Average Error: 0.1 → 0.1
Time: 4.1s
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 r189970 = 1.0;
        double r189971 = 2.0;
        double r189972 = r189970 / r189971;
        double r189973 = x;
        double r189974 = y;
        double r189975 = z;
        double r189976 = sqrt(r189975);
        double r189977 = r189974 * r189976;
        double r189978 = r189973 + r189977;
        double r189979 = r189972 * r189978;
        return r189979;
}
↓
double f(double x, double y, double z) {
        double r189980 = z;
        double r189981 = sqrt(r189980);
        double r189982 = y;
        double r189983 = x;
        double r189984 = fma(r189981, r189982, r189983);
        double r189985 = 1.0;
        double r189986 = r189984 * r189985;
        double r189987 = 2.0;
        double r189988 = r189986 / r189987;
        return r189988;
}

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 2020100 +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)))))