Average Error: 0.1 → 0.1
Time: 16.5s
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 r149597 = 1.0;
        double r149598 = 2.0;
        double r149599 = r149597 / r149598;
        double r149600 = x;
        double r149601 = y;
        double r149602 = z;
        double r149603 = sqrt(r149602);
        double r149604 = r149601 * r149603;
        double r149605 = r149600 + r149604;
        double r149606 = r149599 * r149605;
        return r149606;
}
↓
double f(double x, double y, double z) {
        double r149607 = 1.0;
        double r149608 = 2.0;
        double r149609 = r149607 / r149608;
        double r149610 = z;
        double r149611 = sqrt(r149610);
        double r149612 = y;
        double r149613 = x;
        double r149614 = fma(r149611, r149612, r149613);
        double r149615 = r149609 * r149614;
        return r149615;
}

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 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)))))