Average Error: 0.2 → 0.1
Time: 1.1m
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{z}, y, x\right)\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{z}, y, x\right)
double f(double x, double y, double z) {
        double r318581 = 1.0;
        double r318582 = 2.0;
        double r318583 = r318581 / r318582;
        double r318584 = x;
        double r318585 = y;
        double r318586 = z;
        double r318587 = sqrt(r318586);
        double r318588 = r318585 * r318587;
        double r318589 = r318584 + r318588;
        double r318590 = r318583 * r318589;
        return r318590;
}
↓
double f(double x, double y, double z) {
        double r318591 = 1.0;
        double r318592 = 2.0;
        double r318593 = r318591 / r318592;
        double r318594 = z;
        double r318595 = sqrt(r318594);
        double r318596 = y;
        double r318597 = x;
        double r318598 = fma(r318595, r318596, r318597);
        double r318599 = r318593 * r318598;
        return r318599;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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)))))