Average Error: 0.2 → 0.2
Time: 12.5s
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 r154254 = 1.0;
        double r154255 = 2.0;
        double r154256 = r154254 / r154255;
        double r154257 = x;
        double r154258 = y;
        double r154259 = z;
        double r154260 = sqrt(r154259);
        double r154261 = r154258 * r154260;
        double r154262 = r154257 + r154261;
        double r154263 = r154256 * r154262;
        return r154263;
}
↓
double f(double x, double y, double z) {
        double r154264 = 1.0;
        double r154265 = 2.0;
        double r154266 = r154264 / r154265;
        double r154267 = z;
        double r154268 = sqrt(r154267);
        double r154269 = y;
        double r154270 = x;
        double r154271 = fma(r154268, r154269, r154270);
        double r154272 = r154266 * r154271;
        return r154272;
}

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 2019209 +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)))))