Average Error: 0.0 → 0.0
Time: 997.0ms
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 r128322 = x;
        double r128323 = y;
        double r128324 = z;
        double r128325 = r128324 + r128322;
        double r128326 = r128323 * r128325;
        double r128327 = r128322 + r128326;
        return r128327;
}
↓
double f(double x, double y, double z) {
        double r128328 = y;
        double r128329 = z;
        double r128330 = x;
        double r128331 = fma(r128328, r128330, r128330);
        double r128332 = fma(r128328, r128329, r128331);
        return r128332;
}

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 2020021 +o rules:numerics
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))