?

Average Error: 24.53% → 0.01%
Time: 32.4s
Precision: binary64

?

\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]
\[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
↓
\[\mathsf{fma}\left(2 \cdot \varepsilon, x, {\varepsilon}^{2}\right) \]
(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 2.0) (pow x 2.0)))
↓
(FPCore (x eps) :precision binary64 (fma (* 2.0 eps) x (pow eps 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((2.0 * eps), x, pow(eps, 2.0));
}

function code(x, eps)
	return Float64((Float64(x + eps) ^ 2.0) - (x ^ 2.0))
end
↓
function code(x, eps)
	return fma(Float64(2.0 * eps), x, (eps ^ 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[(N[(2.0 * eps), $MachinePrecision] * x + N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]

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

Error?

Derivation?

  1. Initial program 24.53

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

  2. Taylor expanded in x around 0 0.04

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

  3. Applied egg-rr0.01

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

Reproduce?

herbie shell --seed 2023136 
(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)))