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

Percentage Accurate: 88.5% → 97.7%

Time: 9.1s

Alternatives: 9

Speedup: 2.0×

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: 88.5% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-41}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot {x}^{3} + {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-39}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot 6\right)\right), \mathsf{fma}\left(\varepsilon, {x}^{4} \cdot 5, {\varepsilon}^{3} \cdot \left(10 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.4e-41)
   (+ (* (* eps (* eps 10.0)) (pow x 3.0)) (* (pow x 4.0) (* eps 5.0)))
   (if (<= x 2.65e-39)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (fma
      (* eps eps)
      (fma 4.0 (pow x 3.0) (* x (* (* x x) 6.0)))
      (fma eps (* (pow x 4.0) 5.0) (* (pow eps 3.0) (* 10.0 (* x x))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.4e-41) {
		tmp = ((eps * (eps * 10.0)) * pow(x, 3.0)) + (pow(x, 4.0) * (eps * 5.0));
	} else if (x <= 2.65e-39) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = fma((eps * eps), fma(4.0, pow(x, 3.0), (x * ((x * x) * 6.0))), fma(eps, (pow(x, 4.0) * 5.0), (pow(eps, 3.0) * (10.0 * (x * x)))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4.4e-41)
		tmp = Float64(Float64(Float64(eps * Float64(eps * 10.0)) * (x ^ 3.0)) + Float64((x ^ 4.0) * Float64(eps * 5.0)));
	elseif (x <= 2.65e-39)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = fma(Float64(eps * eps), fma(4.0, (x ^ 3.0), Float64(x * Float64(Float64(x * x) * 6.0))), fma(eps, Float64((x ^ 4.0) * 5.0), Float64((eps ^ 3.0) * Float64(10.0 * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.4e-41], N[(N[(N[(eps * N[(eps * 10.0), $MachinePrecision]), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.65e-39], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * N[(4.0 * N[Power[x, 3.0], $MachinePrecision] + N[(x * N[(N[(x * x), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4e-41

   1. Initial program 30.0%

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

   2. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{\left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}} \]
   3. Step-by-step derivation

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

      3. distribute-rgt-out 99.7%

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

      4. unpow2 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-*l* 99.7%

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

      7. distribute-lft1-in 99.7%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right), {x}^{3}, \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \cdot {x}^{4}\right) \]

      8. metadata-eval 99.7%

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

      9. *-commutative 99.7%

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

   4. Simplified 99.7%

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

      1. fma-udef 99.7%

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

      2. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   if -4.4e-41 < x < 2.65000000000000002e-39

   1. Initial program 99.1%

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

   if 2.65000000000000002e-39 < x

   1. Initial program 22.1%

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

   2. Taylor expanded in eps around 0 99.5%

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot \left(4 \cdot {x}^{3} + \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right) \cdot x\right) + \left(\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right) + {\varepsilon}^{3} \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. fma-def 99.5%

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

      2. unpow2 99.5%

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

      3. fma-def 99.5%

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

      4. *-commutative 99.5%

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

      5. distribute-rgt-out 99.5%

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

      6. unpow2 99.5%

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

      7. metadata-eval 99.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{6}\right)\right), \varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right) + {\varepsilon}^{3} \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right)\right) \]

      8. fma-def 99.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot 6\right)\right), \color{blue}{\mathsf{fma}\left(\varepsilon, 4 \cdot {x}^{4} + {x}^{4}, {\varepsilon}^{3} \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right)\right)}\right) \]

      9. distribute-lft1-in 99.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot 6\right)\right), \mathsf{fma}\left(\varepsilon, \color{blue}{\left(4 + 1\right) \cdot {x}^{4}}, {\varepsilon}^{3} \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right)\right)\right) \]

      10. metadata-eval 99.5%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot 6\right)\right), \mathsf{fma}\left(\varepsilon, \color{blue}{5} \cdot {x}^{4}, {\varepsilon}^{3} \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right)\right)\right) \]

      11. distribute-rgt-out 99.5%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot 6\right)\right), \mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, {\varepsilon}^{3} \cdot \color{blue}{\left({x}^{2} \cdot \left(2 + 8\right)\right)}\right)\right) \]

      12. unpow2 99.5%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot 6\right)\right), \mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, {\varepsilon}^{3} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(2 + 8\right)\right)\right)\right) \]

      13. metadata-eval 99.5%

          \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot 6\right)\right), \mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, {\varepsilon}^{3} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{10}\right)\right)\right) \]

   4. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot 6\right)\right), \mathsf{fma}\left(\varepsilon, 5 \cdot {x}^{4}, {\varepsilon}^{3} \cdot \left(\left(x \cdot x\right) \cdot 10\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-41}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot {x}^{3} + {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-39}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(4, {x}^{3}, x \cdot \left(\left(x \cdot x\right) \cdot 6\right)\right), \mathsf{fma}\left(\varepsilon, {x}^{4} \cdot 5, {\varepsilon}^{3} \cdot \left(10 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-39} \lor \neg \left(x \leq 2.65 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot {x}^{3} + {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.5e-39) (not (<= x 2.65e-39)))
   (+ (* (* eps (* eps 10.0)) (pow x 3.0)) (* (pow x 4.0) (* eps 5.0)))
   (- (pow (+ x eps) 5.0) (pow x 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.5e-39) || !(x <= 2.65e-39)) {
		tmp = ((eps * (eps * 10.0)) * pow(x, 3.0)) + (pow(x, 4.0) * (eps * 5.0));
	} else {
		tmp = pow((x + eps), 5.0) - pow(x, 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 <= (-2.5d-39)) .or. (.not. (x <= 2.65d-39))) then
        tmp = ((eps * (eps * 10.0d0)) * (x ** 3.0d0)) + ((x ** 4.0d0) * (eps * 5.0d0))
    else
        tmp = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.5e-39) || !(x <= 2.65e-39)) {
		tmp = ((eps * (eps * 10.0)) * Math.pow(x, 3.0)) + (Math.pow(x, 4.0) * (eps * 5.0));
	} else {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.5e-39) or not (x <= 2.65e-39):
		tmp = ((eps * (eps * 10.0)) * math.pow(x, 3.0)) + (math.pow(x, 4.0) * (eps * 5.0))
	else:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.5e-39) || !(x <= 2.65e-39))
		tmp = Float64(Float64(Float64(eps * Float64(eps * 10.0)) * (x ^ 3.0)) + Float64((x ^ 4.0) * Float64(eps * 5.0)));
	else
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.5e-39) || ~((x <= 2.65e-39)))
		tmp = ((eps * (eps * 10.0)) * (x ^ 3.0)) + ((x ^ 4.0) * (eps * 5.0));
	else
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.5e-39], N[Not[LessEqual[x, 2.65e-39]], $MachinePrecision]], N[(N[(N[(eps * N[(eps * 10.0), $MachinePrecision]), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4999999999999999e-39 or 2.65000000000000002e-39 < x

   1. Initial program 26.8%

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

   2. Taylor expanded in x around inf 99.5%

      \[\leadsto \color{blue}{\left(4 \cdot \varepsilon + \varepsilon\right) \cdot {x}^{4} + \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right) \cdot {x}^{3}} \]
   3. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. distribute-rgt-out 99.5%

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

      4. unpow2 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-*l* 99.5%

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

      7. distribute-lft1-in 99.5%

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right), {x}^{3}, \color{blue}{\left(\left(4 + 1\right) \cdot \varepsilon\right)} \cdot {x}^{4}\right) \]

      8. metadata-eval 99.5%

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

      9. *-commutative 99.5%

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

   4. Simplified 99.5%

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

      1. fma-udef 99.5%

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

      2. *-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   if -2.4999999999999999e-39 < x < 2.65000000000000002e-39

   1. Initial program 99.1%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-39} \lor \neg \left(x \leq 2.65 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\varepsilon \cdot \left(\varepsilon \cdot 10\right)\right) \cdot {x}^{3} + {x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternative 3: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-40}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-39}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.3e-40)
   (* (pow x 4.0) (* eps 5.0))
   (if (<= x 2.65e-39)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (* eps (* (pow x 4.0) 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.3e-40) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 2.65e-39) {
		tmp = pow((x + eps), 5.0) - pow(x, 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 <= (-3.3d-40)) then
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    else if (x <= 2.65d-39) then
        tmp = ((x + eps) ** 5.0d0) - (x ** 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 <= -3.3e-40) {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 2.65e-39) {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	} else {
		tmp = eps * (Math.pow(x, 4.0) * 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.3e-40:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	elif x <= 2.65e-39:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	else:
		tmp = eps * (math.pow(x, 4.0) * 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.3e-40)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	elseif (x <= 2.65e-39)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 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 <= -3.3e-40)
		tmp = (x ^ 4.0) * (eps * 5.0);
	elseif (x <= 2.65e-39)
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	else
		tmp = eps * ((x ^ 4.0) * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.3e-40], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.65e-39], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.29999999999999993e-40

   1. Initial program 30.0%

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

   2. Taylor expanded in x around inf 97.8%

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

      1. distribute-lft1-in 97.8%

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

      2. metadata-eval 97.8%

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

      3. *-commutative 97.8%

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

   4. Simplified 97.8%

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

   if -3.29999999999999993e-40 < x < 2.65000000000000002e-39

   1. Initial program 99.1%

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

   if 2.65000000000000002e-39 < x

   1. Initial program 22.1%

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

   2. Taylor expanded in eps around 0 94.1%

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

   3. Taylor expanded in x around 0 94.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-40}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-39}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \end{array} \]

Alternative 4: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-41}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-43}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.3e-41)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 8.2e-43) (pow eps 5.0) (* eps (* (pow x 4.0) 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.3e-41) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 8.2e-43) {
		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 <= (-4.3d-41)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 8.2d-43) 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 <= -4.3e-41) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 8.2e-43) {
		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 <= -4.3e-41:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 8.2e-43:
		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 <= -4.3e-41)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 8.2e-43)
		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 <= -4.3e-41)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 8.2e-43)
		tmp = eps ^ 5.0;
	else
		tmp = eps * ((x ^ 4.0) * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.3e-41], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.2e-43], 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 < -4.2999999999999999e-41

   1. Initial program 30.0%

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

   2. Taylor expanded in eps around 0 97.4%

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

   3. Taylor expanded in x around 0 97.5%

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

   if -4.2999999999999999e-41 < x < 8.1999999999999996e-43

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 98.2%

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

   if 8.1999999999999996e-43 < x

   1. Initial program 36.6%

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

   2. Taylor expanded in eps around 0 90.0%

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

   3. Taylor expanded in x around 0 90.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-41}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-43}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \end{array} \]

Alternative 5: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-41}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-43}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.3e-41)
   (* (pow x 4.0) (* eps 5.0))
   (if (<= x 8.2e-43) (pow eps 5.0) (* eps (* (pow x 4.0) 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.3e-41) {
		tmp = pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 8.2e-43) {
		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 <= (-4.3d-41)) then
        tmp = (x ** 4.0d0) * (eps * 5.0d0)
    else if (x <= 8.2d-43) 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 <= -4.3e-41) {
		tmp = Math.pow(x, 4.0) * (eps * 5.0);
	} else if (x <= 8.2e-43) {
		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 <= -4.3e-41:
		tmp = math.pow(x, 4.0) * (eps * 5.0)
	elif x <= 8.2e-43:
		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 <= -4.3e-41)
		tmp = Float64((x ^ 4.0) * Float64(eps * 5.0));
	elseif (x <= 8.2e-43)
		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 <= -4.3e-41)
		tmp = (x ^ 4.0) * (eps * 5.0);
	elseif (x <= 8.2e-43)
		tmp = eps ^ 5.0;
	else
		tmp = eps * ((x ^ 4.0) * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.3e-41], N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.2e-43], 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 < -4.2999999999999999e-41

   1. Initial program 30.0%

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

   2. Taylor expanded in x around inf 97.8%

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

      1. distribute-lft1-in 97.8%

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

      2. metadata-eval 97.8%

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

      3. *-commutative 97.8%

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

   4. Simplified 97.8%

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

   if -4.2999999999999999e-41 < x < 8.1999999999999996e-43

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 98.2%

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

   if 8.1999999999999996e-43 < x

   1. Initial program 36.6%

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

   2. Taylor expanded in eps around 0 90.0%

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

   3. Taylor expanded in x around 0 90.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-41}:\\ \;\;\;\;{x}^{4} \cdot \left(\varepsilon \cdot 5\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-43}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \end{array} \]

Alternative 6: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-41}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-43}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -7.8e-41)
   (* 5.0 (* eps (pow x 4.0)))
   (if (<= x 8.2e-43) (pow eps 5.0) (* eps (* (* x x) (* 5.0 (* x x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -7.8e-41) {
		tmp = 5.0 * (eps * pow(x, 4.0));
	} else if (x <= 8.2e-43) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-7.8d-41)) then
        tmp = 5.0d0 * (eps * (x ** 4.0d0))
    else if (x <= 8.2d-43) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * ((x * x) * (5.0d0 * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -7.8e-41) {
		tmp = 5.0 * (eps * Math.pow(x, 4.0));
	} else if (x <= 8.2e-43) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -7.8e-41:
		tmp = 5.0 * (eps * math.pow(x, 4.0))
	elif x <= 8.2e-43:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * ((x * x) * (5.0 * (x * x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -7.8e-41)
		tmp = Float64(5.0 * Float64(eps * (x ^ 4.0)));
	elseif (x <= 8.2e-43)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(Float64(x * x) * Float64(5.0 * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -7.8e-41)
		tmp = 5.0 * (eps * (x ^ 4.0));
	elseif (x <= 8.2e-43)
		tmp = eps ^ 5.0;
	else
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -7.8e-41], N[(5.0 * N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.2e-43], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(N[(x * x), $MachinePrecision] * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.79999999999999982e-41

   1. Initial program 30.0%

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

   2. Taylor expanded in eps around 0 97.4%

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

   3. Taylor expanded in x around 0 97.5%

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

   if -7.79999999999999982e-41 < x < 8.1999999999999996e-43

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 98.2%

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

   if 8.1999999999999996e-43 < x

   1. Initial program 36.6%

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

   2. Taylor expanded in eps around 0 90.0%

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

      1. +-commutative 90.0%

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

      2. metadata-eval 90.0%

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

      3. pow-prod-up 89.9%

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

      4. pow2 89.9%

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

      5. pow2 89.9%

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

      6. fma-def 90.0%

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

   4. Applied egg-rr 90.0%

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

      1. fma-udef 89.9%

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

      2. pow2 89.9%

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

      3. pow-prod-down 89.9%

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

      4. pow-sqr 90.0%

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

      5. metadata-eval 90.0%

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

      6. distribute-rgt1-in 90.0%

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

      7. metadata-eval 90.0%

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

      8. metadata-eval 90.0%

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

      9. pow-sqr 89.8%

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

      10. pow2 89.8%

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

      11. pow2 89.8%

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

      12. associate-*r* 89.8%

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

   6. Applied egg-rr 89.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-41}:\\ \;\;\;\;5 \cdot \left(\varepsilon \cdot {x}^{4}\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-43}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]

Alternative 7: 97.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-41} \lor \neg \left(x \leq 1.12 \cdot 10^{-42}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -4.3e-41) (not (<= x 1.12e-42)))
   (* eps (* (* x x) (* 5.0 (* x x))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -4.3e-41) || !(x <= 1.12e-42)) {
		tmp = eps * ((x * x) * (5.0 * (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 <= (-4.3d-41)) .or. (.not. (x <= 1.12d-42))) then
        tmp = eps * ((x * x) * (5.0d0 * (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 <= -4.3e-41) || !(x <= 1.12e-42)) {
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -4.3e-41) or not (x <= 1.12e-42):
		tmp = eps * ((x * x) * (5.0 * (x * x)))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -4.3e-41) || !(x <= 1.12e-42))
		tmp = Float64(eps * Float64(Float64(x * x) * Float64(5.0 * Float64(x * x))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -4.3e-41) || ~((x <= 1.12e-42)))
		tmp = eps * ((x * x) * (5.0 * (x * x)));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -4.3e-41], N[Not[LessEqual[x, 1.12e-42]], $MachinePrecision]], N[(eps * N[(N[(x * x), $MachinePrecision] * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2999999999999999e-41 or 1.1199999999999999e-42 < x

   1. Initial program 33.0%

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

   2. Taylor expanded in eps around 0 94.1%

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

      1. +-commutative 94.1%

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

      2. metadata-eval 94.1%

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

      3. pow-prod-up 94.0%

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

      4. pow2 94.0%

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

      5. pow2 94.0%

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

      6. fma-def 94.1%

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

   4. Applied egg-rr 94.1%

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

      1. fma-udef 94.0%

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

      2. pow2 94.0%

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

      3. pow-prod-down 94.0%

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

      4. pow-sqr 94.1%

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

      5. metadata-eval 94.1%

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

      6. distribute-rgt1-in 94.1%

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

      7. metadata-eval 94.1%

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

      8. metadata-eval 94.1%

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

      9. pow-sqr 93.9%

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

      10. pow2 93.9%

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

      11. pow2 93.9%

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

      12. associate-*r* 93.8%

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

   6. Applied egg-rr 93.8%

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

   if -4.2999999999999999e-41 < x < 1.1199999999999999e-42

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 98.2%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-41} \lor \neg \left(x \leq 1.12 \cdot 10^{-42}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 8: 82.0% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

2. Taylor expanded in eps around 0 84.3%

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

   1. +-commutative 84.3%

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

   2. metadata-eval 84.3%

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

   3. pow-prod-up 84.3%

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

   4. pow2 84.3%

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

   5. pow2 84.3%

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

   6. fma-def 84.3%

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

4. Applied egg-rr 84.3%

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

   1. fma-udef 84.3%

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

   2. pow2 84.3%

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

   3. pow-prod-down 84.3%

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

   4. pow-sqr 84.3%

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

   5. metadata-eval 84.3%

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

   6. distribute-rgt1-in 84.3%

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

   7. metadata-eval 84.3%

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

   8. metadata-eval 84.3%

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

   9. pow-sqr 84.3%

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

   10. pow2 84.3%

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

   11. pow2 84.3%

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

   12. associate-*r* 84.3%

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

6. Applied egg-rr 84.3%

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

7. Final simplification 84.3%

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

Alternative 9: 70.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 90.6%

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

   1. sqr-pow 44.1%

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

   2. metadata-eval 44.1%

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

   3. metadata-eval 44.1%

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

3. Applied egg-rr 44.1%

   \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{2.5} \cdot {\left(x + \varepsilon\right)}^{2.5}} - {x}^{5} \]
4. Step-by-step derivation

   1. pow2 44.1%

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

5. Applied egg-rr 44.1%

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

6. Taylor expanded in x around inf 74.5%

   \[\leadsto \color{blue}{{x}^{5} + -1 \cdot {x}^{5}} \]
7. Step-by-step derivation

   1. distribute-rgt1-in 74.5%

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

   2. metadata-eval 74.5%

      \[\leadsto \color{blue}{0} \cdot {x}^{5} \]

   3. mul0-lft 74.5%

      \[\leadsto \color{blue}{0} \]

8. Simplified 74.5%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 74.5%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023194 
(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)))