Average Error: 0.1 → 0.1
Time: 6.7s
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 r228666 = 1.0;
        double r228667 = 2.0;
        double r228668 = r228666 / r228667;
        double r228669 = x;
        double r228670 = y;
        double r228671 = z;
        double r228672 = sqrt(r228671);
        double r228673 = r228670 * r228672;
        double r228674 = r228669 + r228673;
        double r228675 = r228668 * r228674;
        return r228675;
}
↓
double f(double x, double y, double z) {
        double r228676 = z;
        double r228677 = sqrt(r228676);
        double r228678 = y;
        double r228679 = x;
        double r228680 = fma(r228677, r228678, r228679);
        double r228681 = 1.0;
        double r228682 = r228680 * r228681;
        double r228683 = 2.0;
        double r228684 = r228682 / r228683;
        return r228684;
}

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 2020018 +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)))))