Average Error: 0.1 → 0.1
Time: 42.6s
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 r7885373 = 1.0;
        double r7885374 = 2.0;
        double r7885375 = r7885373 / r7885374;
        double r7885376 = x;
        double r7885377 = y;
        double r7885378 = z;
        double r7885379 = sqrt(r7885378);
        double r7885380 = r7885377 * r7885379;
        double r7885381 = r7885376 + r7885380;
        double r7885382 = r7885375 * r7885381;
        return r7885382;
}
↓
double f(double x, double y, double z) {
        double r7885383 = y;
        double r7885384 = z;
        double r7885385 = sqrt(r7885384);
        double r7885386 = x;
        double r7885387 = fma(r7885383, r7885385, r7885386);
        double r7885388 = 1.0;
        double r7885389 = r7885387 * r7885388;
        double r7885390 = 2.0;
        double r7885391 = r7885389 / r7885390;
        return r7885391;
}

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019165 +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)))))