Average Error: 0.1 → 0.1
Time: 7.0s
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 r270168 = 1.0;
        double r270169 = 2.0;
        double r270170 = r270168 / r270169;
        double r270171 = x;
        double r270172 = y;
        double r270173 = z;
        double r270174 = sqrt(r270173);
        double r270175 = r270172 * r270174;
        double r270176 = r270171 + r270175;
        double r270177 = r270170 * r270176;
        return r270177;
}
↓
double f(double x, double y, double z) {
        double r270178 = z;
        double r270179 = sqrt(r270178);
        double r270180 = y;
        double r270181 = x;
        double r270182 = fma(r270179, r270180, r270181);
        double r270183 = 1.0;
        double r270184 = r270182 * r270183;
        double r270185 = 2.0;
        double r270186 = r270184 / r270185;
        return r270186;
}

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 2020035 +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)))))