Average Error: 0.1 → 0.1
Time: 6.2s
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 r905 = 1.0;
        double r906 = 2.0;
        double r907 = r905 / r906;
        double r908 = x;
        double r909 = y;
        double r910 = z;
        double r911 = sqrt(r910);
        double r912 = r909 * r911;
        double r913 = r908 + r912;
        double r914 = r907 * r913;
        return r914;
}
↓
double f(double x, double y, double z) {
        double r915 = z;
        double r916 = sqrt(r915);
        double r917 = y;
        double r918 = x;
        double r919 = fma(r916, r917, r918);
        double r920 = 1.0;
        double r921 = r919 * r920;
        double r922 = 2.0;
        double r923 = r921 / r922;
        return r923;
}

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 2020025 +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)))))