Average Error: 0.1 → 0.1
Time: 17.0s
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 r145302 = 1.0;
        double r145303 = 2.0;
        double r145304 = r145302 / r145303;
        double r145305 = x;
        double r145306 = y;
        double r145307 = z;
        double r145308 = sqrt(r145307);
        double r145309 = r145306 * r145308;
        double r145310 = r145305 + r145309;
        double r145311 = r145304 * r145310;
        return r145311;
}
↓
double f(double x, double y, double z) {
        double r145312 = 1.0;
        double r145313 = 2.0;
        double r145314 = r145312 / r145313;
        double r145315 = z;
        double r145316 = sqrt(r145315);
        double r145317 = y;
        double r145318 = x;
        double r145319 = fma(r145316, r145317, r145318);
        double r145320 = r145314 * r145319;
        return r145320;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019323 +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)))))