Average Error: 0.1 → 0.1
Time: 6.4s
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 r215844 = 1.0;
        double r215845 = 2.0;
        double r215846 = r215844 / r215845;
        double r215847 = x;
        double r215848 = y;
        double r215849 = z;
        double r215850 = sqrt(r215849);
        double r215851 = r215848 * r215850;
        double r215852 = r215847 + r215851;
        double r215853 = r215846 * r215852;
        return r215853;
}
↓
double f(double x, double y, double z) {
        double r215854 = z;
        double r215855 = sqrt(r215854);
        double r215856 = y;
        double r215857 = x;
        double r215858 = fma(r215855, r215856, r215857);
        double r215859 = 1.0;
        double r215860 = r215858 * r215859;
        double r215861 = 2.0;
        double r215862 = r215860 / r215861;
        return r215862;
}

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 2020062 +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)))))