Average Error: 0.2 → 0.2
Time: 7.6s
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 r220334 = 1.0;
        double r220335 = 2.0;
        double r220336 = r220334 / r220335;
        double r220337 = x;
        double r220338 = y;
        double r220339 = z;
        double r220340 = sqrt(r220339);
        double r220341 = r220338 * r220340;
        double r220342 = r220337 + r220341;
        double r220343 = r220336 * r220342;
        return r220343;
}
↓
double f(double x, double y, double z) {
        double r220344 = z;
        double r220345 = sqrt(r220344);
        double r220346 = y;
        double r220347 = x;
        double r220348 = fma(r220345, r220346, r220347);
        double r220349 = 1.0;
        double r220350 = r220348 * r220349;
        double r220351 = 2.0;
        double r220352 = r220350 / r220351;
        return r220352;
}

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. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2020036 +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)))))