Average Error: 0.2 → 0.2
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 r222830 = 1.0;
        double r222831 = 2.0;
        double r222832 = r222830 / r222831;
        double r222833 = x;
        double r222834 = y;
        double r222835 = z;
        double r222836 = sqrt(r222835);
        double r222837 = r222834 * r222836;
        double r222838 = r222833 + r222837;
        double r222839 = r222832 * r222838;
        return r222839;
}
↓
double f(double x, double y, double z) {
        double r222840 = z;
        double r222841 = sqrt(r222840);
        double r222842 = y;
        double r222843 = x;
        double r222844 = fma(r222841, r222842, r222843);
        double r222845 = 1.0;
        double r222846 = r222844 * r222845;
        double r222847 = 2.0;
        double r222848 = r222846 / r222847;
        return r222848;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]

  2. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2020036 +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)))))