Average Error: 0.0 → 0.0
Time: 22.6s
Precision: 64
\[\left(x \cdot y + x\right) + y\]
↓
\[y + \mathsf{fma}\left(y, x, x\right)\]
\left(x \cdot y + x\right) + y
↓
y + \mathsf{fma}\left(y, x, x\right)
double f(double x, double y) {
        double r2026338 = x;
        double r2026339 = y;
        double r2026340 = r2026338 * r2026339;
        double r2026341 = r2026340 + r2026338;
        double r2026342 = r2026341 + r2026339;
        return r2026342;
}
↓
double f(double x, double y) {
        double r2026343 = y;
        double r2026344 = x;
        double r2026345 = fma(r2026343, r2026344, r2026344);
        double r2026346 = r2026343 + r2026345;
        return r2026346;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  (+ (+ (* x y) x) y))