Average Error: 0.0 → 0.0
Time: 777.0ms
Precision: 64
\[x + y \cdot \left(z + x\right)\]
↓
\[\mathsf{fma}\left(y, z, \mathsf{fma}\left(y, x, x\right)\right)\]
x + y \cdot \left(z + x\right)
↓
\mathsf{fma}\left(y, z, \mathsf{fma}\left(y, x, x\right)\right)
double f(double x, double y, double z) {
        double r175895 = x;
        double r175896 = y;
        double r175897 = z;
        double r175898 = r175897 + r175895;
        double r175899 = r175896 * r175898;
        double r175900 = r175895 + r175899;
        return r175900;
}
↓
double f(double x, double y, double z) {
        double r175901 = y;
        double r175902 = z;
        double r175903 = x;
        double r175904 = fma(r175901, r175903, r175903);
        double r175905 = fma(r175901, r175902, r175904);
        return r175905;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[x + y \cdot \left(z + x\right)\]

  2. Simplified0.0

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

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{z \cdot y + \left(x + x \cdot y\right)}\]

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))