Average Error: 0.0 → 0.0
Time: 653.0ms
Precision: 64
\[\left(x \cdot y + x\right) + y\]
↓
\[\mathsf{fma}\left(x + 1, y, x\right)\]
\left(x \cdot y + x\right) + y
↓
\mathsf{fma}\left(x + 1, y, x\right)
double f(double x, double y) {
        double r87508 = x;
        double r87509 = y;
        double r87510 = r87508 * r87509;
        double r87511 = r87510 + r87508;
        double r87512 = r87511 + r87509;
        return r87512;
}
↓
double f(double x, double y) {
        double r87513 = x;
        double r87514 = 1.0;
        double r87515 = r87513 + r87514;
        double r87516 = y;
        double r87517 = fma(r87515, r87516, r87513);
        return r87517;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x + 1, y, x\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x + 1, y, x\right)\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))