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

Average Accuracy: 88.5% → 98.9%

Time: 11.3s

Precision: binary64

Cost: 40264

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(10 \cdot {x}^{3}\right) + 5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-305)
     (+ (pow eps 5.0) (* x (* 5.0 (pow eps 4.0))))
     (if (<= t_0 0.0)
       (* eps (+ (* eps (* 10.0 (pow x 3.0))) (* 5.0 (pow x 4.0))))
       t_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 t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-305) {
		tmp = pow(eps, 5.0) + (x * (5.0 * pow(eps, 4.0)));
	} else if (t_0 <= 0.0) {
		tmp = eps * ((eps * (10.0 * pow(x, 3.0))) + (5.0 * pow(x, 4.0)));
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-5d-305)) then
        tmp = (eps ** 5.0d0) + (x * (5.0d0 * (eps ** 4.0d0)))
    else if (t_0 <= 0.0d0) then
        tmp = eps * ((eps * (10.0d0 * (x ** 3.0d0))) + (5.0d0 * (x ** 4.0d0)))
    else
        tmp = t_0
    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 t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-305) {
		tmp = Math.pow(eps, 5.0) + (x * (5.0 * Math.pow(eps, 4.0)));
	} else if (t_0 <= 0.0) {
		tmp = eps * ((eps * (10.0 * Math.pow(x, 3.0))) + (5.0 * Math.pow(x, 4.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -5e-305:
		tmp = math.pow(eps, 5.0) + (x * (5.0 * math.pow(eps, 4.0)))
	elif t_0 <= 0.0:
		tmp = eps * ((eps * (10.0 * math.pow(x, 3.0))) + (5.0 * math.pow(x, 4.0)))
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-305)
		tmp = Float64((eps ^ 5.0) + Float64(x * Float64(5.0 * (eps ^ 4.0))));
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64(Float64(eps * Float64(10.0 * (x ^ 3.0))) + Float64(5.0 * (x ^ 4.0))));
	else
		tmp = t_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)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -5e-305)
		tmp = (eps ^ 5.0) + (x * (5.0 * (eps ^ 4.0)));
	elseif (t_0 <= 0.0)
		tmp = eps * ((eps * (10.0 * (x ^ 3.0))) + (5.0 * (x ^ 4.0)));
	else
		tmp = t_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_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-305], N[(N[Power[eps, 5.0], $MachinePrecision] + N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(N[(eps * N[(10.0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -4.99999999999999985e-305

   1. Initial program 97.6%

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

   2. Taylor expanded in x around 0 93.2%

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

   3. Simplified 93.2%

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

      [Start] 93.2	\[ {\varepsilon}^{5} + \left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}\right) \cdot x \]
      +-commutative [<=] 93.2	\[ {\varepsilon}^{5} + \color{blue}{\left({\varepsilon}^{4} + 4 \cdot {\varepsilon}^{4}\right)} \cdot x \]
      *-commutative [=>] 93.2	\[ {\varepsilon}^{5} + \color{blue}{x \cdot \left({\varepsilon}^{4} + 4 \cdot {\varepsilon}^{4}\right)} \]
      distribute-rgt1-in [=>] 93.2	\[ {\varepsilon}^{5} + x \cdot \color{blue}{\left(\left(4 + 1\right) \cdot {\varepsilon}^{4}\right)} \]
      metadata-eval [=>] 93.2	\[ {\varepsilon}^{5} + x \cdot \left(\color{blue}{5} \cdot {\varepsilon}^{4}\right) \]

   if -4.99999999999999985e-305 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 86.4%

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

   2. Taylor expanded in eps around 0 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot 6\right)\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right)} \]
      Proof

      [Start] 99.7	\[ {\varepsilon}^{2} \cdot \left(4 \cdot {x}^{3} + \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right) \cdot x\right) + \varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right) \]
      fma-def [=>] 99.7	\[ \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, 4 \cdot {x}^{3} + \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right) \cdot x, \varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)\right)} \]
      unpow2 [=>] 99.7	\[ \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, 4 \cdot {x}^{3} + \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right) \cdot x, \varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)\right) \]
      fma-def [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left(4, {x}^{3}, \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right) \cdot x\right)}, \varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)\right) \]
      *-commutative [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, \color{blue}{x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}\right), \varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)\right) \]
      distribute-rgt-out [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \color{blue}{\left({x}^{2} \cdot \left(2 + 4\right)\right)}\right), \varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)\right) \]
      unpow2 [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(2 + 4\right)\right)\right), \varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)\right) \]
      metadata-eval [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{6}\right)\right), \varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)\right) \]
      distribute-lft1-in [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot 6\right)\right), \varepsilon \cdot \color{blue}{\left(\left(4 + 1\right) \cdot {x}^{4}\right)}\right) \]
      metadata-eval [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot 6\right)\right), \varepsilon \cdot \left(\color{blue}{5} \cdot {x}^{4}\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot 6 + 4 \cdot x\right)}, \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      Proof

      [Start] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot 6\right)\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      fma-udef [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{4 \cdot {x}^{3} + x \cdot \left(\left(x \cdot x\right) \cdot 6\right)}, \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      +-commutative [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot 6\right) + 4 \cdot {x}^{3}}, \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      *-commutative [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot \color{blue}{\left(6 \cdot \left(x \cdot x\right)\right)} + 4 \cdot {x}^{3}, \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      associate-*r* [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\left(x \cdot 6\right) \cdot \left(x \cdot x\right)} + 4 \cdot {x}^{3}, \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      cube-mult [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \left(x \cdot 6\right) \cdot \left(x \cdot x\right) + 4 \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}, \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      associate-*r* [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \left(x \cdot 6\right) \cdot \left(x \cdot x\right) + \color{blue}{\left(4 \cdot x\right) \cdot \left(x \cdot x\right)}, \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      distribute-rgt-out [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot 6 + 4 \cdot x\right)}, \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]

   5. Applied egg-rr 99.7%

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

      [Start] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \left(x \cdot x\right) \cdot \left(x \cdot 6 + 4 \cdot x\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      *-commutative [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \left(x \cdot x\right) \cdot \left(x \cdot 6 + \color{blue}{x \cdot 4}\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      distribute-lft-out [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(6 + 4\right)\right)}, \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      metadata-eval [=>] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{10}\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]

   6. Applied egg-rr 99.7%

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

      [Start] 99.7	\[ \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \left(x \cdot x\right) \cdot \left(x \cdot 10\right), \varepsilon \cdot \left(5 \cdot {x}^{4}\right)\right) \]
      fma-udef [=>] 99.7	\[ \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 10\right)\right) + \varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]
      associate-*l* [=>] 99.7	\[ \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 10\right)\right)\right)} + \varepsilon \cdot \left(5 \cdot {x}^{4}\right) \]
      distribute-lft-out [=>] 99.7	\[ \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot 10\right)\right) + 5 \cdot {x}^{4}\right)} \]
      associate-*r* [=>] 99.7	\[ \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 10\right)} + 5 \cdot {x}^{4}\right) \]
      *-commutative [=>] 99.7	\[ \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(10 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)} + 5 \cdot {x}^{4}\right) \]
      pow3 [=>] 99.7	\[ \varepsilon \cdot \left(\varepsilon \cdot \left(10 \cdot \color{blue}{{x}^{3}}\right) + 5 \cdot {x}^{4}\right) \]

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 97.7%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -5 \cdot 10^{-305}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(10 \cdot {x}^{3}\right) + 5 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	39881

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-305} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	98.9%
Cost	39880

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	97.7%
Cost	7684

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-61}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot 10\right)\right)\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-41}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot 10\right) + \left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	97.6%
Cost	6793

\[\begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{-61} \lor \neg \left(x \leq 7.5 \cdot 10^{-41}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot 10\right) + \left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 5
Accuracy	83.1%
Cost	1216

\[\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot 10\right) + \left(\varepsilon \cdot 5\right) \cdot \left(x \cdot x\right)\right) \]

Alternative 6
Accuracy	82.8%
Cost	704

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

Alternative 7
Accuracy	82.8%
Cost	704

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

Alternative 8
Accuracy	82.8%
Cost	704

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

Reproduce

herbie shell --seed 2023138 
(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)))