Average Error: 0.2 → 0.2
Time: 18.1s
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 r298966 = 1.0;
        double r298967 = 2.0;
        double r298968 = r298966 / r298967;
        double r298969 = x;
        double r298970 = y;
        double r298971 = z;
        double r298972 = sqrt(r298971);
        double r298973 = r298970 * r298972;
        double r298974 = r298969 + r298973;
        double r298975 = r298968 * r298974;
        return r298975;
}
↓
double f(double x, double y, double z) {
        double r298976 = 1.0;
        double r298977 = 2.0;
        double r298978 = r298976 / r298977;
        double r298979 = z;
        double r298980 = sqrt(r298979);
        double r298981 = y;
        double r298982 = x;
        double r298983 = fma(r298980, r298981, r298982);
        double r298984 = r298978 * r298983;
        return r298984;
}

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. Simplified0.2

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{z}, y, x\right)}\]

  3. Final simplification0.2

    \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{z}, y, x\right)\]

Reproduce

herbie shell --seed 2020042 +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)))))