Average Error: 7.6 → 1.2
Time: 3.8s
Precision: binary64
\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]
\[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.194 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, {x}^{3} \cdot 10, \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left({\varepsilon}^{3} \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.82064444690524 \cdot 10^{-37}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)\\ \end{array} \]
(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 5.0) (pow x 5.0)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= x -1.194e-57)
   (fma
    (* eps eps)
    (* (pow x 3.0) 10.0)
    (fma
     (* eps 5.0)
     (pow x 4.0)
     (* x (* (pow eps 3.0) (fma 10.0 x (* eps 5.0))))))
   (if (<= x 1.82064444690524e-37)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (* (pow x 3.0) (* eps (fma 10.0 eps (* x 5.0)))))))
double code(double x, double eps) {
	return pow((x + eps), 5.0) - pow(x, 5.0);
}
↓
double code(double x, double eps) {
	double tmp;
	if (x <= -1.194e-57) {
		tmp = fma((eps * eps), (pow(x, 3.0) * 10.0), fma((eps * 5.0), pow(x, 4.0), (x * (pow(eps, 3.0) * fma(10.0, x, (eps * 5.0))))));
	} else if (x <= 1.82064444690524e-37) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = pow(x, 3.0) * (eps * fma(10.0, eps, (x * 5.0)));
	}
	return tmp;
}

function code(x, eps)
	return Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
end
↓
function code(x, eps)
	tmp = 0.0
	if (x <= -1.194e-57)
		tmp = fma(Float64(eps * eps), Float64((x ^ 3.0) * 10.0), fma(Float64(eps * 5.0), (x ^ 4.0), Float64(x * Float64((eps ^ 3.0) * fma(10.0, x, Float64(eps * 5.0))))));
	elseif (x <= 1.82064444690524e-37)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64((x ^ 3.0) * Float64(eps * fma(10.0, eps, Float64(x * 5.0))));
	end
	return tmp
end

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]
↓
code[x_, eps_] := If[LessEqual[x, -1.194e-57], N[(N[(eps * eps), $MachinePrecision] * N[(N[Power[x, 3.0], $MachinePrecision] * 10.0), $MachinePrecision] + N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision] + N[(x * N[(N[Power[eps, 3.0], $MachinePrecision] * N[(10.0 * x + N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.82064444690524e-37], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 3.0], $MachinePrecision] * N[(eps * N[(10.0 * eps + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

{\left(x + \varepsilon\right)}^{5} - {x}^{5}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.194 \cdot 10^{-57}:\\
\;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, {x}^{3} \cdot 10, \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left({\varepsilon}^{3} \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)\right)\right)\\

\mathbf{elif}\;x \leq 1.82064444690524 \cdot 10^{-37}:\\
\;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\

\mathbf{else}:\\
\;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)\\


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if x < -1.1940000000000001e-57

    1. Initial program 36.5

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

    2. Taylor expanded in x around inf 4.9

      \[\leadsto \color{blue}{\left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(\left({\varepsilon}^{4} + 4 \cdot {\varepsilon}^{4}\right) \cdot x + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)\right)} \]

    3. Simplified4.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, {x}^{3} \cdot 10, \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left(\left({\varepsilon}^{3} \cdot 10\right) \cdot x - {\varepsilon}^{4} \cdot -5\right)\right)\right)} \]

    4. Taylor expanded in x around 0 4.9

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, {x}^{3} \cdot 10, \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 5 \cdot \left({\varepsilon}^{4} \cdot x\right)}\right)\right) \]

    5. Simplified4.9

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, {x}^{3} \cdot 10, \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{x \cdot \left({\varepsilon}^{3} \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)}\right)\right) \]

    if -1.1940000000000001e-57 < x < 1.82064444690524e-37

    1. Initial program 0.5

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

    if 1.82064444690524e-37 < x

    1. Initial program 43.3

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

    2. Taylor expanded in x around inf 3.5

      \[\leadsto \color{blue}{\left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}} \]

    3. Simplified3.5

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \left({x}^{3} \cdot 10\right) \cdot \varepsilon\right)} \]

    4. Taylor expanded in eps around 0 3.6

      \[\leadsto \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]

    5. Simplified3.6

      \[\leadsto \color{blue}{\varepsilon \cdot \left({x}^{3} \cdot \mathsf{fma}\left(5, x, \varepsilon \cdot 10\right)\right)} \]

    6. Taylor expanded in eps around 0 3.6

      \[\leadsto \color{blue}{10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]

    7. Simplified3.6

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(\varepsilon \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.194 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, {x}^{3} \cdot 10, \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, x \cdot \left({\varepsilon}^{3} \cdot \mathsf{fma}\left(10, x, \varepsilon \cdot 5\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.82064444690524 \cdot 10^{-37}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot \left(\varepsilon \cdot \mathsf{fma}\left(10, \varepsilon, x \cdot 5\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022190 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=5"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 5.0) (pow x 5.0)))