Average Error: 0.0 → 0.0
Time: 1.1s
Precision: binary64
\[\left(x \cdot y + x\right) + y \]
↓
\[y + \mathsf{fma}\left(y, x, x\right) \]
(FPCore (x y) :precision binary64 (+ (+ (* x y) x) y))
↓
(FPCore (x y) :precision binary64 (+ y (fma y x x)))
double code(double x, double y) {
	return ((x * y) + x) + y;
}
↓
double code(double x, double y) {
	return y + fma(y, x, x);
}

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

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

\left(x \cdot y + x\right) + y
↓
y + \mathsf{fma}\left(y, x, x\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. Taylor expanded in x around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022150 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))