Average Error: 38.8 → 0
Time: 1.8s
Precision: binary64
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
↓
\[\mathsf{fma}\left(x, x, x \cdot 2\right) \]
(FPCore (x) :precision binary64 (- (* (+ x 1.0) (+ x 1.0)) 1.0))
↓
(FPCore (x) :precision binary64 (fma x x (* x 2.0)))
double code(double x) {
	return ((x + 1.0) * (x + 1.0)) - 1.0;
}
↓
double code(double x) {
	return fma(x, x, (x * 2.0));
}

function code(x)
	return Float64(Float64(Float64(x + 1.0) * Float64(x + 1.0)) - 1.0)
end
↓
function code(x)
	return fma(x, x, Float64(x * 2.0))
end

code[x_] := N[(N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]
↓
code[x_] := N[(x * x + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]

\left(x + 1\right) \cdot \left(x + 1\right) - 1
↓
\mathsf{fma}\left(x, x, x \cdot 2\right)

Error

Bits error versus x

Derivation

  1. Initial program 38.8

    \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]

  2. Simplified0.0

    \[\leadsto \color{blue}{x \cdot \left(x + 2\right)} \]

  3. Taylor expanded in x around 0 0.0

    \[\leadsto \color{blue}{{x}^{2} + 2 \cdot x} \]

  4. Applied unpow2_binary640.0

    \[\leadsto \color{blue}{x \cdot x} + 2 \cdot x \]

  5. Applied fma-def_binary640

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 2 \cdot x\right)} \]

  6. Final simplification0

    \[\leadsto \mathsf{fma}\left(x, x, x \cdot 2\right) \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))