Average Error: 0.1 → 0.1
Time: 40.2s
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\frac{\mathsf{fma}\left(y, \sqrt{z}, x\right) \cdot 1}{2}\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\frac{\mathsf{fma}\left(y, \sqrt{z}, x\right) \cdot 1}{2}
double f(double x, double y, double z) {
        double r10281619 = 1.0;
        double r10281620 = 2.0;
        double r10281621 = r10281619 / r10281620;
        double r10281622 = x;
        double r10281623 = y;
        double r10281624 = z;
        double r10281625 = sqrt(r10281624);
        double r10281626 = r10281623 * r10281625;
        double r10281627 = r10281622 + r10281626;
        double r10281628 = r10281621 * r10281627;
        return r10281628;
}
↓
double f(double x, double y, double z) {
        double r10281629 = y;
        double r10281630 = z;
        double r10281631 = sqrt(r10281630);
        double r10281632 = x;
        double r10281633 = fma(r10281629, r10281631, r10281632);
        double r10281634 = 1.0;
        double r10281635 = r10281633 * r10281634;
        double r10281636 = 2.0;
        double r10281637 = r10281635 / r10281636;
        return r10281637;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))