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 r128117 = x;
        double r128118 = r128117 * r128117;
        double r128119 = y;
        double r128120 = 4.0;
        double r128121 = r128119 * r128120;
        double r128122 = z;
        double r128123 = r128121 * r128122;
        double r128124 = r128118 - r128123;
        return r128124;
}
↓
double f(double x, double y, double z) {
        double r128125 = x;
        double r128126 = y;
        double r128127 = 4.0;
        double r128128 = r128126 * r128127;
        double r128129 = z;
        double r128130 = r128128 * r128129;
        double r128131 = -r128130;
        double r128132 = fma(r128125, r128125, r128131);
        return r128132;
}

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)))