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 r259904 = 1.0;
        double r259905 = 2.0;
        double r259906 = r259904 / r259905;
        double r259907 = x;
        double r259908 = y;
        double r259909 = z;
        double r259910 = sqrt(r259909);
        double r259911 = r259908 * r259910;
        double r259912 = r259907 + r259911;
        double r259913 = r259906 * r259912;
        return r259913;
}
↓
double f(double x, double y, double z) {
        double r259914 = z;
        double r259915 = sqrt(r259914);
        double r259916 = y;
        double r259917 = x;
        double r259918 = fma(r259915, r259916, r259917);
        double r259919 = 1.0;
        double r259920 = r259918 * r259919;
        double r259921 = 2.0;
        double r259922 = r259920 / r259921;
        return r259922;
}

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 2020081 +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)))))