Average Error: 0.2 → 0.2
Time: 13.3s
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 r237237 = 1.0;
        double r237238 = 2.0;
        double r237239 = r237237 / r237238;
        double r237240 = x;
        double r237241 = y;
        double r237242 = z;
        double r237243 = sqrt(r237242);
        double r237244 = r237241 * r237243;
        double r237245 = r237240 + r237244;
        double r237246 = r237239 * r237245;
        return r237246;
}
↓
double f(double x, double y, double z) {
        double r237247 = 1.0;
        double r237248 = 2.0;
        double r237249 = r237247 / r237248;
        double r237250 = z;
        double r237251 = sqrt(r237250);
        double r237252 = y;
        double r237253 = x;
        double r237254 = fma(r237251, r237252, r237253);
        double r237255 = r237249 * r237254;
        return r237255;
}

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 2020046 +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)))))