Average Error: 0.1 → 0.1
Time: 6.8s
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 r310042 = 1.0;
        double r310043 = 2.0;
        double r310044 = r310042 / r310043;
        double r310045 = x;
        double r310046 = y;
        double r310047 = z;
        double r310048 = sqrt(r310047);
        double r310049 = r310046 * r310048;
        double r310050 = r310045 + r310049;
        double r310051 = r310044 * r310050;
        return r310051;
}
↓
double f(double x, double y, double z) {
        double r310052 = z;
        double r310053 = sqrt(r310052);
        double r310054 = y;
        double r310055 = x;
        double r310056 = fma(r310053, r310054, r310055);
        double r310057 = 1.0;
        double r310058 = r310056 * r310057;
        double r310059 = 2.0;
        double r310060 = r310058 / r310059;
        return r310060;
}

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 2020024 +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)))))