Average Error: 0.2 → 0.2
Time: 18.8s
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{z}, y, x\right)\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{z}, y, x\right)
double f(double x, double y, double z) {
        double r132362 = 1.0;
        double r132363 = 2.0;
        double r132364 = r132362 / r132363;
        double r132365 = x;
        double r132366 = y;
        double r132367 = z;
        double r132368 = sqrt(r132367);
        double r132369 = r132366 * r132368;
        double r132370 = r132365 + r132369;
        double r132371 = r132364 * r132370;
        return r132371;
}
↓
double f(double x, double y, double z) {
        double r132372 = 1.0;
        double r132373 = 2.0;
        double r132374 = r132372 / r132373;
        double r132375 = z;
        double r132376 = sqrt(r132375);
        double r132377 = y;
        double r132378 = x;
        double r132379 = fma(r132376, r132377, r132378);
        double r132380 = r132374 * r132379;
        return r132380;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{z}, y, x\right)}\]

  3. Final simplification0.2

    \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{z}, y, x\right)\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  :precision binary64
  (* (/ 1 2) (+ x (* y (sqrt z)))))