Average Error: 0.1 → 0.1
Time: 20.8s
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\frac{\mathsf{fma}\left(y, \sqrt{z}, x\right) \cdot 1}{2}\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\frac{\mathsf{fma}\left(y, \sqrt{z}, x\right) \cdot 1}{2}
double f(double x, double y, double z) {
        double r6793208 = 1.0;
        double r6793209 = 2.0;
        double r6793210 = r6793208 / r6793209;
        double r6793211 = x;
        double r6793212 = y;
        double r6793213 = z;
        double r6793214 = sqrt(r6793213);
        double r6793215 = r6793212 * r6793214;
        double r6793216 = r6793211 + r6793215;
        double r6793217 = r6793210 * r6793216;
        return r6793217;
}
↓
double f(double x, double y, double z) {
        double r6793218 = y;
        double r6793219 = z;
        double r6793220 = sqrt(r6793219);
        double r6793221 = x;
        double r6793222 = fma(r6793218, r6793220, r6793221);
        double r6793223 = 1.0;
        double r6793224 = r6793222 * r6793223;
        double r6793225 = 2.0;
        double r6793226 = r6793224 / r6793225;
        return r6793226;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019170 +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)))))