Average Error: 0.1 → 0.1
Time: 15.2s
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 r241341 = 1.0;
        double r241342 = 2.0;
        double r241343 = r241341 / r241342;
        double r241344 = x;
        double r241345 = y;
        double r241346 = z;
        double r241347 = sqrt(r241346);
        double r241348 = r241345 * r241347;
        double r241349 = r241344 + r241348;
        double r241350 = r241343 * r241349;
        return r241350;
}
↓
double f(double x, double y, double z) {
        double r241351 = 1.0;
        double r241352 = 2.0;
        double r241353 = r241351 / r241352;
        double r241354 = z;
        double r241355 = sqrt(r241354);
        double r241356 = y;
        double r241357 = x;
        double r241358 = fma(r241355, r241356, r241357);
        double r241359 = r241353 * r241358;
        return r241359;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019351 +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)))))