Average Error: 0.2 → 0.1
Time: 1.0m
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 r250742 = 1.0;
        double r250743 = 2.0;
        double r250744 = r250742 / r250743;
        double r250745 = x;
        double r250746 = y;
        double r250747 = z;
        double r250748 = sqrt(r250747);
        double r250749 = r250746 * r250748;
        double r250750 = r250745 + r250749;
        double r250751 = r250744 * r250750;
        return r250751;
}
↓
double f(double x, double y, double z) {
        double r250752 = 1.0;
        double r250753 = 2.0;
        double r250754 = r250752 / r250753;
        double r250755 = z;
        double r250756 = sqrt(r250755);
        double r250757 = y;
        double r250758 = x;
        double r250759 = fma(r250756, r250757, r250758);
        double r250760 = r250754 * r250759;
        return r250760;
}

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. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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)))))