Average Error: 0.0 → 0.0
Time: 827.0ms
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 r122189 = x;
        double r122190 = y;
        double r122191 = r122189 * r122190;
        double r122192 = r122191 + r122189;
        double r122193 = r122192 + r122190;
        return r122193;
}
↓
double f(double x, double y) {
        double r122194 = x;
        double r122195 = 1.0;
        double r122196 = r122194 + r122195;
        double r122197 = y;
        double r122198 = fma(r122196, r122197, r122194);
        return r122198;
}

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. Using strategy rm

  4. Applied *-un-lft-identity0.0

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

  5. Applied *-un-lft-identity0.0

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

  6. Applied distribute-lft-out0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))