Average Error: 0.0 → 0.0
Time: 1.9s
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 r133968 = x;
        double r133969 = y;
        double r133970 = z;
        double r133971 = r133970 + r133968;
        double r133972 = r133969 * r133971;
        double r133973 = r133968 + r133972;
        return r133973;
}
↓
double f(double x, double y, double z) {
        double r133974 = y;
        double r133975 = z;
        double r133976 = x;
        double r133977 = fma(r133974, r133976, r133976);
        double r133978 = fma(r133974, r133975, r133977);
        return r133978;
}

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 2020057 +o rules:numerics
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))