Average Error: 0.1 → 0.1
Time: 6.6s
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 r208694 = 1.0;
        double r208695 = 2.0;
        double r208696 = r208694 / r208695;
        double r208697 = x;
        double r208698 = y;
        double r208699 = z;
        double r208700 = sqrt(r208699);
        double r208701 = r208698 * r208700;
        double r208702 = r208697 + r208701;
        double r208703 = r208696 * r208702;
        return r208703;
}
↓
double f(double x, double y, double z) {
        double r208704 = z;
        double r208705 = sqrt(r208704);
        double r208706 = y;
        double r208707 = x;
        double r208708 = fma(r208705, r208706, r208707);
        double r208709 = 1.0;
        double r208710 = r208708 * r208709;
        double r208711 = 2.0;
        double r208712 = r208710 / r208711;
        return r208712;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2020065 +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)))))