Average Error: 0.2 → 0.2
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 r217199 = 1.0;
        double r217200 = 2.0;
        double r217201 = r217199 / r217200;
        double r217202 = x;
        double r217203 = y;
        double r217204 = z;
        double r217205 = sqrt(r217204);
        double r217206 = r217203 * r217205;
        double r217207 = r217202 + r217206;
        double r217208 = r217201 * r217207;
        return r217208;
}
↓
double f(double x, double y, double z) {
        double r217209 = z;
        double r217210 = sqrt(r217209);
        double r217211 = y;
        double r217212 = x;
        double r217213 = fma(r217210, r217211, r217212);
        double r217214 = 1.0;
        double r217215 = r217213 * r217214;
        double r217216 = 2.0;
        double r217217 = r217215 / r217216;
        return r217217;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2020046 +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)))))