Average Error: 0.0 → 0.1
Time: 1.0s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot z\]
↓
\[\mathsf{fma}\left(x, x, -y \cdot \left(4 \cdot z\right)\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot z
↓
\mathsf{fma}\left(x, x, -y \cdot \left(4 \cdot z\right)\right)
double f(double x, double y, double z) {
        double r181609 = x;
        double r181610 = r181609 * r181609;
        double r181611 = y;
        double r181612 = 4.0;
        double r181613 = r181611 * r181612;
        double r181614 = z;
        double r181615 = r181613 * r181614;
        double r181616 = r181610 - r181615;
        return r181616;
}
↓
double f(double x, double y, double z) {
        double r181617 = x;
        double r181618 = y;
        double r181619 = 4.0;
        double r181620 = z;
        double r181621 = r181619 * r181620;
        double r181622 = r181618 * r181621;
        double r181623 = -r181622;
        double r181624 = fma(r181617, r181617, r181623);
        return r181624;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

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

  3. Applied associate-*l*0.1

    \[\leadsto x \cdot x - \color{blue}{y \cdot \left(4 \cdot z\right)}\]
  4. Using strategy rm

  5. Applied fma-neg0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4) z)))