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

Average Error: 0.1 → 0

Time: 6.7s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r110260 = x;
        double r110261 = 3.0;
        double r110262 = 8.0;
        double r110263 = r110261 / r110262;
        double r110264 = y;
        double r110265 = r110263 * r110264;
        double r110266 = r110260 - r110265;
        return r110266;
}

↓

double f(double x, double y) {
        double r110267 = 3.0;
        double r110268 = 8.0;
        double r110269 = r110267 / r110268;
        double r110270 = y;
        double r110271 = -r110270;
        double r110272 = x;
        double r110273 = fma(r110269, r110271, r110272);
        return r110273;
}

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

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

6. Simplified 0

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

7. Final simplification 0

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3 8) y)))