Average Error: 0.2 → 0.2
Time: 6.3s
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 r226833 = 1.0;
        double r226834 = 2.0;
        double r226835 = r226833 / r226834;
        double r226836 = x;
        double r226837 = y;
        double r226838 = z;
        double r226839 = sqrt(r226838);
        double r226840 = r226837 * r226839;
        double r226841 = r226836 + r226840;
        double r226842 = r226835 * r226841;
        return r226842;
}
↓
double f(double x, double y, double z) {
        double r226843 = z;
        double r226844 = sqrt(r226843);
        double r226845 = y;
        double r226846 = x;
        double r226847 = fma(r226844, r226845, r226846);
        double r226848 = 1.0;
        double r226849 = r226847 * r226848;
        double r226850 = 2.0;
        double r226851 = r226849 / r226850;
        return r226851;
}

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 2020064 +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)))))