Average Error: 0.1 → 0.1
Time: 6.8s
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 r249955 = 1.0;
        double r249956 = 2.0;
        double r249957 = r249955 / r249956;
        double r249958 = x;
        double r249959 = y;
        double r249960 = z;
        double r249961 = sqrt(r249960);
        double r249962 = r249959 * r249961;
        double r249963 = r249958 + r249962;
        double r249964 = r249957 * r249963;
        return r249964;
}
↓
double f(double x, double y, double z) {
        double r249965 = z;
        double r249966 = sqrt(r249965);
        double r249967 = y;
        double r249968 = x;
        double r249969 = fma(r249966, r249967, r249968);
        double r249970 = 1.0;
        double r249971 = r249969 * r249970;
        double r249972 = 2.0;
        double r249973 = r249971 / r249972;
        return r249973;
}

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 2019354 +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)))))