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

Average Error: 0.1 → 0

Time: 4.5s

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 r136736 = x;
        double r136737 = 3.0;
        double r136738 = 8.0;
        double r136739 = r136737 / r136738;
        double r136740 = y;
        double r136741 = r136739 * r136740;
        double r136742 = r136736 - r136741;
        return r136742;
}

↓

double f(double x, double y) {
        double r136743 = 3.0;
        double r136744 = 8.0;
        double r136745 = r136743 / r136744;
        double r136746 = y;
        double r136747 = -r136746;
        double r136748 = x;
        double r136749 = fma(r136745, r136747, r136748);
        return r136749;
}

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 2019305 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3 8) y)))