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

Average Error: 6.5 → 2.1

Time: 5.5s

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 r702512 = x;
        double r702513 = y;
        double r702514 = r702512 * r702513;
        double r702515 = z;
        double r702516 = r702514 / r702515;
        return r702516;
}

↓

double f(double x, double y, double z) {
        double r702517 = x;
        double r702518 = cbrt(r702517);
        double r702519 = r702518 * r702518;
        double r702520 = z;
        double r702521 = cbrt(r702520);
        double r702522 = r702521 * r702521;
        double r702523 = y;
        double r702524 = cbrt(r702523);
        double r702525 = r702524 * r702524;
        double r702526 = r702522 / r702525;
        double r702527 = r702519 / r702526;
        double r702528 = r702521 / r702524;
        double r702529 = r702518 / r702528;
        double r702530 = r702527 * r702529;
        return r702530;
}

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))