Average Error: 0.1 → 0.1
Time: 6.9s
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 r213828 = 1.0;
        double r213829 = 2.0;
        double r213830 = r213828 / r213829;
        double r213831 = x;
        double r213832 = y;
        double r213833 = z;
        double r213834 = sqrt(r213833);
        double r213835 = r213832 * r213834;
        double r213836 = r213831 + r213835;
        double r213837 = r213830 * r213836;
        return r213837;
}
↓
double f(double x, double y, double z) {
        double r213838 = z;
        double r213839 = sqrt(r213838);
        double r213840 = y;
        double r213841 = x;
        double r213842 = fma(r213839, r213840, r213841);
        double r213843 = 1.0;
        double r213844 = r213842 * r213843;
        double r213845 = 2.0;
        double r213846 = r213844 / r213845;
        return r213846;
}

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 2020089 +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)))))