Average Error: 0.1 → 0.1
Time: 46.2s
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\mathsf{fma}\left(\sqrt{z}, y, x\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\mathsf{fma}\left(\sqrt{z}, y, x\right) \cdot \frac{1}{2}
double f(double x, double y, double z) {
        double r12472628 = 1.0;
        double r12472629 = 2.0;
        double r12472630 = r12472628 / r12472629;
        double r12472631 = x;
        double r12472632 = y;
        double r12472633 = z;
        double r12472634 = sqrt(r12472633);
        double r12472635 = r12472632 * r12472634;
        double r12472636 = r12472631 + r12472635;
        double r12472637 = r12472630 * r12472636;
        return r12472637;
}
↓
double f(double x, double y, double z) {
        double r12472638 = z;
        double r12472639 = sqrt(r12472638);
        double r12472640 = y;
        double r12472641 = x;
        double r12472642 = fma(r12472639, r12472640, r12472641);
        double r12472643 = 1.0;
        double r12472644 = 2.0;
        double r12472645 = r12472643 / r12472644;
        double r12472646 = r12472642 * r12472645;
        return r12472646;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))