Average Error: 0.1 → 0.1
Time: 17.3s
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(1, x, y \cdot \sqrt{z}\right)\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(1, x, y \cdot \sqrt{z}\right)
double f(double x, double y, double z) {
        double r111624 = 1.0;
        double r111625 = 2.0;
        double r111626 = r111624 / r111625;
        double r111627 = x;
        double r111628 = y;
        double r111629 = z;
        double r111630 = sqrt(r111629);
        double r111631 = r111628 * r111630;
        double r111632 = r111627 + r111631;
        double r111633 = r111626 * r111632;
        return r111633;
}
↓
double f(double x, double y, double z) {
        double r111634 = 1.0;
        double r111635 = 2.0;
        double r111636 = r111634 / r111635;
        double r111637 = 1.0;
        double r111638 = x;
        double r111639 = y;
        double r111640 = z;
        double r111641 = sqrt(r111640);
        double r111642 = r111639 * r111641;
        double r111643 = fma(r111637, r111638, r111642);
        double r111644 = r111636 * r111643;
        return r111644;
}

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. Using strategy rm

  3. Applied *-un-lft-identity0.1

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

  4. Applied fma-def0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019326 +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)))))