Average Error: 0.1 → 0.1
Time: 1.1s
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 r137187 = x;
        double r137188 = r137187 * r137187;
        double r137189 = y;
        double r137190 = 4.0;
        double r137191 = r137189 * r137190;
        double r137192 = z;
        double r137193 = r137191 * r137192;
        double r137194 = r137188 - r137193;
        return r137194;
}
↓
double f(double x, double y, double z) {
        double r137195 = x;
        double r137196 = y;
        double r137197 = 4.0;
        double r137198 = r137196 * r137197;
        double r137199 = z;
        double r137200 = r137198 * r137199;
        double r137201 = -r137200;
        double r137202 = fma(r137195, r137195, r137201);
        return r137202;
}

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 2020001 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4) z)))