Average Error: 0.2 → 0.2
Time: 7.1s
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\frac{\mathsf{fma}\left(\sqrt{z}, y, x\right) \cdot 1}{2}\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\frac{\mathsf{fma}\left(\sqrt{z}, y, x\right) \cdot 1}{2}
double f(double x, double y, double z) {
        double r226451 = 1.0;
        double r226452 = 2.0;
        double r226453 = r226451 / r226452;
        double r226454 = x;
        double r226455 = y;
        double r226456 = z;
        double r226457 = sqrt(r226456);
        double r226458 = r226455 * r226457;
        double r226459 = r226454 + r226458;
        double r226460 = r226453 * r226459;
        return r226460;
}
↓
double f(double x, double y, double z) {
        double r226461 = z;
        double r226462 = sqrt(r226461);
        double r226463 = y;
        double r226464 = x;
        double r226465 = fma(r226462, r226463, r226464);
        double r226466 = 1.0;
        double r226467 = r226465 * r226466;
        double r226468 = 2.0;
        double r226469 = r226467 / r226468;
        return r226469;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2020057 +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)))))