Average Error: 0.2 → 0.1
Time: 6.6s
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 r205575 = 1.0;
        double r205576 = 2.0;
        double r205577 = r205575 / r205576;
        double r205578 = x;
        double r205579 = y;
        double r205580 = z;
        double r205581 = sqrt(r205580);
        double r205582 = r205579 * r205581;
        double r205583 = r205578 + r205582;
        double r205584 = r205577 * r205583;
        return r205584;
}
↓
double f(double x, double y, double z) {
        double r205585 = z;
        double r205586 = sqrt(r205585);
        double r205587 = y;
        double r205588 = x;
        double r205589 = fma(r205586, r205587, r205588);
        double r205590 = 1.0;
        double r205591 = r205589 * r205590;
        double r205592 = 2.0;
        double r205593 = r205591 / r205592;
        return r205593;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]

  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 2020047 +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)))))