Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, F

Average Error: 0.3 → 0.3

Time: 10.0s

Precision: 64

Math

\[\left(x \cdot 27\right) \cdot y\]

↓

\[\left(y \cdot x\right) \cdot 27\]

C

double f(double x, double y) {
        double r8389110 = x;
        double r8389111 = 27.0;
        double r8389112 = r8389110 * r8389111;
        double r8389113 = y;
        double r8389114 = r8389112 * r8389113;
        return r8389114;
}

↓

double f(double x, double y) {
        double r8389115 = y;
        double r8389116 = x;
        double r8389117 = r8389115 * r8389116;
        double r8389118 = 27.0;
        double r8389119 = r8389117 * r8389118;
        return r8389119;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Initial program 0.3

   \[\left(x \cdot 27\right) \cdot y\]

2. Taylor expanded around 0 0.3

   \[\leadsto \color{blue}{27 \cdot \left(x \cdot y\right)}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 0.3

   \[\leadsto \color{blue}{\left(\sqrt{27} \cdot \sqrt{27}\right)} \cdot \left(x \cdot y\right)\]

5. Applied associate-*l* 0.4

   \[\leadsto \color{blue}{\sqrt{27} \cdot \left(\sqrt{27} \cdot \left(x \cdot y\right)\right)}\]
6. Using strategy rm

7. Applied add-sqr-sqrt 0.4

   \[\leadsto \sqrt{\color{blue}{\sqrt{27} \cdot \sqrt{27}}} \cdot \left(\sqrt{27} \cdot \left(x \cdot y\right)\right)\]

8. Applied sqrt-prod 0.7

   \[\leadsto \color{blue}{\left(\sqrt{\sqrt{27}} \cdot \sqrt{\sqrt{27}}\right)} \cdot \left(\sqrt{27} \cdot \left(x \cdot y\right)\right)\]

9. Applied associate-*l* 0.6

   \[\leadsto \color{blue}{\sqrt{\sqrt{27}} \cdot \left(\sqrt{\sqrt{27}} \cdot \left(\sqrt{27} \cdot \left(x \cdot y\right)\right)\right)}\]

10. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{x \cdot \left(y \cdot {\left(\sqrt{27}\right)}^{2}\right)}\]

11. Simplified 0.3

    \[\leadsto \color{blue}{27 \cdot \left(x \cdot y\right)}\]

12. Final simplification 0.3

    \[\leadsto \left(y \cdot x\right) \cdot 27\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, F"
  (* (* x 27.0) y))