Average Error: 0.1 → 0.1
Time: 9.0s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot z\]
↓
\[\mathsf{fma}\left(x, x, -z \cdot \left(y \cdot 4\right)\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot z
↓
\mathsf{fma}\left(x, x, -z \cdot \left(y \cdot 4\right)\right)
double f(double x, double y, double z) {
        double r102643 = x;
        double r102644 = r102643 * r102643;
        double r102645 = y;
        double r102646 = 4.0;
        double r102647 = r102645 * r102646;
        double r102648 = z;
        double r102649 = r102647 * r102648;
        double r102650 = r102644 - r102649;
        return r102650;
}
↓
double f(double x, double y, double z) {
        double r102651 = x;
        double r102652 = z;
        double r102653 = y;
        double r102654 = 4.0;
        double r102655 = r102653 * r102654;
        double r102656 = r102652 * r102655;
        double r102657 = -r102656;
        double r102658 = fma(r102651, r102651, r102657);
        return r102658;
}

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. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  (- (* x x) (* (* y 4.0) z)))