Average Error: 0.1 → 0.1
Time: 6.6s
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 r197725 = 1.0;
        double r197726 = 2.0;
        double r197727 = r197725 / r197726;
        double r197728 = x;
        double r197729 = y;
        double r197730 = z;
        double r197731 = sqrt(r197730);
        double r197732 = r197729 * r197731;
        double r197733 = r197728 + r197732;
        double r197734 = r197727 * r197733;
        return r197734;
}
↓
double f(double x, double y, double z) {
        double r197735 = z;
        double r197736 = sqrt(r197735);
        double r197737 = y;
        double r197738 = x;
        double r197739 = fma(r197736, r197737, r197738);
        double r197740 = 1.0;
        double r197741 = r197739 * r197740;
        double r197742 = 2.0;
        double r197743 = r197741 / r197742;
        return r197743;
}

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)))))