Average Error: 0.1 → 0.1
Time: 3.6s
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 r180672 = 1.0;
        double r180673 = 2.0;
        double r180674 = r180672 / r180673;
        double r180675 = x;
        double r180676 = y;
        double r180677 = z;
        double r180678 = sqrt(r180677);
        double r180679 = r180676 * r180678;
        double r180680 = r180675 + r180679;
        double r180681 = r180674 * r180680;
        return r180681;
}
↓
double f(double x, double y, double z) {
        double r180682 = z;
        double r180683 = sqrt(r180682);
        double r180684 = y;
        double r180685 = x;
        double r180686 = fma(r180683, r180684, r180685);
        double r180687 = 1.0;
        double r180688 = r180686 * r180687;
        double r180689 = 2.0;
        double r180690 = r180688 / r180689;
        return r180690;
}

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 2020039 +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)))))