Average Error: 0.0 → 0.0
Time: 949.0ms
Precision: 64
\[\left(x \cdot y + x\right) + y\]
↓
\[y + \mathsf{fma}\left(x, y, x\right)\]
\left(x \cdot y + x\right) + y
↓
y + \mathsf{fma}\left(x, y, x\right)
double f(double x, double y) {
        double r91196 = x;
        double r91197 = y;
        double r91198 = r91196 * r91197;
        double r91199 = r91198 + r91196;
        double r91200 = r91199 + r91197;
        return r91200;
}
↓
double f(double x, double y) {
        double r91201 = y;
        double r91202 = x;
        double r91203 = fma(r91202, r91201, r91202);
        double r91204 = r91201 + r91203;
        return r91204;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))