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 r143042 = x;
        double r143043 = r143042 * r143042;
        double r143044 = y;
        double r143045 = 4.0;
        double r143046 = r143044 * r143045;
        double r143047 = z;
        double r143048 = r143046 * r143047;
        double r143049 = r143043 - r143048;
        return r143049;
}
↓
double f(double x, double y, double z) {
        double r143050 = x;
        double r143051 = y;
        double r143052 = 4.0;
        double r143053 = r143051 * r143052;
        double r143054 = z;
        double r143055 = r143053 * r143054;
        double r143056 = -r143055;
        double r143057 = fma(r143050, r143050, r143056);
        return r143057;
}

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)))