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

Average Error: 0.1 → 0

Time: 3.5s

Precision: 64

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

Math

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

↓

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

C

double f(double x, double y) {
        double r35412 = x;
        double r35413 = 3.0;
        double r35414 = 8.0;
        double r35415 = r35413 / r35414;
        double r35416 = y;
        double r35417 = r35415 * r35416;
        double r35418 = r35412 - r35417;
        return r35418;
}

↓

double f(double x, double y) {
        double r35419 = 3.0;
        double r35420 = 8.0;
        double r35421 = r35419 / r35420;
        double r35422 = -r35421;
        double r35423 = y;
        double r35424 = x;
        double r35425 = fma(r35422, r35423, r35424);
        return r35425;
}

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 2020045 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3 8) y)))