Average Error: 0.1 → 0.1
Time: 14.4s
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 r286478 = 1.0;
        double r286479 = 2.0;
        double r286480 = r286478 / r286479;
        double r286481 = x;
        double r286482 = y;
        double r286483 = z;
        double r286484 = sqrt(r286483);
        double r286485 = r286482 * r286484;
        double r286486 = r286481 + r286485;
        double r286487 = r286480 * r286486;
        return r286487;
}
↓
double f(double x, double y, double z) {
        double r286488 = 1.0;
        double r286489 = 2.0;
        double r286490 = r286488 / r286489;
        double r286491 = z;
        double r286492 = sqrt(r286491);
        double r286493 = y;
        double r286494 = x;
        double r286495 = fma(r286492, r286493, r286494);
        double r286496 = r286490 * r286495;
        return r286496;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019350 +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)))))