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

Average Error: 0.1 → 0

Time: 4.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x, double y) {
        double r181816 = x;
        double r181817 = 3.0;
        double r181818 = 8.0;
        double r181819 = r181817 / r181818;
        double r181820 = y;
        double r181821 = r181819 * r181820;
        double r181822 = r181816 - r181821;
        return r181822;
}

↓

double f(double x, double y) {
        double r181823 = y;
        double r181824 = -r181823;
        double r181825 = 3.0;
        double r181826 = 8.0;
        double r181827 = r181825 / r181826;
        double r181828 = x;
        double r181829 = fma(r181824, r181827, r181828);
        return r181829;
}

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

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

6. Simplified 0

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

7. Final simplification 0

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3 8) y)))