Average Error: 0.2 → 0.2
Time: 17.1s
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 r209307 = 1.0;
        double r209308 = 2.0;
        double r209309 = r209307 / r209308;
        double r209310 = x;
        double r209311 = y;
        double r209312 = z;
        double r209313 = sqrt(r209312);
        double r209314 = r209311 * r209313;
        double r209315 = r209310 + r209314;
        double r209316 = r209309 * r209315;
        return r209316;
}
↓
double f(double x, double y, double z) {
        double r209317 = 1.0;
        double r209318 = 2.0;
        double r209319 = r209317 / r209318;
        double r209320 = z;
        double r209321 = sqrt(r209320);
        double r209322 = y;
        double r209323 = x;
        double r209324 = fma(r209321, r209322, r209323);
        double r209325 = r209319 * r209324;
        return r209325;
}

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 2019325 +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)))))