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

Percentage Accurate: 88.3% → 99.2%

Time: 6.5s

Alternatives: 5

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: 88.3% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-304} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -1e-304) (not (<= t_0 0.0)))
     t_0
     (* eps (* (* x x) (* x (* x 5.0)))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if ((t_0 <= -1e-304) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * ((x * x) * (x * (x * 5.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-1d-304)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = eps * ((x * x) * (x * (x * 5.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if ((t_0 <= -1e-304) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = eps * ((x * x) * (x * (x * 5.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if (t_0 <= -1e-304) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = eps * ((x * x) * (x * (x * 5.0)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if ((t_0 <= -1e-304) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(eps * Float64(Float64(x * x) * Float64(x * Float64(x * 5.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if ((t_0 <= -1e-304) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = eps * ((x * x) * (x * (x * 5.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-304], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(eps * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -9.99999999999999971e-305 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 96.9%

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

   if -9.99999999999999971e-305 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 87.1%

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

   2. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt1-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

   4. Simplified 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. sqrt-unprod 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. swap-sqr 99.6%

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

      6. pow-prod-up 99.6%

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

      7. metadata-eval 99.6%

         \[\leadsto \varepsilon \cdot \sqrt{{x}^{\color{blue}{8}} \cdot \left(5 \cdot 5\right)} \]

      8. metadata-eval 99.6%

         \[\leadsto \varepsilon \cdot \sqrt{{x}^{8} \cdot \color{blue}{25}} \]

   6. Applied egg-rr 99.6%

      \[\leadsto \varepsilon \cdot \color{blue}{\sqrt{{x}^{8} \cdot 25}} \]
   7. Step-by-step derivation

      1. sqrt-prod 99.6%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\sqrt{{x}^{8}} \cdot \sqrt{25}\right)} \]

      2. metadata-eval 99.6%

         \[\leadsto \varepsilon \cdot \left(\sqrt{{x}^{8}} \cdot \color{blue}{5}\right) \]

      3. sqrt-pow1 99.9%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{{x}^{\left(\frac{8}{2}\right)}} \cdot 5\right) \]

      4. metadata-eval 99.9%

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

      5. metadata-eval 99.9%

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

      6. pow-sqr 99.9%

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

      7. pow2 99.9%

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

      8. pow2 99.9%

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

      9. add-sqr-sqrt 99.8%

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

      10. swap-sqr 99.8%

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

      11. unpow2 99.8%

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

      12. associate-*l* 99.9%

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

      13. unpow-prod-down 99.9%

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

      14. pow2 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 99.8%

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

       1. unpow2 99.8%

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

       2. associate-*l* 99.8%

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

       3. unpow2 99.8%

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

       4. rem-square-sqrt 99.9%

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

   11. Simplified 99.9%

       \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot 5\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-304} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \end{array} \]

Alternative 2: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-46}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-43}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \sqrt{{x}^{8} \cdot 25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.8e-46)
   (* eps (* (* x x) (* x (* x 5.0))))
   (if (<= x 4.6e-43) (pow eps 5.0) (* eps (sqrt (* (pow x 8.0) 25.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.8e-46) {
		tmp = eps * ((x * x) * (x * (x * 5.0)));
	} else if (x <= 4.6e-43) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * sqrt((pow(x, 8.0) * 25.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-3.8d-46)) then
        tmp = eps * ((x * x) * (x * (x * 5.0d0)))
    else if (x <= 4.6d-43) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * sqrt(((x ** 8.0d0) * 25.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.8e-46) {
		tmp = eps * ((x * x) * (x * (x * 5.0)));
	} else if (x <= 4.6e-43) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * Math.sqrt((Math.pow(x, 8.0) * 25.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.8e-46:
		tmp = eps * ((x * x) * (x * (x * 5.0)))
	elif x <= 4.6e-43:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * math.sqrt((math.pow(x, 8.0) * 25.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.8e-46)
		tmp = Float64(eps * Float64(Float64(x * x) * Float64(x * Float64(x * 5.0))));
	elseif (x <= 4.6e-43)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * sqrt(Float64((x ^ 8.0) * 25.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.8e-46)
		tmp = eps * ((x * x) * (x * (x * 5.0)));
	elseif (x <= 4.6e-43)
		tmp = eps ^ 5.0;
	else
		tmp = eps * sqrt(((x ^ 8.0) * 25.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.8e-46], N[(eps * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.6e-43], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[Sqrt[N[(N[Power[x, 8.0], $MachinePrecision] * 25.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.7999999999999997e-46

   1. Initial program 30.9%

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

   2. Taylor expanded in x around inf 89.6%

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

      1. *-commutative 89.6%

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

      2. distribute-rgt1-in 89.6%

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

      3. metadata-eval 89.6%

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

      4. *-commutative 89.6%

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

      5. associate-*r* 89.6%

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

   4. Simplified 89.6%

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

      1. add-sqr-sqrt 89.4%

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

      2. sqrt-unprod 86.1%

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

      3. *-commutative 86.1%

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

      4. *-commutative 86.1%

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

      5. swap-sqr 86.3%

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

      6. pow-prod-up 86.5%

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

      7. metadata-eval 86.5%

         \[\leadsto \varepsilon \cdot \sqrt{{x}^{\color{blue}{8}} \cdot \left(5 \cdot 5\right)} \]

      8. metadata-eval 86.5%

         \[\leadsto \varepsilon \cdot \sqrt{{x}^{8} \cdot \color{blue}{25}} \]

   6. Applied egg-rr 86.5%

      \[\leadsto \varepsilon \cdot \color{blue}{\sqrt{{x}^{8} \cdot 25}} \]
   7. Step-by-step derivation

      1. sqrt-prod 86.2%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\sqrt{{x}^{8}} \cdot \sqrt{25}\right)} \]

      2. metadata-eval 86.2%

         \[\leadsto \varepsilon \cdot \left(\sqrt{{x}^{8}} \cdot \color{blue}{5}\right) \]

      3. sqrt-pow1 89.6%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{{x}^{\left(\frac{8}{2}\right)}} \cdot 5\right) \]

      4. metadata-eval 89.6%

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

      5. metadata-eval 89.6%

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

      6. pow-sqr 89.3%

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

      7. pow2 89.3%

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

      8. pow2 89.3%

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

      9. add-sqr-sqrt 88.8%

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

      10. swap-sqr 89.1%

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

      11. unpow2 89.1%

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

      12. associate-*l* 89.5%

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

      13. unpow-prod-down 89.2%

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

      14. pow2 89.2%

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

   8. Applied egg-rr 89.2%

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

   9. Taylor expanded in x around 0 88.9%

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

       1. unpow2 88.9%

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

       2. associate-*l* 88.9%

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

       3. unpow2 88.9%

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

       4. rem-square-sqrt 89.7%

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

   11. Simplified 89.7%

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

   if -3.7999999999999997e-46 < x < 4.5999999999999998e-43

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 4.5999999999999998e-43 < x

   1. Initial program 33.1%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-rgt1-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.7%

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

   4. Simplified 99.7%

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

      1. add-sqr-sqrt 99.4%

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

      2. sqrt-unprod 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. swap-sqr 99.6%

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

      6. pow-prod-up 99.8%

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

      7. metadata-eval 99.8%

         \[\leadsto \varepsilon \cdot \sqrt{{x}^{\color{blue}{8}} \cdot \left(5 \cdot 5\right)} \]

      8. metadata-eval 99.8%

         \[\leadsto \varepsilon \cdot \sqrt{{x}^{8} \cdot \color{blue}{25}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\sqrt{{x}^{8} \cdot 25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-46}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-43}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \sqrt{{x}^{8} \cdot 25}\\ \end{array} \]

Alternative 3: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-43}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.8e-47)
   (* eps (* (* x x) (* x (* x 5.0))))
   (if (<= x 1.4e-43) (pow eps 5.0) (* eps (* 5.0 (pow x 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.8e-47) {
		tmp = eps * ((x * x) * (x * (x * 5.0)));
	} else if (x <= 1.4e-43) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * (5.0 * 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 <= (-4.8d-47)) then
        tmp = eps * ((x * x) * (x * (x * 5.0d0)))
    else if (x <= 1.4d-43) then
        tmp = eps ** 5.0d0
    else
        tmp = eps * (5.0d0 * (x ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -4.8e-47) {
		tmp = eps * ((x * x) * (x * (x * 5.0)));
	} else if (x <= 1.4e-43) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = eps * (5.0 * Math.pow(x, 4.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4.8e-47:
		tmp = eps * ((x * x) * (x * (x * 5.0)))
	elif x <= 1.4e-43:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = eps * (5.0 * math.pow(x, 4.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4.8e-47)
		tmp = Float64(eps * Float64(Float64(x * x) * Float64(x * Float64(x * 5.0))));
	elseif (x <= 1.4e-43)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(5.0 * (x ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4.8e-47)
		tmp = eps * ((x * x) * (x * (x * 5.0)));
	elseif (x <= 1.4e-43)
		tmp = eps ^ 5.0;
	else
		tmp = eps * (5.0 * (x ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.8e-47], N[(eps * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-43], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.7999999999999999e-47

   1. Initial program 30.9%

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

   2. Taylor expanded in x around inf 89.6%

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

      1. *-commutative 89.6%

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

      2. distribute-rgt1-in 89.6%

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

      3. metadata-eval 89.6%

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

      4. *-commutative 89.6%

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

      5. associate-*r* 89.6%

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

   4. Simplified 89.6%

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

      1. add-sqr-sqrt 89.4%

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

      2. sqrt-unprod 86.1%

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

      3. *-commutative 86.1%

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

      4. *-commutative 86.1%

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

      5. swap-sqr 86.3%

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

      6. pow-prod-up 86.5%

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

      7. metadata-eval 86.5%

         \[\leadsto \varepsilon \cdot \sqrt{{x}^{\color{blue}{8}} \cdot \left(5 \cdot 5\right)} \]

      8. metadata-eval 86.5%

         \[\leadsto \varepsilon \cdot \sqrt{{x}^{8} \cdot \color{blue}{25}} \]

   6. Applied egg-rr 86.5%

      \[\leadsto \varepsilon \cdot \color{blue}{\sqrt{{x}^{8} \cdot 25}} \]
   7. Step-by-step derivation

      1. sqrt-prod 86.2%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\sqrt{{x}^{8}} \cdot \sqrt{25}\right)} \]

      2. metadata-eval 86.2%

         \[\leadsto \varepsilon \cdot \left(\sqrt{{x}^{8}} \cdot \color{blue}{5}\right) \]

      3. sqrt-pow1 89.6%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{{x}^{\left(\frac{8}{2}\right)}} \cdot 5\right) \]

      4. metadata-eval 89.6%

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

      5. metadata-eval 89.6%

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

      6. pow-sqr 89.3%

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

      7. pow2 89.3%

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

      8. pow2 89.3%

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

      9. add-sqr-sqrt 88.8%

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

      10. swap-sqr 89.1%

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

      11. unpow2 89.1%

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

      12. associate-*l* 89.5%

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

      13. unpow-prod-down 89.2%

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

      14. pow2 89.2%

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

   8. Applied egg-rr 89.2%

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

   9. Taylor expanded in x around 0 88.9%

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

       1. unpow2 88.9%

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

       2. associate-*l* 88.9%

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

       3. unpow2 88.9%

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

       4. rem-square-sqrt 89.7%

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

   11. Simplified 89.7%

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

   if -4.7999999999999999e-47 < x < 1.3999999999999999e-43

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 1.3999999999999999e-43 < x

   1. Initial program 33.1%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-rgt1-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-47}:\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-43}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(5 \cdot {x}^{4}\right)\\ \end{array} \]

Alternative 4: 97.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-47} \lor \neg \left(x \leq 4.8 \cdot 10^{-44}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -8e-47) (not (<= x 4.8e-44)))
   (* eps (* (* x x) (* x (* x 5.0))))
   (pow eps 5.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -8e-47) || !(x <= 4.8e-44)) {
		tmp = eps * ((x * x) * (x * (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 <= (-8d-47)) .or. (.not. (x <= 4.8d-44))) then
        tmp = eps * ((x * x) * (x * (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 <= -8e-47) || !(x <= 4.8e-44)) {
		tmp = eps * ((x * x) * (x * (x * 5.0)));
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -8e-47) or not (x <= 4.8e-44):
		tmp = eps * ((x * x) * (x * (x * 5.0)))
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -8e-47) || !(x <= 4.8e-44))
		tmp = Float64(eps * Float64(Float64(x * x) * Float64(x * Float64(x * 5.0))));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -8e-47) || ~((x <= 4.8e-44)))
		tmp = eps * ((x * x) * (x * (x * 5.0)));
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -8e-47], N[Not[LessEqual[x, 4.8e-44]], $MachinePrecision]], N[(eps * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.9999999999999998e-47 or 4.80000000000000017e-44 < x

   1. Initial program 32.0%

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

   2. Taylor expanded in x around inf 94.6%

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

      1. *-commutative 94.6%

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

      2. distribute-rgt1-in 94.6%

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

      3. metadata-eval 94.6%

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

      4. *-commutative 94.6%

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

      5. associate-*r* 94.7%

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

   4. Simplified 94.7%

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

      1. add-sqr-sqrt 94.4%

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

      2. sqrt-unprod 92.9%

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

      3. *-commutative 92.9%

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

      4. *-commutative 92.9%

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

      5. swap-sqr 93.0%

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

      6. pow-prod-up 93.1%

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

      7. metadata-eval 93.1%

         \[\leadsto \varepsilon \cdot \sqrt{{x}^{\color{blue}{8}} \cdot \left(5 \cdot 5\right)} \]

      8. metadata-eval 93.1%

         \[\leadsto \varepsilon \cdot \sqrt{{x}^{8} \cdot \color{blue}{25}} \]

   6. Applied egg-rr 93.1%

      \[\leadsto \varepsilon \cdot \color{blue}{\sqrt{{x}^{8} \cdot 25}} \]
   7. Step-by-step derivation

      1. sqrt-prod 93.0%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(\sqrt{{x}^{8}} \cdot \sqrt{25}\right)} \]

      2. metadata-eval 93.0%

         \[\leadsto \varepsilon \cdot \left(\sqrt{{x}^{8}} \cdot \color{blue}{5}\right) \]

      3. sqrt-pow1 94.7%

         \[\leadsto \varepsilon \cdot \left(\color{blue}{{x}^{\left(\frac{8}{2}\right)}} \cdot 5\right) \]

      4. metadata-eval 94.7%

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

      5. metadata-eval 94.7%

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

      6. pow-sqr 94.4%

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

      7. pow2 94.4%

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

      8. pow2 94.4%

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

      9. add-sqr-sqrt 94.0%

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

      10. swap-sqr 94.3%

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

      11. unpow2 94.3%

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

      12. associate-*l* 94.3%

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

      13. unpow-prod-down 94.3%

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

      14. pow2 94.3%

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

   8. Applied egg-rr 94.3%

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

   9. Taylor expanded in x around 0 94.0%

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

       1. unpow2 94.0%

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

       2. associate-*l* 94.1%

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

       3. unpow2 94.1%

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

       4. rem-square-sqrt 94.7%

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

   11. Simplified 94.7%

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

   if -7.9999999999999998e-47 < x < 4.80000000000000017e-44

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-47} \lor \neg \left(x \leq 4.8 \cdot 10^{-44}\right):\\ \;\;\;\;\varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

Alternative 5: 82.6% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.8%

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

2. Taylor expanded in x around inf 84.1%

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

   1. *-commutative 84.1%

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

   2. distribute-rgt1-in 84.1%

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

   3. metadata-eval 84.1%

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

   4. *-commutative 84.1%

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

   5. associate-*r* 84.1%

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

4. Simplified 84.1%

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

   1. add-sqr-sqrt 84.1%

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

   2. sqrt-unprod 83.8%

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

   3. *-commutative 83.8%

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

   4. *-commutative 83.8%

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

   5. swap-sqr 83.8%

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

   6. pow-prod-up 83.8%

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

   7. metadata-eval 83.8%

      \[\leadsto \varepsilon \cdot \sqrt{{x}^{\color{blue}{8}} \cdot \left(5 \cdot 5\right)} \]

   8. metadata-eval 83.8%

      \[\leadsto \varepsilon \cdot \sqrt{{x}^{8} \cdot \color{blue}{25}} \]

6. Applied egg-rr 83.8%

   \[\leadsto \varepsilon \cdot \color{blue}{\sqrt{{x}^{8} \cdot 25}} \]
7. Step-by-step derivation

   1. sqrt-prod 83.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\sqrt{{x}^{8}} \cdot \sqrt{25}\right)} \]

   2. metadata-eval 83.8%

      \[\leadsto \varepsilon \cdot \left(\sqrt{{x}^{8}} \cdot \color{blue}{5}\right) \]

   3. sqrt-pow1 84.1%

      \[\leadsto \varepsilon \cdot \left(\color{blue}{{x}^{\left(\frac{8}{2}\right)}} \cdot 5\right) \]

   4. metadata-eval 84.1%

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

   5. metadata-eval 84.1%

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

   6. pow-sqr 84.1%

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

   7. pow2 84.1%

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

   8. pow2 84.1%

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

   9. add-sqr-sqrt 84.0%

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

   10. swap-sqr 84.0%

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

   11. unpow2 84.0%

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

   12. associate-*l* 84.0%

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

   13. unpow-prod-down 84.0%

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

   14. pow2 84.0%

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

8. Applied egg-rr 84.0%

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

9. Taylor expanded in x around 0 84.0%

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

    1. unpow2 84.0%

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

    2. associate-*l* 84.0%

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

    3. unpow2 84.0%

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

    4. rem-square-sqrt 84.1%

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

11. Simplified 84.1%

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

12. Final simplification 84.1%

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

Reproduce

herbie shell --seed 2023271 
(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)))