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 r150097 = x;
        double r150098 = r150097 * r150097;
        double r150099 = y;
        double r150100 = 4.0;
        double r150101 = r150099 * r150100;
        double r150102 = z;
        double r150103 = r150101 * r150102;
        double r150104 = r150098 - r150103;
        return r150104;
}
↓
double f(double x, double y, double z) {
        double r150105 = x;
        double r150106 = y;
        double r150107 = 4.0;
        double r150108 = z;
        double r150109 = r150107 * r150108;
        double r150110 = r150106 * r150109;
        double r150111 = -r150110;
        double r150112 = fma(r150105, r150105, r150111);
        return r150112;
}

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)))