Average Error: 0.1 → 0.1
Time: 986.0ms
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 r161191 = x;
        double r161192 = r161191 * r161191;
        double r161193 = y;
        double r161194 = 4.0;
        double r161195 = r161193 * r161194;
        double r161196 = z;
        double r161197 = r161195 * r161196;
        double r161198 = r161192 - r161197;
        return r161198;
}
↓
double f(double x, double y, double z) {
        double r161199 = x;
        double r161200 = y;
        double r161201 = 4.0;
        double r161202 = r161200 * r161201;
        double r161203 = z;
        double r161204 = r161202 * r161203;
        double r161205 = -r161204;
        double r161206 = fma(r161199, r161199, r161205);
        return r161206;
}

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 2020039 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4) z)))