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 r122640 = x;
        double r122641 = 3.0;
        double r122642 = 8.0;
        double r122643 = r122641 / r122642;
        double r122644 = y;
        double r122645 = r122643 * r122644;
        double r122646 = r122640 - r122645;
        return r122646;
}

↓

double f(double x, double y) {
        double r122647 = 3.0;
        double r122648 = 8.0;
        double r122649 = r122647 / r122648;
        double r122650 = y;
        double r122651 = -r122650;
        double r122652 = x;
        double r122653 = fma(r122649, r122651, r122652);
        return r122653;
}

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 2019322 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3 8) y)))