Average Error: 0.2 → 0.2
Time: 14.5s
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 r335221 = 1.0;
        double r335222 = 2.0;
        double r335223 = r335221 / r335222;
        double r335224 = x;
        double r335225 = y;
        double r335226 = z;
        double r335227 = sqrt(r335226);
        double r335228 = r335225 * r335227;
        double r335229 = r335224 + r335228;
        double r335230 = r335223 * r335229;
        return r335230;
}
↓
double f(double x, double y, double z) {
        double r335231 = 1.0;
        double r335232 = 2.0;
        double r335233 = r335231 / r335232;
        double r335234 = z;
        double r335235 = sqrt(r335234);
        double r335236 = y;
        double r335237 = x;
        double r335238 = fma(r335235, r335236, r335237);
        double r335239 = r335233 * r335238;
        return r335239;
}

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)))))