Average Error: 0.2 → 0.1
Time: 1.0m
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 r274998 = 1.0;
        double r274999 = 2.0;
        double r275000 = r274998 / r274999;
        double r275001 = x;
        double r275002 = y;
        double r275003 = z;
        double r275004 = sqrt(r275003);
        double r275005 = r275002 * r275004;
        double r275006 = r275001 + r275005;
        double r275007 = r275000 * r275006;
        return r275007;
}
↓
double f(double x, double y, double z) {
        double r275008 = 1.0;
        double r275009 = 2.0;
        double r275010 = r275008 / r275009;
        double r275011 = z;
        double r275012 = sqrt(r275011);
        double r275013 = y;
        double r275014 = x;
        double r275015 = fma(r275012, r275013, r275014);
        double r275016 = r275010 * r275015;
        return r275016;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]

  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 2020047 +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)))))