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

Percentage Accurate: 92.4% → 98.8%

Time: 12.3s

Precision: binary64

Cost: 13512

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} + 4 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-56}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot 10\right) \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.7e-48)
   (* eps (+ (pow x 4.0) (* 4.0 (pow x 4.0))))
   (if (<= x 1.3e-56)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (+
      (* x (* (* x 10.0) (* (+ x eps) (* eps eps))))
      (* (pow x 4.0) (* eps 5.0))))))

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 <= -4.7e-48) {
		tmp = eps * (pow(x, 4.0) + (4.0 * pow(x, 4.0)));
	} else if (x <= 1.3e-56) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = (x * ((x * 10.0) * ((x + eps) * (eps * eps)))) + (pow(x, 4.0) * (eps * 5.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-4.7d-48)) then
        tmp = eps * ((x ** 4.0d0) + (4.0d0 * (x ** 4.0d0)))
    else if (x <= 1.3d-56) then
        tmp = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    else
        tmp = (x * ((x * 10.0d0) * ((x + eps) * (eps * eps)))) + ((x ** 4.0d0) * (eps * 5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
}

↓

public static double code(double x, double eps) {
	double tmp;
	if (x <= -4.7e-48) {
		tmp = eps * (Math.pow(x, 4.0) + (4.0 * Math.pow(x, 4.0)));
	} else if (x <= 1.3e-56) {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	} else {
		tmp = (x * ((x * 10.0) * ((x + eps) * (eps * eps)))) + (Math.pow(x, 4.0) * (eps * 5.0));
	}
	return tmp;
}

Python

def code(x, eps):
	return math.pow((x + eps), 5.0) - math.pow(x, 5.0)

↓

def code(x, eps):
	tmp = 0
	if x <= -4.7e-48:
		tmp = eps * (math.pow(x, 4.0) + (4.0 * math.pow(x, 4.0)))
	elif x <= 1.3e-56:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	else:
		tmp = (x * ((x * 10.0) * ((x + eps) * (eps * eps)))) + (math.pow(x, 4.0) * (eps * 5.0))
	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 <= -4.7e-48)
		tmp = Float64(eps * Float64((x ^ 4.0) + Float64(4.0 * (x ^ 4.0))));
	elseif (x <= 1.3e-56)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64(Float64(x * Float64(Float64(x * 10.0) * Float64(Float64(x + eps) * Float64(eps * eps)))) + Float64((x ^ 4.0) * Float64(eps * 5.0)));
	end
	return tmp
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
end

↓

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4.7e-48)
		tmp = eps * ((x ^ 4.0) + (4.0 * (x ^ 4.0)));
	elseif (x <= 1.3e-56)
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	else
		tmp = (x * ((x * 10.0) * ((x + eps) * (eps * eps)))) + ((x ^ 4.0) * (eps * 5.0));
	end
	tmp_2 = 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, -4.7e-48], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] + N[(4.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e-56], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(x * 10.0), $MachinePrecision] * N[(N[(x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6999999999999998e-48

   1. Initial program 61.2%

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

   2. Taylor expanded in eps around 0 99.9%

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

   if -4.6999999999999998e-48 < x < 1.29999999999999998e-56

   1. Initial program 100.0%

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

   if 1.29999999999999998e-56 < x

   1. Initial program 48.4%

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

   2. Taylor expanded in x around inf 99.7%

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

   3. Simplified 99.7%

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

      [Start] 99.7%	\[ \left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(\left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      fma-def [=>] 99.7%	\[ \color{blue}{\mathsf{fma}\left(4 \cdot \varepsilon + \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right)} \]
      distribute-lft1-in [=>] 99.7%	\[ \mathsf{fma}\left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      metadata-eval [=>] 99.7%	\[ \mathsf{fma}\left(\color{blue}{5} \cdot \varepsilon, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      *-commutative [=>] 99.7%	\[ \mathsf{fma}\left(\color{blue}{\varepsilon \cdot 5}, {x}^{4}, \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3} + \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2}\right) \]
      +-commutative [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) \cdot {x}^{2} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}}\right) \]
      *-commutative [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right)} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}\right) \]
      *-commutative [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{3} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)}\right) \]
      unpow3 [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]
      unpow2 [<=] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right) \]
      associate-*l* [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, {x}^{2} \cdot \left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]
      distribute-lft-out [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{{x}^{2} \cdot \left(\left(\left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + 4 \cdot {\varepsilon}^{3}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)}\right) \]

   4. Taylor expanded in x around 0 99.7%

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

   5. Simplified 99.7%

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

      [Start] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right) + 10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)\right) \]
      +-commutative [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{10 \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)}\right) \]
      unpow2 [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, 10 \cdot \left({\varepsilon}^{3} \cdot \color{blue}{\left(x \cdot x\right)}\right) + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]
      associate-*r* [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right)} + 10 \cdot \left({\varepsilon}^{2} \cdot {x}^{3}\right)\right) \]
      unpow2 [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + 10 \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot {x}^{3}\right)\right) \]
      associate-*r* [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left(10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot {x}^{3}}\right) \]
      cube-mult [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \left(10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right) \]
      associate-*r* [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(10 \cdot {\varepsilon}^{3}\right) \cdot \left(x \cdot x\right) + \color{blue}{\left(\left(10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right) \cdot \left(x \cdot x\right)}\right) \]
      distribute-rgt-in [<=] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \color{blue}{\left(x \cdot x\right) \cdot \left(10 \cdot {\varepsilon}^{3} + \left(10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right)}\right) \]
      associate-*r* [<=] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(x \cdot x\right) \cdot \left(10 \cdot {\varepsilon}^{3} + \color{blue}{10 \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right)}\right)\right) \]
      unpow2 [<=] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(x \cdot x\right) \cdot \left(10 \cdot {\varepsilon}^{3} + 10 \cdot \left(\color{blue}{{\varepsilon}^{2}} \cdot x\right)\right)\right) \]
      distribute-lft-out [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(x \cdot x\right) \cdot \color{blue}{\left(10 \cdot \left({\varepsilon}^{3} + {\varepsilon}^{2} \cdot x\right)\right)}\right) \]
      unpow2 [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(x \cdot x\right) \cdot \left(10 \cdot \left({\varepsilon}^{3} + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot x\right)\right)\right) \]
      associate-*l* [=>] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(x \cdot x\right) \cdot \left(10 \cdot \left({\varepsilon}^{3} + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot x\right)}\right)\right)\right) \]

   6. Applied egg-rr 99.7%

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

      [Start] 99.7%	\[ \mathsf{fma}\left(\varepsilon \cdot 5, {x}^{4}, \left(x \cdot x\right) \cdot \left(10 \cdot \left({\varepsilon}^{3} + \varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right)\right) \]
      fma-udef [=>] 99.7%	\[ \color{blue}{\left(\varepsilon \cdot 5\right) \cdot {x}^{4} + \left(x \cdot x\right) \cdot \left(10 \cdot \left({\varepsilon}^{3} + \varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right)} \]
      *-commutative [=>] 99.7%	\[ \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5\right)} + \left(x \cdot x\right) \cdot \left(10 \cdot \left({\varepsilon}^{3} + \varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right) \]
      *-commutative [=>] 99.7%	\[ {x}^{4} \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} + \left(x \cdot x\right) \cdot \left(10 \cdot \left({\varepsilon}^{3} + \varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right) \]
      associate-*l* [=>] 99.7%	\[ {x}^{4} \cdot \left(5 \cdot \varepsilon\right) + \color{blue}{x \cdot \left(x \cdot \left(10 \cdot \left({\varepsilon}^{3} + \varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]
      associate-*r* [=>] 99.7%	\[ {x}^{4} \cdot \left(5 \cdot \varepsilon\right) + x \cdot \color{blue}{\left(\left(x \cdot 10\right) \cdot \left({\varepsilon}^{3} + \varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right)} \]
      unpow3 [=>] 99.7%	\[ {x}^{4} \cdot \left(5 \cdot \varepsilon\right) + x \cdot \left(\left(x \cdot 10\right) \cdot \left(\color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon} + \varepsilon \cdot \left(\varepsilon \cdot x\right)\right)\right) \]
      associate-*r* [=>] 99.7%	\[ {x}^{4} \cdot \left(5 \cdot \varepsilon\right) + x \cdot \left(\left(x \cdot 10\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon + \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot x}\right)\right) \]
      distribute-lft-out [=>] 99.7%	\[ {x}^{4} \cdot \left(5 \cdot \varepsilon\right) + x \cdot \left(\left(x \cdot 10\right) \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} + 4 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-56}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot 10\right) \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.8%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} + 4 \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-56}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot 10\right) \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \end{array} \]

Alternative 2
Accuracy	98.8%
Cost	13512

\[\begin{array}{l} t_0 := {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{-57}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot 10\right) \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + t_0\\ \end{array} \]

Alternative 3
Accuracy	98.5%
Cost	7944

\[\begin{array}{l} t_0 := {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-62}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot 10\right) \cdot \left(\left(x + \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + t_0\\ \end{array} \]

Alternative 4
Accuracy	98.4%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-49}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-61}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	98.4%
Cost	6793

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-50} \lor \neg \left(x \leq 10^{-61}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 6
Accuracy	86.1%
Cost	704

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

Reproduce

herbie shell --seed 2023263 
(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)))