Average Error: 0.1 → 0.1
Time: 11.7s
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 r213542 = 1.0;
        double r213543 = 2.0;
        double r213544 = r213542 / r213543;
        double r213545 = x;
        double r213546 = y;
        double r213547 = z;
        double r213548 = sqrt(r213547);
        double r213549 = r213546 * r213548;
        double r213550 = r213545 + r213549;
        double r213551 = r213544 * r213550;
        return r213551;
}
↓
double f(double x, double y, double z) {
        double r213552 = 1.0;
        double r213553 = 2.0;
        double r213554 = r213552 / r213553;
        double r213555 = z;
        double r213556 = sqrt(r213555);
        double r213557 = y;
        double r213558 = x;
        double r213559 = fma(r213556, r213557, r213558);
        double r213560 = r213554 * r213559;
        return r213560;
}

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 2019212 +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)))))