Average Error: 0.0 → 0.0
Time: 1.1s
Precision: binary64
\[\left(x \cdot y + x\right) + y \]
↓
\[x + \mathsf{fma}\left(x, y, y\right) \]
\left(x \cdot y + x\right) + y
↓
x + \mathsf{fma}\left(x, y, y\right)

(FPCore (x y) :precision binary64 (+ (+ (* x y) x) y))
↓
(FPCore (x y) :precision binary64 (+ x (fma x y y)))
double code(double x, double y) {
	return ((x * y) + x) + y;
}
↓
double code(double x, double y) {
	return x + fma(x, y, 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. Applied fma-udef_binary640.0

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

  4. Applied associate-+r+_binary640.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022076 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))