Average Error: 0.1 → 0.1
Time: 6.8s
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 r252449 = 1.0;
        double r252450 = 2.0;
        double r252451 = r252449 / r252450;
        double r252452 = x;
        double r252453 = y;
        double r252454 = z;
        double r252455 = sqrt(r252454);
        double r252456 = r252453 * r252455;
        double r252457 = r252452 + r252456;
        double r252458 = r252451 * r252457;
        return r252458;
}
↓
double f(double x, double y, double z) {
        double r252459 = z;
        double r252460 = sqrt(r252459);
        double r252461 = y;
        double r252462 = x;
        double r252463 = fma(r252460, r252461, r252462);
        double r252464 = 1.0;
        double r252465 = r252463 * r252464;
        double r252466 = 2.0;
        double r252467 = r252465 / r252466;
        return r252467;
}

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 2019353 +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)))))