Average Error: 0.1 → 0.1
Time: 5.3s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot z\]
↓
\[\mathsf{fma}\left(x, x, -z \cdot \left(y \cdot 4\right)\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot z
↓
\mathsf{fma}\left(x, x, -z \cdot \left(y \cdot 4\right)\right)
double f(double x, double y, double z) {
        double r110472 = x;
        double r110473 = r110472 * r110472;
        double r110474 = y;
        double r110475 = 4.0;
        double r110476 = r110474 * r110475;
        double r110477 = z;
        double r110478 = r110476 * r110477;
        double r110479 = r110473 - r110478;
        return r110479;
}
↓
double f(double x, double y, double z) {
        double r110480 = x;
        double r110481 = z;
        double r110482 = y;
        double r110483 = 4.0;
        double r110484 = r110482 * r110483;
        double r110485 = r110481 * r110484;
        double r110486 = -r110485;
        double r110487 = fma(r110480, r110480, r110486);
        return r110487;
}

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. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{-z \cdot \left(y \cdot 4\right)}\right)\]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, x, -z \cdot \left(y \cdot 4\right)\right)\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4) z)))