Average Error: 0.1 → 0.1
Time: 7.0s
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 r215577 = 1.0;
        double r215578 = 2.0;
        double r215579 = r215577 / r215578;
        double r215580 = x;
        double r215581 = y;
        double r215582 = z;
        double r215583 = sqrt(r215582);
        double r215584 = r215581 * r215583;
        double r215585 = r215580 + r215584;
        double r215586 = r215579 * r215585;
        return r215586;
}
↓
double f(double x, double y, double z) {
        double r215587 = z;
        double r215588 = sqrt(r215587);
        double r215589 = y;
        double r215590 = x;
        double r215591 = fma(r215588, r215589, r215590);
        double r215592 = 1.0;
        double r215593 = r215591 * r215592;
        double r215594 = 2.0;
        double r215595 = r215593 / r215594;
        return r215595;
}

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 2020033 +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)))))