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

Percentage Accurate: 88.3% → 97.7%

Time: 9.2s

Alternatives: 8

Speedup: 1.8×

Specification

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

Math

\[\begin{array}{l} \\ {\left(x + \varepsilon\right)}^{5} - {x}^{5} \end{array} \]

FPCore

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

C

double code(double x, double eps) {
	return pow((x + eps), 5.0) - pow(x, 5.0);
}

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(x + \varepsilon\right)}^{5} - {x}^{5} \end{array} \]

FPCore

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

C

double code(double x, double eps) {
	return pow((x + eps), 5.0) - pow(x, 5.0);
}

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(\frac{\varepsilon \cdot 10}{x} + 5\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-69}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5 + \varepsilon \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(x \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.65e-48)
   (* eps (* (pow x 4.0) (+ (/ (* eps 10.0) x) 5.0)))
   (if (<= x 1.55e-69)
     (pow eps 5.0)
     (*
      eps
      (+
       (* (pow x 4.0) 5.0)
       (*
        eps
        (+ (* 10.0 (pow x 3.0)) (* eps (* x (+ (* eps 5.0) (* x 10.0)))))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.65e-48) {
		tmp = eps * (pow(x, 4.0) * (((eps * 10.0) / x) + 5.0));
	} else if (x <= 1.55e-69) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * ((pow(x, 4.0) * 5.0) + (eps * ((10.0 * pow(x, 3.0)) + (eps * (x * ((eps * 5.0) + (x * 10.0)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.65d-48)) then
        tmp = eps * ((x ** 4.0d0) * (((eps * 10.0d0) / x) + 5.0d0))
    else if (x <= 1.55d-69) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * (((x ** 4.0d0) * 5.0d0) + (eps * ((10.0d0 * (x ** 3.0d0)) + (eps * (x * ((eps * 5.0d0) + (x * 10.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.65e-48) {
		tmp = eps * (Math.pow(x, 4.0) * (((eps * 10.0) / x) + 5.0));
	} else if (x <= 1.55e-69) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * ((Math.pow(x, 4.0) * 5.0) + (eps * ((10.0 * Math.pow(x, 3.0)) + (eps * (x * ((eps * 5.0) + (x * 10.0)))))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.65e-48:
		tmp = eps * (math.pow(x, 4.0) * (((eps * 10.0) / x) + 5.0))
	elif x <= 1.55e-69:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * ((math.pow(x, 4.0) * 5.0) + (eps * ((10.0 * math.pow(x, 3.0)) + (eps * (x * ((eps * 5.0) + (x * 10.0)))))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.65e-48)
		tmp = Float64(eps * Float64((x ^ 4.0) * Float64(Float64(Float64(eps * 10.0) / x) + 5.0)));
	elseif (x <= 1.55e-69)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(Float64((x ^ 4.0) * 5.0) + Float64(eps * Float64(Float64(10.0 * (x ^ 3.0)) + Float64(eps * Float64(x * Float64(Float64(eps * 5.0) + Float64(x * 10.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.65e-48)
		tmp = eps * ((x ^ 4.0) * (((eps * 10.0) / x) + 5.0));
	elseif (x <= 1.55e-69)
		tmp = eps ^ 5.0;
	else
		tmp = eps * (((x ^ 4.0) * 5.0) + (eps * ((10.0 * (x ^ 3.0)) + (eps * (x * ((eps * 5.0) + (x * 10.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.65e-48], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(N[(eps * 10.0), $MachinePrecision] / x), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-69], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] + N[(eps * N[(N[(10.0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(x * N[(N[(eps * 5.0), $MachinePrecision] + N[(x * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.65e-48

   1. Initial program 28.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 99.5%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \color{blue}{\left(4 \cdot \varepsilon + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      2. associate-+r+ 99.5%

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      3. mul-1-neg 99.5%

         \[\leadsto {x}^{4} \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{\left(-\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      4. unsub-neg 99.5%

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      5. distribute-rgt1-in 99.5%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      6. metadata-eval 99.5%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      7. *-commutative 99.5%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{{\varepsilon}^{2} \cdot -10}{x}\right)} \]
   6. Step-by-step derivation

      1. associate-/l* 99.5%

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

      2. unpow2 99.5%

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

      3. associate-*l* 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in x around inf 99.5%

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

   10. Simplified 99.6%

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

   if -2.65e-48 < x < 1.55e-69

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

   if 1.55e-69 < x

   1. Initial program 42.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 97.5%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in x around 0 97.4%

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left({x}^{3} \cdot 10 + \varepsilon \cdot \color{blue}{\left(x \cdot \left(5 \cdot \varepsilon + 10 \cdot x\right)\right)}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(\frac{\varepsilon \cdot 10}{x} + 5\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-69}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5 + \varepsilon \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(x \cdot \left(\varepsilon \cdot 5 + x \cdot 10\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(\frac{\varepsilon \cdot 10}{x} + 5\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-69}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5 + \varepsilon \cdot \left(10 \cdot \left(\left(x \cdot x\right) \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.8e-48)
   (* eps (* (pow x 4.0) (+ (/ (* eps 10.0) x) 5.0)))
   (if (<= x 1.55e-69)
     (pow eps 5.0)
     (* eps (+ (* (pow x 4.0) 5.0) (* eps (* 10.0 (* (* x x) (+ x eps)))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.8e-48) {
		tmp = eps * (pow(x, 4.0) * (((eps * 10.0) / x) + 5.0));
	} else if (x <= 1.55e-69) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * ((pow(x, 4.0) * 5.0) + (eps * (10.0 * ((x * x) * (x + eps)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-3.8d-48)) then
        tmp = eps * ((x ** 4.0d0) * (((eps * 10.0d0) / x) + 5.0d0))
    else if (x <= 1.55d-69) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * (((x ** 4.0d0) * 5.0d0) + (eps * (10.0d0 * ((x * x) * (x + eps)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.8e-48) {
		tmp = eps * (Math.pow(x, 4.0) * (((eps * 10.0) / x) + 5.0));
	} else if (x <= 1.55e-69) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * ((Math.pow(x, 4.0) * 5.0) + (eps * (10.0 * ((x * x) * (x + eps)))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.8e-48:
		tmp = eps * (math.pow(x, 4.0) * (((eps * 10.0) / x) + 5.0))
	elif x <= 1.55e-69:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * ((math.pow(x, 4.0) * 5.0) + (eps * (10.0 * ((x * x) * (x + eps)))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.8e-48)
		tmp = Float64(eps * Float64((x ^ 4.0) * Float64(Float64(Float64(eps * 10.0) / x) + 5.0)));
	elseif (x <= 1.55e-69)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(Float64((x ^ 4.0) * 5.0) + Float64(eps * Float64(10.0 * Float64(Float64(x * x) * Float64(x + eps))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.8e-48)
		tmp = eps * ((x ^ 4.0) * (((eps * 10.0) / x) + 5.0));
	elseif (x <= 1.55e-69)
		tmp = eps ^ 5.0;
	else
		tmp = eps * (((x ^ 4.0) * 5.0) + (eps * (10.0 * ((x * x) * (x + eps)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.8e-48], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(N[(eps * 10.0), $MachinePrecision] / x), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-69], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] + N[(eps * N[(10.0 * N[(N[(x * x), $MachinePrecision] * N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.80000000000000002e-48

   1. Initial program 28.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 99.5%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \color{blue}{\left(4 \cdot \varepsilon + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      2. associate-+r+ 99.5%

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      3. mul-1-neg 99.5%

         \[\leadsto {x}^{4} \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{\left(-\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      4. unsub-neg 99.5%

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      5. distribute-rgt1-in 99.5%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      6. metadata-eval 99.5%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      7. *-commutative 99.5%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{{\varepsilon}^{2} \cdot -10}{x}\right)} \]
   6. Step-by-step derivation

      1. associate-/l* 99.5%

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

      2. unpow2 99.5%

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

      3. associate-*l* 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in x around inf 99.5%

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

   10. Simplified 99.6%

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

   if -3.80000000000000002e-48 < x < 1.55e-69

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

   if 1.55e-69 < x

   1. Initial program 42.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 97.5%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in eps around 0 96.8%

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \color{blue}{\varepsilon \cdot \left(10 \cdot \left(\varepsilon \cdot {x}^{2}\right) + 10 \cdot {x}^{3}\right)}\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 96.8%

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

      2. cube-mult 96.8%

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

      3. unpow2 96.8%

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

      4. distribute-rgt-out 96.8%

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

   8. Simplified 96.8%

      \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \color{blue}{\varepsilon \cdot \left(10 \cdot \left({x}^{2} \cdot \left(\varepsilon + x\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. unpow2 96.8%

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

   10. Applied egg-rr 96.8%

       \[\leadsto \varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\varepsilon + x\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-48}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(\frac{\varepsilon \cdot 10}{x} + 5\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-69}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5 + \varepsilon \cdot \left(10 \cdot \left(\left(x \cdot x\right) \cdot \left(x + \varepsilon\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-47} \lor \neg \left(x \leq 1.55 \cdot 10^{-69}\right):\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(\frac{\varepsilon \cdot 10}{x} + 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.4e-47) (not (<= x 1.55e-69)))
   (* eps (* (pow x 4.0) (+ (/ (* eps 10.0) x) 5.0)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.4e-47) || !(x <= 1.55e-69)) {
		tmp = eps * (pow(x, 4.0) * (((eps * 10.0) / x) + 5.0));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-1.4d-47)) .or. (.not. (x <= 1.55d-69))) then
        tmp = eps * ((x ** 4.0d0) * (((eps * 10.0d0) / x) + 5.0d0))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -1.4e-47) || !(x <= 1.55e-69)) {
		tmp = eps * (Math.pow(x, 4.0) * (((eps * 10.0) / x) + 5.0));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.4e-47) or not (x <= 1.55e-69):
		tmp = eps * (math.pow(x, 4.0) * (((eps * 10.0) / x) + 5.0))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.4e-47) || !(x <= 1.55e-69))
		tmp = Float64(eps * Float64((x ^ 4.0) * Float64(Float64(Float64(eps * 10.0) / x) + 5.0)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -1.4e-47) || ~((x <= 1.55e-69)))
		tmp = eps * ((x ^ 4.0) * (((eps * 10.0) / x) + 5.0));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.4e-47], N[Not[LessEqual[x, 1.55e-69]], $MachinePrecision]], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(N[(eps * 10.0), $MachinePrecision] / x), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.39999999999999996e-47 or 1.55e-69 < x

   1. Initial program 37.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 96.8%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 96.8%

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \color{blue}{\left(4 \cdot \varepsilon + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      2. associate-+r+ 96.8%

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) + -1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      3. mul-1-neg 96.8%

         \[\leadsto {x}^{4} \cdot \left(\left(\varepsilon + 4 \cdot \varepsilon\right) + \color{blue}{\left(-\frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}\right) \]

      4. unsub-neg 96.8%

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(\varepsilon + 4 \cdot \varepsilon\right) - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)} \]

      5. distribute-rgt1-in 96.8%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\left(4 + 1\right) \cdot \varepsilon} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      6. metadata-eval 96.8%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{5} \cdot \varepsilon - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

      7. *-commutative 96.8%

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{\varepsilon \cdot 5} - \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right) \]

   5. Simplified 96.8%

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon \cdot 5 - \frac{{\varepsilon}^{2} \cdot -10}{x}\right)} \]
   6. Step-by-step derivation

      1. associate-/l* 96.8%

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

      2. unpow2 96.8%

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

      3. associate-*l* 96.8%

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

   7. Applied egg-rr 96.8%

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

   8. Taylor expanded in x around inf 96.8%

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

   10. Simplified 97.0%

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

   if -1.39999999999999996e-47 < x < 1.55e-69

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-47} \lor \neg \left(x \leq 1.55 \cdot 10^{-69}\right):\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(\frac{\varepsilon \cdot 10}{x} + 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-48} \lor \neg \left(x \leq 1.55 \cdot 10^{-69}\right):\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -6.2e-48) (not (<= x 1.55e-69)))
   (* eps (* (pow x 4.0) 5.0))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -6.2e-48) || !(x <= 1.55e-69)) {
		tmp = eps * (pow(x, 4.0) * 5.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-6.2d-48)) .or. (.not. (x <= 1.55d-69))) then
        tmp = eps * ((x ** 4.0d0) * 5.0d0)
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -6.2e-48) || !(x <= 1.55e-69)) {
		tmp = eps * (Math.pow(x, 4.0) * 5.0);
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -6.2e-48) or not (x <= 1.55e-69):
		tmp = eps * (math.pow(x, 4.0) * 5.0)
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -6.2e-48) || !(x <= 1.55e-69))
		tmp = Float64(eps * Float64((x ^ 4.0) * 5.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -6.2e-48) || ~((x <= 1.55e-69)))
		tmp = eps * ((x ^ 4.0) * 5.0);
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -6.2e-48], N[Not[LessEqual[x, 1.55e-69]], $MachinePrecision]], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000033e-48 or 1.55e-69 < x

   1. Initial program 37.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 95.7%

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

      1. *-commutative 95.7%

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

      2. distribute-rgt1-in 95.7%

         \[\leadsto \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \cdot {x}^{4} \]

      3. metadata-eval 95.7%

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

      4. *-commutative 95.7%

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

      5. associate-*r* 95.9%

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

   5. Simplified 95.9%

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

   if -6.20000000000000033e-48 < x < 1.55e-69

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-48} \lor \neg \left(x \leq 1.55 \cdot 10^{-69}\right):\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-48}:\\ \;\;\;\;{x}^{5} \cdot \left(\frac{\varepsilon}{x} \cdot \left(--5\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-69}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.5e-48)
   (* (pow x 5.0) (* (/ eps x) (- -5.0)))
   (if (<= x 1.55e-69) (pow eps 5.0) (* eps (* (pow x 4.0) 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.5e-48) {
		tmp = pow(x, 5.0) * ((eps / x) * -(-5.0));
	} else if (x <= 1.55e-69) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * (pow(x, 4.0) * 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-3.5d-48)) then
        tmp = (x ** 5.0d0) * ((eps / x) * -(-5.0d0))
    else if (x <= 1.55d-69) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * ((x ** 4.0d0) * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.5e-48) {
		tmp = Math.pow(x, 5.0) * ((eps / x) * -(-5.0));
	} else if (x <= 1.55e-69) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * (Math.pow(x, 4.0) * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.5e-48:
		tmp = math.pow(x, 5.0) * ((eps / x) * -(-5.0))
	elif x <= 1.55e-69:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * (math.pow(x, 4.0) * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.5e-48)
		tmp = Float64((x ^ 5.0) * Float64(Float64(eps / x) * Float64(-(-5.0))));
	elseif (x <= 1.55e-69)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64((x ^ 4.0) * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.5e-48)
		tmp = (x ^ 5.0) * ((eps / x) * -(-5.0));
	elseif (x <= 1.55e-69)
		tmp = eps ^ 5.0;
	else
		tmp = eps * ((x ^ 4.0) * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.5e-48], N[(N[Power[x, 5.0], $MachinePrecision] * N[(N[(eps / x), $MachinePrecision] * (--5.0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-69], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.49999999999999991e-48

   1. Initial program 28.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sqr-pow 0.0%

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{\left(\frac{5}{2}\right)} \cdot {x}^{\left(\frac{5}{2}\right)}} \]

      2. metadata-eval 0.0%

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{\color{blue}{2.5}} \cdot {x}^{\left(\frac{5}{2}\right)} \]

      3. metadata-eval 0.0%

         \[\leadsto {\left(x + \varepsilon\right)}^{5} - {x}^{2.5} \cdot {x}^{\color{blue}{2.5}} \]

   4. Applied egg-rr 0.0%

      \[\leadsto {\left(x + \varepsilon\right)}^{5} - \color{blue}{{x}^{2.5} \cdot {x}^{2.5}} \]

   5. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{5} \cdot \left(-1 \cdot \frac{\varepsilon + 4 \cdot \varepsilon}{x} - \left(1 + {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 0.0%

         \[\leadsto \color{blue}{-{x}^{5} \cdot \left(-1 \cdot \frac{\varepsilon + 4 \cdot \varepsilon}{x} - \left(1 + {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      2. distribute-rgt-neg-in 0.0%

         \[\leadsto \color{blue}{{x}^{5} \cdot \left(-\left(-1 \cdot \frac{\varepsilon + 4 \cdot \varepsilon}{x} - \left(1 + {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)} \]

      3. sub-neg 0.0%

         \[\leadsto {x}^{5} \cdot \left(-\color{blue}{\left(-1 \cdot \frac{\varepsilon + 4 \cdot \varepsilon}{x} + \left(-\left(1 + {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)}\right) \]

      4. mul-1-neg 0.0%

         \[\leadsto {x}^{5} \cdot \left(-\left(\color{blue}{\left(-\frac{\varepsilon + 4 \cdot \varepsilon}{x}\right)} + \left(-\left(1 + {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)\right) \]

      5. distribute-rgt1-in 0.0%

         \[\leadsto {x}^{5} \cdot \left(-\left(\left(-\frac{\color{blue}{\left(4 + 1\right) \cdot \varepsilon}}{x}\right) + \left(-\left(1 + {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)\right) \]

      6. metadata-eval 0.0%

         \[\leadsto {x}^{5} \cdot \left(-\left(\left(-\frac{\color{blue}{5} \cdot \varepsilon}{x}\right) + \left(-\left(1 + {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)\right) \]

      7. associate-*r/ 0.0%

         \[\leadsto {x}^{5} \cdot \left(-\left(\left(-\color{blue}{5 \cdot \frac{\varepsilon}{x}}\right) + \left(-\left(1 + {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative 0.0%

         \[\leadsto {x}^{5} \cdot \left(-\left(\left(-\color{blue}{\frac{\varepsilon}{x} \cdot 5}\right) + \left(-\left(1 + {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)\right) \]

      9. distribute-rgt-neg-in 0.0%

         \[\leadsto {x}^{5} \cdot \left(-\left(\color{blue}{\frac{\varepsilon}{x} \cdot \left(-5\right)} + \left(-\left(1 + {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)\right) \]

      10. metadata-eval 0.0%

          \[\leadsto {x}^{5} \cdot \left(-\left(\frac{\varepsilon}{x} \cdot \color{blue}{-5} + \left(-\left(1 + {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)\right) \]

      11. unpow2 0.0%

          \[\leadsto {x}^{5} \cdot \left(-\left(\frac{\varepsilon}{x} \cdot -5 + \left(-\left(1 + \color{blue}{\sqrt{-1} \cdot \sqrt{-1}}\right)\right)\right)\right) \]

      12. rem-square-sqrt 98.8%

          \[\leadsto {x}^{5} \cdot \left(-\left(\frac{\varepsilon}{x} \cdot -5 + \left(-\left(1 + \color{blue}{-1}\right)\right)\right)\right) \]

      13. metadata-eval 98.8%

          \[\leadsto {x}^{5} \cdot \left(-\left(\frac{\varepsilon}{x} \cdot -5 + \left(-\color{blue}{0}\right)\right)\right) \]

      14. metadata-eval 98.8%

          \[\leadsto {x}^{5} \cdot \left(-\left(\frac{\varepsilon}{x} \cdot -5 + \color{blue}{0}\right)\right) \]

   7. Simplified 98.8%

      \[\leadsto \color{blue}{{x}^{5} \cdot \left(-\left(\frac{\varepsilon}{x} \cdot -5 + 0\right)\right)} \]

   if -3.49999999999999991e-48 < x < 1.55e-69

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

   if 1.55e-69 < x

   1. Initial program 42.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 94.0%

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

      1. *-commutative 94.0%

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

      2. distribute-rgt1-in 94.0%

         \[\leadsto \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \cdot {x}^{4} \]

      3. metadata-eval 94.0%

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

      4. *-commutative 94.0%

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

      5. associate-*r* 94.2%

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

   5. Simplified 94.2%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-48}:\\ \;\;\;\;{x}^{5} \cdot \left(\frac{\varepsilon}{x} \cdot \left(--5\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-69}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-48}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-69}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.6e-48)
   (* (* eps 5.0) (* (* x x) (* x x)))
   (if (<= x 1.55e-69) (pow eps 5.0) (* 5.0 (* eps (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.6e-48) {
		tmp = (eps * 5.0) * ((x * x) * (x * x));
	} else if (x <= 1.55e-69) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * pow(x, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.6d-48)) then
        tmp = (eps * 5.0d0) * ((x * x) * (x * x))
    else if (x <= 1.55d-69) then
        tmp = eps ** 5.0d0
    else
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.6e-48) {
		tmp = (eps * 5.0) * ((x * x) * (x * x));
	} else if (x <= 1.55e-69) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.6e-48:
		tmp = (eps * 5.0) * ((x * x) * (x * x))
	elif x <= 1.55e-69:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.6e-48)
		tmp = Float64(Float64(eps * 5.0) * Float64(Float64(x * x) * Float64(x * x)));
	elseif (x <= 1.55e-69)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.6e-48)
		tmp = (eps * 5.0) * ((x * x) * (x * x));
	elseif (x <= 1.55e-69)
		tmp = eps ^ 5.0;
	else
		tmp = 5.0 * (eps * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.6e-48], N[(N[(eps * 5.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-69], N[Power[eps, 5.0], $MachinePrecision], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5999999999999999e-48

   1. Initial program 28.6%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.6%

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

      1. distribute-rgt1-in 98.6%

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \]

      2. metadata-eval 98.6%

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

   5. Simplified 98.6%

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

      1. metadata-eval 98.6%

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

      2. pow-prod-up 98.6%

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

   7. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left(5 \cdot \varepsilon\right) \]
   8. Step-by-step derivation

      1. unpow2 99.5%

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

   9. Applied egg-rr 98.6%

      \[\leadsto \left({x}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \varepsilon\right) \]
   10. Step-by-step derivation

       1. unpow2 99.5%

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

   11. Applied egg-rr 98.6%

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

   if -1.5999999999999999e-48 < x < 1.55e-69

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

   if 1.55e-69 < x

   1. Initial program 42.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 94.0%

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

      1. distribute-rgt1-in 94.0%

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \]

      2. metadata-eval 94.0%

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

   5. Simplified 94.0%

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

   6. Taylor expanded in x around 0 94.1%

      \[\leadsto \color{blue}{5 \cdot \left(\varepsilon \cdot {x}^{4}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-48}:\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-69}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-48} \lor \neg \left(x \leq 1.55 \cdot 10^{-69}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.8e-48) (not (<= x 1.55e-69)))
   (* (* eps 5.0) (* (* x x) (* x x)))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.8e-48) || !(x <= 1.55e-69)) {
		tmp = (eps * 5.0) * ((x * x) * (x * x));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.8d-48)) .or. (.not. (x <= 1.55d-69))) then
        tmp = (eps * 5.0d0) * ((x * x) * (x * x))
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.8e-48) || !(x <= 1.55e-69)) {
		tmp = (eps * 5.0) * ((x * x) * (x * x));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.8e-48) or not (x <= 1.55e-69):
		tmp = (eps * 5.0) * ((x * x) * (x * x))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.8e-48) || !(x <= 1.55e-69))
		tmp = Float64(Float64(eps * 5.0) * Float64(Float64(x * x) * Float64(x * x)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.8e-48) || ~((x <= 1.55e-69)))
		tmp = (eps * 5.0) * ((x * x) * (x * x));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.8e-48], N[Not[LessEqual[x, 1.55e-69]], $MachinePrecision]], N[(N[(eps * 5.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.80000000000000005e-48 or 1.55e-69 < x

   1. Initial program 37.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 95.7%

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

      1. distribute-rgt1-in 95.7%

         \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \]

      2. metadata-eval 95.7%

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

   5. Simplified 95.7%

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

      1. metadata-eval 95.7%

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

      2. pow-prod-up 95.6%

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

   7. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left(5 \cdot \varepsilon\right) \]
   8. Step-by-step derivation

      1. unpow2 97.8%

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

   9. Applied egg-rr 95.6%

      \[\leadsto \left({x}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \varepsilon\right) \]
   10. Step-by-step derivation

       1. unpow2 97.8%

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

   11. Applied egg-rr 95.6%

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

   if -2.80000000000000005e-48 < x < 1.55e-69

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-48} \lor \neg \left(x \leq 1.55 \cdot 10^{-69}\right):\\ \;\;\;\;\left(\varepsilon \cdot 5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.3% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot 5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* eps 5.0) (* (* x x) (* x x))))

C

double code(double x, double eps) {
	return (eps * 5.0) * ((x * x) * (x * x));
}

Fortran

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

Java

public static double code(double x, double eps) {
	return (eps * 5.0) * ((x * x) * (x * x));
}

Python

def code(x, eps):
	return (eps * 5.0) * ((x * x) * (x * x))

Julia

function code(x, eps)
	return Float64(Float64(eps * 5.0) * Float64(Float64(x * x) * Float64(x * x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * 5.0) * ((x * x) * (x * x));
end

Wolfram

code[x_, eps_] := N[(N[(eps * 5.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.3%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 84.3%

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

   1. distribute-rgt1-in 84.3%

      \[\leadsto {x}^{4} \cdot \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \]

   2. metadata-eval 84.3%

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

5. Simplified 84.3%

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

   1. metadata-eval 84.3%

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

   2. pow-prod-up 84.3%

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

7. Applied egg-rr 84.3%

   \[\leadsto \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left(5 \cdot \varepsilon\right) \]
8. Step-by-step derivation

   1. unpow2 84.9%

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

9. Applied egg-rr 84.3%

   \[\leadsto \left({x}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(5 \cdot \varepsilon\right) \]
10. Step-by-step derivation

    1. unpow2 84.9%

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

11. Applied egg-rr 84.3%

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

12. Final simplification 84.3%

    \[\leadsto \left(\varepsilon \cdot 5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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)))