsto \frac{x}{\frac{z}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}\]

6. Applied add-cube-cbrt 7.0

   \[\leadsto \frac{x}{\frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\]

7. Applied times-frac 7.0

   \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}}}\]

8. Applied add-cube-cbrt 7.1

   \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\]

9. Applied times-frac 2.1

   \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}}\]

10. Final simplification 2.1

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\]

Reproduce

herbie shell --seed 2020047 +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))