Average Error: 0.0 → 0.0
Time: 1.2s
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 r100959 = x;
        double r100960 = y;
        double r100961 = z;
        double r100962 = r100961 + r100959;
        double r100963 = r100960 * r100962;
        double r100964 = r100959 + r100963;
        return r100964;
}
↓
double f(double x, double y, double z) {
        double r100965 = y;
        double r100966 = z;
        double r100967 = x;
        double r100968 = fma(r100965, r100967, r100967);
        double r100969 = fma(r100965, r100966, r100968);
        return r100969;
}

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 2020036 +o rules:numerics
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))