Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, A

Average Error: 0.1 → 0

Time: 4.5s

Precision: 64

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

Math

\[x - \frac{3}{8} \cdot y\]

↓

\[\mathsf{fma}\left(y, -\frac{3}{8}, x\right)\]

C

double f(double x, double y) {
        double r204846 = x;
        double r204847 = 3.0;
        double r204848 = 8.0;
        double r204849 = r204847 / r204848;
        double r204850 = y;
        double r204851 = r204849 * r204850;
        double r204852 = r204846 - r204851;
        return r204852;
}

↓

double f(double x, double y) {
        double r204853 = y;
        double r204854 = 3.0;
        double r204855 = 8.0;
        double r204856 = r204854 / r204855;
        double r204857 = -r204856;
        double r204858 = x;
        double r204859 = fma(r204853, r204857, r204858);
        return r204859;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Initial program 0.1

   \[x - \frac{3}{8} \cdot y\]
2. Using strategy rm

3. Applied add-cube-cbrt 0.8

   \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} - \frac{3}{8} \cdot y\]

4. Applied prod-diff 0.8

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, -y \cdot \frac{3}{8}\right) + \mathsf{fma}\left(-y, \frac{3}{8}, y \cdot \frac{3}{8}\right)}\]

5. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, -\frac{3}{8}, x\right)} + \mathsf{fma}\left(-y, \frac{3}{8}, y \cdot \frac{3}{8}\right)\]

6. Simplified 0

   \[\leadsto \mathsf{fma}\left(y, -\frac{3}{8}, x\right) + \color{blue}{0}\]

7. Final simplification 0

   \[\leadsto \mathsf{fma}\left(y, -\frac{3}{8}, x\right)\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3 8) y)))