Average Error: 0.0 → 0.0
Time: 711.0ms
Precision: binary64
\[\left(x \cdot y + x\right) + y \]
↓
\[\mathsf{fma}\left(x, y, x + y\right) \]
\left(x \cdot y + x\right) + y
↓
\mathsf{fma}\left(x, y, x + y\right)

(FPCore (x y) :precision binary64 (+ (+ (* x y) x) y))
↓
(FPCore (x y) :precision binary64 (fma x y (+ x y)))
double code(double x, double y) {
	return ((x * y) + x) + y;
}
↓
double code(double x, double y) {
	return fma(x, y, (x + y));
}

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 in y around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022067 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))