Average Error: 0.0 → 0.0
Time: 1.0s
Precision: binary64
\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]
\[\left(x \cdot y + x\right) + y \]
↓
\[x + \mathsf{fma}\left(x, y, y\right) \]
(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);
}

function code(x, y)
	return Float64(Float64(Float64(x * y) + x) + y)
end
↓
function code(x, y)
	return Float64(x + fma(x, y, y))
end

code[x_, y_] := N[(N[(N[(x * y), $MachinePrecision] + x), $MachinePrecision] + y), $MachinePrecision]
↓
code[x_, y_] := N[(x + N[(x * y + y), $MachinePrecision]), $MachinePrecision]

\left(x \cdot y + x\right) + y
↓
x + \mathsf{fma}\left(x, y, y\right)

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2022153 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))