Average Error: 0.1 → 0.1
Time: 1.9s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot z\]
↓
\[\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot z\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot z
↓
\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot z\right)
double f(double x, double y, double z) {
        double r181608 = x;
        double r181609 = r181608 * r181608;
        double r181610 = y;
        double r181611 = 4.0;
        double r181612 = r181610 * r181611;
        double r181613 = z;
        double r181614 = r181612 * r181613;
        double r181615 = r181609 - r181614;
        return r181615;
}
↓
double f(double x, double y, double z) {
        double r181616 = x;
        double r181617 = y;
        double r181618 = 4.0;
        double r181619 = r181617 * r181618;
        double r181620 = z;
        double r181621 = r181619 * r181620;
        double r181622 = -r181621;
        double r181623 = fma(r181616, r181616, r181622);
        return r181623;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[x \cdot x - \left(y \cdot 4\right) \cdot z\]
  2. Using strategy rm

  3. Applied fma-neg0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot z\right)}\]

  4. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot z\right)\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4) z)))