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

Average Error: 0.1 → 0

Time: 6.8s

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 r132173 = x;
        double r132174 = 3.0;
        double r132175 = 8.0;
        double r132176 = r132174 / r132175;
        double r132177 = y;
        double r132178 = r132176 * r132177;
        double r132179 = r132173 - r132178;
        return r132179;
}

↓

double f(double x, double y) {
        double r132180 = 3.0;
        double r132181 = 8.0;
        double r132182 = r132180 / r132181;
        double r132183 = y;
        double r132184 = -r132183;
        double r132185 = x;
        double r132186 = fma(r132182, r132184, r132185);
        return r132186;
}

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 2019304 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3 8) y)))