Average Error: 0.1 → 0.1
Time: 1.5s
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 r164312 = x;
        double r164313 = r164312 * r164312;
        double r164314 = y;
        double r164315 = 4.0;
        double r164316 = r164314 * r164315;
        double r164317 = z;
        double r164318 = r164316 * r164317;
        double r164319 = r164313 - r164318;
        return r164319;
}
↓
double f(double x, double y, double z) {
        double r164320 = x;
        double r164321 = y;
        double r164322 = 4.0;
        double r164323 = r164321 * r164322;
        double r164324 = z;
        double r164325 = r164323 * r164324;
        double r164326 = -r164325;
        double r164327 = fma(r164320, r164320, r164326);
        return r164327;
}

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 2020035 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4) z)))