Average Error: 0.1 → 0.1
Time: 12.1s
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\frac{\mathsf{fma}\left(y, \sqrt{z}, x\right) \cdot 1}{2}\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\frac{\mathsf{fma}\left(y, \sqrt{z}, x\right) \cdot 1}{2}
double f(double x, double y, double z) {
        double r134736 = 1.0;
        double r134737 = 2.0;
        double r134738 = r134736 / r134737;
        double r134739 = x;
        double r134740 = y;
        double r134741 = z;
        double r134742 = sqrt(r134741);
        double r134743 = r134740 * r134742;
        double r134744 = r134739 + r134743;
        double r134745 = r134738 * r134744;
        return r134745;
}
↓
double f(double x, double y, double z) {
        double r134746 = y;
        double r134747 = z;
        double r134748 = sqrt(r134747);
        double r134749 = x;
        double r134750 = fma(r134746, r134748, r134749);
        double r134751 = 1.0;
        double r134752 = r134750 * r134751;
        double r134753 = 2.0;
        double r134754 = r134752 / r134753;
        return r134754;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))