Average Error: 0.1 → 0.1
Time: 15.9s
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\mathsf{fma}\left(\sqrt{z}, y, x\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\mathsf{fma}\left(\sqrt{z}, y, x\right) \cdot \frac{1}{2}
double f(double x, double y, double z) {
        double r168421 = 1.0;
        double r168422 = 2.0;
        double r168423 = r168421 / r168422;
        double r168424 = x;
        double r168425 = y;
        double r168426 = z;
        double r168427 = sqrt(r168426);
        double r168428 = r168425 * r168427;
        double r168429 = r168424 + r168428;
        double r168430 = r168423 * r168429;
        return r168430;
}
↓
double f(double x, double y, double z) {
        double r168431 = z;
        double r168432 = sqrt(r168431);
        double r168433 = y;
        double r168434 = x;
        double r168435 = fma(r168432, r168433, r168434);
        double r168436 = 1.0;
        double r168437 = 2.0;
        double r168438 = r168436 / r168437;
        double r168439 = r168435 * r168438;
        return r168439;
}

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. Using strategy rm

  4. Applied *-un-lft-identity0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019323 +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)))))