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

Average Error: 0.1 → 0

Time: 4.1s

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 r141113 = x;
        double r141114 = 3.0;
        double r141115 = 8.0;
        double r141116 = r141114 / r141115;
        double r141117 = y;
        double r141118 = r141116 * r141117;
        double r141119 = r141113 - r141118;
        return r141119;
}

↓

double f(double x, double y) {
        double r141120 = 3.0;
        double r141121 = 8.0;
        double r141122 = r141120 / r141121;
        double r141123 = y;
        double r141124 = -r141123;
        double r141125 = x;
        double r141126 = fma(r141122, r141124, r141125);
        return r141126;
}

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 2019208 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3 8) y)))