ENA, Section 1.4, Exercise 4b, n=5

Percentage Accurate: 88.3% → 98.4%

Time: 8.3s

Precision: binary64

Cost: 20676

\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(x \cdot x\right) \cdot \left({\varepsilon}^{3} \cdot 10 + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-52}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x + \varepsilon\right)\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right)\\ \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 5.0) (pow x 5.0)))

↓

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.02e-51)
   (fma
    (* eps 5.0)
    (pow x 4.0)
    (* (* x x) (+ (* (pow eps 3.0) 10.0) (* x (* eps (* eps 10.0))))))
   (if (<= x 1.55e-52)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (fma
      (* x x)
      (* (* eps eps) (* 10.0 (+ x eps)))
      (* eps (* 5.0 (pow x 4.0)))))))

C

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 <= -1.02e-51) {
		tmp = fma((eps * 5.0), pow(x, 4.0), ((x * x) * ((pow(eps, 3.0) * 10.0) + (x * (eps * (eps * 10.0))))));
	} else if (x <= 1.55e-52) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = fma((x * x), ((eps * eps) * (10.0 * (x + eps))), (eps * (5.0 * pow(x, 4.0))));
	}
	return tmp;
}

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
end

↓

function code(x, eps)
	tmp = 0.0
	if (x <= -1.02e-51)
		tmp = fma(Float64(eps * 5.0), (x ^ 4.0), Float64(Float64(x * x) * Float64(Float64((eps ^ 3.0) * 10.0) + Float64(x * Float64(eps * Float64(eps * 10.0))))));
	elseif (x <= 1.55e-52)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = fma(Float64(x * x), Float64(Float64(eps * eps) * Float64(10.0 * Float64(x + eps))), Float64(eps * Float64(5.0 * (x ^ 4.0))));
	end
	return tmp
end

Wolfram

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, -1.02e-51], N[(N[(eps * 5.0), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(N[Power[eps, 3.0], $MachinePrecision] * 10.0), $MachinePrecision] + N[(x * N[(eps * N[(eps * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-52], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(10.0 * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.01999999999999998e-51

   1. Initial program 39.3%

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

   2. Taylor expanded in x around inf 99.8%

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

   3. Simplified 99.8%

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

      [Start] 99.8%	\[ \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      fma-def [=>] 99.8%	\[ \color{blue}{\mathsf{fma}\left(4 \cdot \varepsilon + \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)} \]
      distribute-lft1-in [=>] 99.8%	\[ \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      metadata-eval [=>] 99.8%	\[ \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      *-commutative [=>] 99.8%	\[ \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      +-commutative [=>] 99.8%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]
      *-commutative [=>] 99.8%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]
      *-commutative [=>] 99.8%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]
      unpow3 [=>] 99.8%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]
      unpow2 [<=] 99.8%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]
      associate-*l* [=>] 99.8%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]
      distribute-lft-out [=>] 99.8%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   if -1.01999999999999998e-51 < x < 1.5499999999999999e-52

   1. Initial program 99.7%

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

   if 1.5499999999999999e-52 < x

   1. Initial program 42.6%

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

   2. Taylor expanded in x around inf 95.7%

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

   3. Simplified 95.7%

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

      [Start] 95.7%	\[ \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      fma-def [=>] 95.7%	\[ \color{blue}{\mathsf{fma}\left(4 \cdot \varepsilon + \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)} \]
      distribute-lft1-in [=>] 95.7%	\[ \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      metadata-eval [=>] 95.7%	\[ \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      *-commutative [=>] 95.7%	\[ \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      +-commutative [=>] 95.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]
      *-commutative [=>] 95.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]
      *-commutative [=>] 95.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]
      unpow3 [=>] 95.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]
      unpow2 [<=] 95.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]
      associate-*l* [=>] 95.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]
      distribute-lft-out [=>] 95.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   4. Taylor expanded in eps around 0 95.6%

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

   5. Simplified 95.7%

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

      [Start] 95.6%	\[ 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + \left(5 \cdot \left(\varepsilon \cdot {x}^{4}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)\right) \]
      associate-*r* [=>] 95.6%	\[ \color{blue}{\left(10 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}} + \left(5 \cdot \left(\varepsilon \cdot {x}^{4}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)\right) \]
      *-commutative [=>] 95.6%	\[ \color{blue}{\left({\varepsilon}^{2} \cdot 10\right)} \cdot {x}^{3} + \left(5 \cdot \left(\varepsilon \cdot {x}^{4}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)\right) \]
      associate-*r* [<=] 95.6%	\[ \color{blue}{{\varepsilon}^{2} \cdot \left(10 \cdot {x}^{3}\right)} + \left(5 \cdot \left(\varepsilon \cdot {x}^{4}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)\right) \]
      unpow2 [=>] 95.6%	\[ \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(10 \cdot {x}^{3}\right) + \left(5 \cdot \left(\varepsilon \cdot {x}^{4}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)\right) \]
      associate-*r* [<=] 95.6%	\[ \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right)} + \left(5 \cdot \left(\varepsilon \cdot {x}^{4}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)\right) \]
      +-commutative [=>] 95.6%	\[ \varepsilon \cdot \left(\varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right) + \color{blue}{\left(10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\right)} \]
      associate-*r* [=>] 95.7%	\[ \varepsilon \cdot \left(\varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right) + \left(10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + \color{blue}{\left(5 \cdot \varepsilon\right) \cdot {x}^{4}}\right) \]
      *-commutative [<=] 95.7%	\[ \varepsilon \cdot \left(\varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right) + \left(10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + \color{blue}{\left(\varepsilon \cdot 5\right)} \cdot {x}^{4}\right) \]
      associate-*r* [<=] 95.7%	\[ \varepsilon \cdot \left(\varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right) + \left(10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)}\right) \]
      associate-+r+ [=>] 95.7%	\[ \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)\right) + \varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]

   6. Taylor expanded in eps around 0 95.7%

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

   7. Simplified 95.7%

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon + x\right) \cdot 10\right)}, \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      Step-by-step derivation

      [Start] 95.7%	\[ \mathsf{fma}\left(x \cdot x, 10 \cdot {\varepsilon}^{3} + 10 \cdot \left({\varepsilon}^{2} \cdot x\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      distribute-lft-out [=>] 95.7%	\[ \mathsf{fma}\left(x \cdot x, \color{blue}{10 \cdot \left({\varepsilon}^{3} + {\varepsilon}^{2} \cdot x\right)}, \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      unpow3 [=>] 95.7%	\[ \mathsf{fma}\left(x \cdot x, 10 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon} + {\varepsilon}^{2} \cdot x\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      unpow2 [=>] 95.7%	\[ \mathsf{fma}\left(x \cdot x, 10 \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot x\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      distribute-lft-in [<=] 95.7%	\[ \mathsf{fma}\left(x \cdot x, 10 \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right)}, \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      *-commutative [=>] 95.7%	\[ \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot 10}, \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      associate-*l* [=>] 95.7%	\[ \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon + x\right) \cdot 10\right)}, \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(x \cdot x\right) \cdot \left({\varepsilon}^{3} \cdot 10 + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-52}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x + \varepsilon\right)\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.4%
Cost	20676

\[\begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(x \cdot x\right) \cdot \left({\varepsilon}^{3} \cdot 10 + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-52}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x + \varepsilon\right)\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	39881

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-322} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \end{array} \]

Alternative 3
Accuracy	98.4%
Cost	14217

\[\begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-49} \lor \neg \left(x \leq 10^{-52}\right):\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(10 \cdot \left(x + \varepsilon\right)\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternative 4
Accuracy	98.3%
Cost	14024

\[\begin{array}{l} t_0 := 5 \cdot {x}^{4}\\ t_1 := 10 \cdot {x}^{3}\\ \mathbf{if}\;x \leq -7 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(t_0 + \varepsilon \cdot t_1\right)\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot t_0 + \left(\varepsilon \cdot \varepsilon\right) \cdot t_1\\ \end{array} \]

Alternative 5
Accuracy	98.2%
Cost	13897

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-53} \lor \neg \left(x \leq 1.1 \cdot 10^{-52}\right):\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternative 6
Accuracy	96.9%
Cost	7049

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-56} \lor \neg \left(x \leq 10^{-84}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 7
Accuracy	96.9%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-55}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-83}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 8
Accuracy	96.8%
Cost	6792

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-57}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-83}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot \left(x \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	82.5%
Cost	704

\[\varepsilon \cdot \left(x \cdot \left(x \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right) \]

Reproduce

herbie shell --seed 2023165 
(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)))