Average Error: 0.1 → 0.1
Time: 17.1s
Precision: 64
\[\frac{1.0}{2.0} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\frac{\mathsf{fma}\left(y, \sqrt{z}, x\right) \cdot 1.0}{2.0}\]
\frac{1.0}{2.0} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\frac{\mathsf{fma}\left(y, \sqrt{z}, x\right) \cdot 1.0}{2.0}
double f(double x, double y, double z) {
        double r9179549 = 1.0;
        double r9179550 = 2.0;
        double r9179551 = r9179549 / r9179550;
        double r9179552 = x;
        double r9179553 = y;
        double r9179554 = z;
        double r9179555 = sqrt(r9179554);
        double r9179556 = r9179553 * r9179555;
        double r9179557 = r9179552 + r9179556;
        double r9179558 = r9179551 * r9179557;
        return r9179558;
}
↓
double f(double x, double y, double z) {
        double r9179559 = y;
        double r9179560 = z;
        double r9179561 = sqrt(r9179560);
        double r9179562 = x;
        double r9179563 = fma(r9179559, r9179561, r9179562);
        double r9179564 = 1.0;
        double r9179565 = r9179563 * r9179564;
        double r9179566 = 2.0;
        double r9179567 = r9179565 / r9179566;
        return r9179567;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[\frac{1.0}{2.0} \cdot \left(x + y \cdot \sqrt{z}\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{1.0 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right)}{2.0}}\]
  3. Using strategy rm

  4. Applied fma-udef0.1

    \[\leadsto \frac{1.0 \cdot \color{blue}{\left(y \cdot \sqrt{z} + x\right)}}{2.0}\]
  5. Using strategy rm

  6. Applied fma-def0.1

    \[\leadsto \frac{1.0 \cdot \color{blue}{\mathsf{fma}\left(y, \sqrt{z}, x\right)}}{2.0}\]

  7. Final simplification0.1

    \[\leadsto \frac{\mathsf{fma}\left(y, \sqrt{z}, x\right) \cdot 1.0}{2.0}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))