Average Error: 0.1 → 0.1
Time: 15.1s
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 r192522 = 1.0;
        double r192523 = 2.0;
        double r192524 = r192522 / r192523;
        double r192525 = x;
        double r192526 = y;
        double r192527 = z;
        double r192528 = sqrt(r192527);
        double r192529 = r192526 * r192528;
        double r192530 = r192525 + r192529;
        double r192531 = r192524 * r192530;
        return r192531;
}
↓
double f(double x, double y, double z) {
        double r192532 = 1.0;
        double r192533 = 2.0;
        double r192534 = r192532 / r192533;
        double r192535 = z;
        double r192536 = sqrt(r192535);
        double r192537 = y;
        double r192538 = x;
        double r192539 = fma(r192536, r192537, r192538);
        double r192540 = r192534 * r192539;
        return r192540;
}

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 2019303 +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)))))