Average Error: 0.1 → 0.1
Time: 3.7s
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 r223677 = 1.0;
        double r223678 = 2.0;
        double r223679 = r223677 / r223678;
        double r223680 = x;
        double r223681 = y;
        double r223682 = z;
        double r223683 = sqrt(r223682);
        double r223684 = r223681 * r223683;
        double r223685 = r223680 + r223684;
        double r223686 = r223679 * r223685;
        return r223686;
}
↓
double f(double x, double y, double z) {
        double r223687 = z;
        double r223688 = sqrt(r223687);
        double r223689 = y;
        double r223690 = x;
        double r223691 = fma(r223688, r223689, r223690);
        double r223692 = 1.0;
        double r223693 = r223691 * r223692;
        double r223694 = 2.0;
        double r223695 = r223693 / r223694;
        return r223695;
}

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 2020025 +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)))))