Average Error: 0.1 → 0.1
Time: 14.7s
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 r11585475 = 1.0;
        double r11585476 = 2.0;
        double r11585477 = r11585475 / r11585476;
        double r11585478 = x;
        double r11585479 = y;
        double r11585480 = z;
        double r11585481 = sqrt(r11585480);
        double r11585482 = r11585479 * r11585481;
        double r11585483 = r11585478 + r11585482;
        double r11585484 = r11585477 * r11585483;
        return r11585484;
}
↓
double f(double x, double y, double z) {
        double r11585485 = y;
        double r11585486 = z;
        double r11585487 = sqrt(r11585486);
        double r11585488 = x;
        double r11585489 = fma(r11585485, r11585487, r11585488);
        double r11585490 = 1.0;
        double r11585491 = r11585489 * r11585490;
        double r11585492 = 2.0;
        double r11585493 = r11585491 / r11585492;
        return r11585493;
}

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 2019169 +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)))))