Average Error: 0.1 → 0.1
Time: 7.1s
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 r221785 = 1.0;
        double r221786 = 2.0;
        double r221787 = r221785 / r221786;
        double r221788 = x;
        double r221789 = y;
        double r221790 = z;
        double r221791 = sqrt(r221790);
        double r221792 = r221789 * r221791;
        double r221793 = r221788 + r221792;
        double r221794 = r221787 * r221793;
        return r221794;
}
↓
double f(double x, double y, double z) {
        double r221795 = z;
        double r221796 = sqrt(r221795);
        double r221797 = y;
        double r221798 = x;
        double r221799 = fma(r221796, r221797, r221798);
        double r221800 = 1.0;
        double r221801 = r221799 * r221800;
        double r221802 = 2.0;
        double r221803 = r221801 / r221802;
        return r221803;
}

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 2019353 +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)))))