Average Error: 0.1 → 0.0
Time: 1.0s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot z\]
↓
\[\mathsf{fma}\left(x, x, -y \cdot \left(4 \cdot z\right)\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot z
↓
\mathsf{fma}\left(x, x, -y \cdot \left(4 \cdot z\right)\right)
double f(double x, double y, double z) {
        double r146507 = x;
        double r146508 = r146507 * r146507;
        double r146509 = y;
        double r146510 = 4.0;
        double r146511 = r146509 * r146510;
        double r146512 = z;
        double r146513 = r146511 * r146512;
        double r146514 = r146508 - r146513;
        return r146514;
}
↓
double f(double x, double y, double z) {
        double r146515 = x;
        double r146516 = y;
        double r146517 = 4.0;
        double r146518 = z;
        double r146519 = r146517 * r146518;
        double r146520 = r146516 * r146519;
        double r146521 = -r146520;
        double r146522 = fma(r146515, r146515, r146521);
        return r146522;
}

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 associate-*l*0.0

    \[\leadsto x \cdot x - \color{blue}{y \cdot \left(4 \cdot z\right)}\]
  4. Using strategy rm

  5. Applied fma-neg0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rasterific.QuadraticFormula:discriminant from Rasterific-0.6.1"
  :precision binary64
  (- (* x x) (* (* y 4) z)))