Average Error: 0.2 → 0.2
Time: 20.4s
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 r174325 = 1.0;
        double r174326 = 2.0;
        double r174327 = r174325 / r174326;
        double r174328 = x;
        double r174329 = y;
        double r174330 = z;
        double r174331 = sqrt(r174330);
        double r174332 = r174329 * r174331;
        double r174333 = r174328 + r174332;
        double r174334 = r174327 * r174333;
        return r174334;
}
↓
double f(double x, double y, double z) {
        double r174335 = 1.0;
        double r174336 = 2.0;
        double r174337 = r174335 / r174336;
        double r174338 = z;
        double r174339 = sqrt(r174338);
        double r174340 = y;
        double r174341 = x;
        double r174342 = fma(r174339, r174340, r174341);
        double r174343 = r174337 * r174342;
        return r174343;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019304 +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)))))