Average Error: 16.6 → 0.0
Time: 2.5s
Precision: binary64
\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]
\[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
↓
\[\mathsf{fma}\left(\varepsilon, \varepsilon, \varepsilon \cdot \left(x \cdot 2\right)\right) \]
(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 2.0) (pow x 2.0)))
↓
(FPCore (x eps) :precision binary64 (fma eps eps (* eps (* x 2.0))))
double code(double x, double eps) {
	return pow((x + eps), 2.0) - pow(x, 2.0);
}
↓
double code(double x, double eps) {
	return fma(eps, eps, (eps * (x * 2.0)));
}

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

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

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

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 16.6

    \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]

  2. Simplified0.0

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

  3. Taylor expanded in eps around 0 0.0

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

  4. Applied egg-rr0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022170 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=2"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 2.0) (pow x 2.0)))