Average Error: 0.1 → 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 r239769 = 1.0;
        double r239770 = 2.0;
        double r239771 = r239769 / r239770;
        double r239772 = x;
        double r239773 = y;
        double r239774 = z;
        double r239775 = sqrt(r239774);
        double r239776 = r239773 * r239775;
        double r239777 = r239772 + r239776;
        double r239778 = r239771 * r239777;
        return r239778;
}
↓
double f(double x, double y, double z) {
        double r239779 = z;
        double r239780 = sqrt(r239779);
        double r239781 = y;
        double r239782 = x;
        double r239783 = fma(r239780, r239781, r239782);
        double r239784 = 1.0;
        double r239785 = r239783 * r239784;
        double r239786 = 2.0;
        double r239787 = r239785 / r239786;
        return r239787;
}

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 2020060 +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)))))