Average Error: 0.1 → 0.1
Time: 19.1s
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 r150743 = 1.0;
        double r150744 = 2.0;
        double r150745 = r150743 / r150744;
        double r150746 = x;
        double r150747 = y;
        double r150748 = z;
        double r150749 = sqrt(r150748);
        double r150750 = r150747 * r150749;
        double r150751 = r150746 + r150750;
        double r150752 = r150745 * r150751;
        return r150752;
}
↓
double f(double x, double y, double z) {
        double r150753 = 1.0;
        double r150754 = 2.0;
        double r150755 = r150753 / r150754;
        double r150756 = z;
        double r150757 = sqrt(r150756);
        double r150758 = y;
        double r150759 = x;
        double r150760 = fma(r150757, r150758, r150759);
        double r150761 = r150755 * r150760;
        return r150761;
}

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

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

Reproduce

herbie shell --seed 2019322 +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)))))