Average Error: 0.1 → 0.1
Time: 6.4s
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\frac{\mathsf{fma}\left(\sqrt{z}, y, x\right) \cdot 1}{2}\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\frac{\mathsf{fma}\left(\sqrt{z}, y, x\right) \cdot 1}{2}
double f(double x, double y, double z) {
        double r225508 = 1.0;
        double r225509 = 2.0;
        double r225510 = r225508 / r225509;
        double r225511 = x;
        double r225512 = y;
        double r225513 = z;
        double r225514 = sqrt(r225513);
        double r225515 = r225512 * r225514;
        double r225516 = r225511 + r225515;
        double r225517 = r225510 * r225516;
        return r225517;
}
↓
double f(double x, double y, double z) {
        double r225518 = z;
        double r225519 = sqrt(r225518);
        double r225520 = y;
        double r225521 = x;
        double r225522 = fma(r225519, r225520, r225521);
        double r225523 = 1.0;
        double r225524 = r225522 * r225523;
        double r225525 = 2.0;
        double r225526 = r225524 / r225525;
        return r225526;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2020060 +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)))))