Average Error: 0.1 → 0.1
Time: 16.6s
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\frac{\mathsf{fma}\left(y, \sqrt{z}, x\right) \cdot 1}{2}\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\frac{\mathsf{fma}\left(y, \sqrt{z}, x\right) \cdot 1}{2}
double f(double x, double y, double z) {
        double r188285 = 1.0;
        double r188286 = 2.0;
        double r188287 = r188285 / r188286;
        double r188288 = x;
        double r188289 = y;
        double r188290 = z;
        double r188291 = sqrt(r188290);
        double r188292 = r188289 * r188291;
        double r188293 = r188288 + r188292;
        double r188294 = r188287 * r188293;
        return r188294;
}
↓
double f(double x, double y, double z) {
        double r188295 = y;
        double r188296 = z;
        double r188297 = sqrt(r188296);
        double r188298 = x;
        double r188299 = fma(r188295, r188297, r188298);
        double r188300 = 1.0;
        double r188301 = r188299 * r188300;
        double r188302 = 2.0;
        double r188303 = r188301 / r188302;
        return r188303;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))