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

Percentage Accurate: 88.4% → 99.0%

Time: 6.7s

Alternatives: 8

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: 88.4% 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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := 2 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot t_1, t_1, \varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))) (t_1 (* 2.0 (* x x))))
   (if (<= t_0 -1e-321)
     (+ (pow eps 5.0) (* x (* 5.0 (pow eps 4.0))))
     (if (<= t_0 0.0) (fma (* eps t_1) t_1 (* eps (pow x 4.0))) t_0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = 2.0 * (x * x);
	double tmp;
	if (t_0 <= -1e-321) {
		tmp = pow(eps, 5.0) + (x * (5.0 * pow(eps, 4.0)));
	} else if (t_0 <= 0.0) {
		tmp = fma((eps * t_1), t_1, (eps * pow(x, 4.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(2.0 * Float64(x * x))
	tmp = 0.0
	if (t_0 <= -1e-321)
		tmp = Float64((eps ^ 5.0) + Float64(x * Float64(5.0 * (eps ^ 4.0))));
	elseif (t_0 <= 0.0)
		tmp = fma(Float64(eps * t_1), t_1, Float64(eps * (x ^ 4.0)));
	else
		tmp = t_0;
	end
	return 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]}, Block[{t$95$1 = N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-321], N[(N[Power[eps, 5.0], $MachinePrecision] + N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(eps * t$95$1), $MachinePrecision] * t$95$1 + N[(eps * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -9.98013e-322

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. distribute-lft1-in 100.0%

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

      2. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   if -9.98013e-322 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-*l* 99.9%

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

   4. Simplified 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. metadata-eval 99.9%

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

      5. distribute-lft1-in 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. add-sqr-sqrt 99.8%

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

      8. associate-*r* 99.9%

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

      9. fma-def 99.9%

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

      10. sqrt-prod 99.9%

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

      11. metadata-eval 99.9%

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

      12. sqrt-pow1 99.9%

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

      13. metadata-eval 99.9%

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

      14. pow2 99.9%

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

      15. sqrt-prod 99.9%

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

      16. metadata-eval 99.9%

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

      17. sqrt-pow1 99.9%

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

      18. metadata-eval 99.9%

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

      19. pow2 99.9%

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

   6. Applied egg-rr 99.9%

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

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

   1. Initial program 97.2%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \left(2 \cdot \left(x \cdot x\right)\right), 2 \cdot \left(x \cdot x\right), \varepsilon \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-321} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (or (<= t_0 -1e-321) (not (<= t_0 0.0)))
     t_0
     (* x (* x (* eps (* 5.0 (* x x))))))))

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-321) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = x * (x * (eps * (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) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if ((t_0 <= (-1d-321)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = x * (x * (eps * (5.0d0 * (x * x))))
    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-321) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = x * (x * (eps * (5.0 * (x * x))));
	}
	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-321) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = x * (x * (eps * (5.0 * (x * x))))
	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-321) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(x * Float64(x * Float64(eps * Float64(5.0 * Float64(x * x)))));
	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-321) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = x * (x * (eps * (5.0 * (x * x))));
	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-321], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(x * N[(x * N[(eps * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -9.98013e-322 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5))

   1. Initial program 98.6%

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

   if -9.98013e-322 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-*l* 99.9%

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

   4. Simplified 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. add-sqr-sqrt 92.3%

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

      4. pow2 92.3%

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

      5. sqrt-prod 51.6%

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

      6. *-commutative 51.6%

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

      7. sqrt-pow1 51.6%

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

      8. metadata-eval 51.6%

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

      9. pow2 51.6%

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

   6. Applied egg-rr 51.6%

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

      1. unpow2 51.6%

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

      2. swap-sqr 51.6%

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

      3. add-sqr-sqrt 99.8%

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

      4. *-commutative 99.8%

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

      5. pow2 99.8%

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

      6. pow2 99.8%

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

      7. pow-prod-up 99.9%

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

      8. metadata-eval 99.9%

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

      9. associate-*r* 99.9%

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

      10. metadata-eval 99.9%

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

      11. pow-sqr 99.8%

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

      12. pow-prod-down 99.8%

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

      13. pow2 99.8%

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

      14. associate-*l* 99.9%

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

      15. associate-*r* 99.9%

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

      16. associate-*r* 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-321} \lor \neg \left({\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0\right):\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]

Alternative 3: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;x \cdot \left(x \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-321)
     (+ (pow eps 5.0) (* x (* 5.0 (pow eps 4.0))))
     (if (<= t_0 0.0) (* x (* x (* eps (* 5.0 (* x x))))) t_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-321) {
		tmp = pow(eps, 5.0) + (x * (5.0 * pow(eps, 4.0)));
	} else if (t_0 <= 0.0) {
		tmp = x * (x * (eps * (5.0 * (x * x))));
	} else {
		tmp = t_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-321)) then
        tmp = (eps ** 5.0d0) + (x * (5.0d0 * (eps ** 4.0d0)))
    else if (t_0 <= 0.0d0) then
        tmp = x * (x * (eps * (5.0d0 * (x * x))))
    else
        tmp = t_0
    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-321) {
		tmp = Math.pow(eps, 5.0) + (x * (5.0 * Math.pow(eps, 4.0)));
	} else if (t_0 <= 0.0) {
		tmp = x * (x * (eps * (5.0 * (x * x))));
	} else {
		tmp = t_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-321:
		tmp = math.pow(eps, 5.0) + (x * (5.0 * math.pow(eps, 4.0)))
	elif t_0 <= 0.0:
		tmp = x * (x * (eps * (5.0 * (x * x))))
	else:
		tmp = t_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-321)
		tmp = Float64((eps ^ 5.0) + Float64(x * Float64(5.0 * (eps ^ 4.0))));
	elseif (t_0 <= 0.0)
		tmp = Float64(x * Float64(x * Float64(eps * Float64(5.0 * Float64(x * x)))));
	else
		tmp = t_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-321)
		tmp = (eps ^ 5.0) + (x * (5.0 * (eps ^ 4.0)));
	elseif (t_0 <= 0.0)
		tmp = x * (x * (eps * (5.0 * (x * x))));
	else
		tmp = t_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[LessEqual[t$95$0, -1e-321], N[(N[Power[eps, 5.0], $MachinePrecision] + N[(x * N[(5.0 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(x * N[(x * N[(eps * N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < -9.98013e-322

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. distribute-lft1-in 100.0%

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

      2. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   if -9.98013e-322 < (-.f64 (pow.f64 (+.f64 x eps) 5) (pow.f64 x 5)) < 0.0

   1. Initial program 85.9%

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

   2. Taylor expanded in x around inf 99.9%

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

      1. distribute-lft1-in 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-*l* 99.9%

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

   4. Simplified 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. add-sqr-sqrt 92.3%

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

      4. pow2 92.3%

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

      5. sqrt-prod 51.6%

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

      6. *-commutative 51.6%

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

      7. sqrt-pow1 51.6%

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

      8. metadata-eval 51.6%

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

      9. pow2 51.6%

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

   6. Applied egg-rr 51.6%

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

      1. unpow2 51.6%

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

      2. swap-sqr 51.6%

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

      3. add-sqr-sqrt 99.8%

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

      4. *-commutative 99.8%

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

      5. pow2 99.8%

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

      6. pow2 99.8%

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

      7. pow-prod-up 99.9%

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

      8. metadata-eval 99.9%

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

      9. associate-*r* 99.9%

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

      10. metadata-eval 99.9%

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

      11. pow-sqr 99.8%

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

      12. pow-prod-down 99.8%

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

      13. pow2 99.8%

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

      14. associate-*l* 99.9%

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

      15. associate-*r* 99.9%

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

      16. associate-*r* 99.9%

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

   8. Applied egg-rr 99.9%

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

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

   1. Initial program 97.2%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq -1 \cdot 10^{-321}:\\ \;\;\;\;{\varepsilon}^{5} + x \cdot \left(5 \cdot {\varepsilon}^{4}\right)\\ \mathbf{elif}\;{\left(x + \varepsilon\right)}^{5} - {x}^{5} \leq 0:\\ \;\;\;\;x \cdot \left(x \cdot \left(\varepsilon \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]

Alternative 4: 97.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-51}:\\ \;\;\;\;t_0 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-48}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot \left(x \cdot t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* 5.0 (* x x))))
   (if (<= x -1.5e-51)
     (* t_0 (* eps (* x x)))
     (if (<= x 1.35e-48) (pow eps 5.0) (* x (* eps (* x t_0)))))))

C

double code(double x, double eps) {
	double t_0 = 5.0 * (x * x);
	double tmp;
	if (x <= -1.5e-51) {
		tmp = t_0 * (eps * (x * x));
	} else if (x <= 1.35e-48) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = x * (eps * (x * t_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 = 5.0d0 * (x * x)
    if (x <= (-1.5d-51)) then
        tmp = t_0 * (eps * (x * x))
    else if (x <= 1.35d-48) then
        tmp = eps ** 5.0d0
    else
        tmp = x * (eps * (x * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 5.0 * (x * x);
	double tmp;
	if (x <= -1.5e-51) {
		tmp = t_0 * (eps * (x * x));
	} else if (x <= 1.35e-48) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = x * (eps * (x * t_0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 5.0 * (x * x)
	tmp = 0
	if x <= -1.5e-51:
		tmp = t_0 * (eps * (x * x))
	elif x <= 1.35e-48:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = x * (eps * (x * t_0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(5.0 * Float64(x * x))
	tmp = 0.0
	if (x <= -1.5e-51)
		tmp = Float64(t_0 * Float64(eps * Float64(x * x)));
	elseif (x <= 1.35e-48)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(x * Float64(eps * Float64(x * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 5.0 * (x * x);
	tmp = 0.0;
	if (x <= -1.5e-51)
		tmp = t_0 * (eps * (x * x));
	elseif (x <= 1.35e-48)
		tmp = eps ^ 5.0;
	else
		tmp = x * (eps * (x * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5e-51], N[(t$95$0 * N[(eps * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e-48], N[Power[eps, 5.0], $MachinePrecision], N[(x * N[(eps * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.50000000000000001e-51

   1. Initial program 35.8%

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

   2. Taylor expanded in x around inf 95.1%

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

      1. distribute-lft1-in 95.1%

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

      2. metadata-eval 95.1%

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

      3. associate-*l* 94.9%

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

   4. Simplified 94.9%

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

      1. associate-*r* 95.1%

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

      2. *-commutative 95.1%

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

      3. add-sqr-sqrt 62.9%

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

      4. pow2 62.9%

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

      5. sqrt-prod 58.7%

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

      6. *-commutative 58.7%

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

      7. sqrt-pow1 58.7%

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

      8. metadata-eval 58.7%

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

      9. pow2 58.7%

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

   6. Applied egg-rr 58.7%

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

      1. unpow2 58.7%

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

      2. swap-sqr 58.9%

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

      3. add-sqr-sqrt 94.7%

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

      4. *-commutative 94.7%

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

      5. pow2 94.7%

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

      6. pow2 94.7%

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

      7. pow-prod-up 95.1%

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

      8. metadata-eval 95.1%

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

      9. associate-*r* 95.0%

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

      10. metadata-eval 95.0%

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

      11. pow-sqr 94.6%

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

      12. pow-prod-down 94.6%

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

      13. pow2 94.6%

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

      14. associate-*l* 94.9%

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

      15. *-commutative 94.9%

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

      16. associate-*r* 95.1%

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

   8. Applied egg-rr 95.1%

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

   if -1.50000000000000001e-51 < x < 1.35000000000000006e-48

   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 1.35000000000000006e-48 < x

   1. Initial program 21.9%

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

   2. Taylor expanded in x around inf 99.3%

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

      1. distribute-lft1-in 99.3%

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

      2. metadata-eval 99.3%

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

      3. associate-*l* 99.4%

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

   4. Simplified 99.4%

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

      1. associate-*r* 99.3%

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

      2. *-commutative 99.3%

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

      3. add-sqr-sqrt 60.5%

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

      4. pow2 60.5%

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

      5. sqrt-prod 49.3%

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

      6. *-commutative 49.3%

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

      7. sqrt-pow1 49.4%

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

      8. metadata-eval 49.4%

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

      9. pow2 49.4%

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

   6. Applied egg-rr 49.4%

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

      1. unpow2 49.4%

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

      2. swap-sqr 49.5%

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

      3. add-sqr-sqrt 99.2%

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

      4. *-commutative 99.2%

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

      5. pow2 99.2%

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

      6. pow2 99.2%

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

      7. pow-prod-up 99.3%

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

      8. metadata-eval 99.3%

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

      9. associate-*r* 99.3%

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

      10. metadata-eval 99.3%

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

      11. pow-sqr 99.1%

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

      12. pow-prod-down 99.1%

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

      13. pow2 99.1%

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

      14. associate-*l* 99.2%

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

      15. associate-*r* 99.2%

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

      16. associate-*r* 99.5%

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

   8. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-51}:\\ \;\;\;\;\left(5 \cdot \left(x \cdot x\right)\right) \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-48}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot \left(x \cdot \left(5 \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]

Alternative 5: 82.4% 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 88.2%

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

2. Taylor expanded in eps around 0 83.4%

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

   1. distribute-lft1-in 83.4%

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

   2. metadata-eval 83.4%

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

   3. sqr-pow 83.3%

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

   4. associate-*r* 83.4%

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

   5. metadata-eval 83.4%

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

   6. pow2 83.4%

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

   7. metadata-eval 83.4%

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

   8. pow2 83.4%

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

4. Applied egg-rr 83.4%

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

5. Final simplification 83.4%

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

Alternative 6: 82.4% accurate, 18.8× speedup

Math

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

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(x * Float64(Float64(x * 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[(x * N[(N[(x * x), $MachinePrecision] * N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.2%

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

2. Taylor expanded in eps around 0 83.4%

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

   1. add-sqr-sqrt 83.3%

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

   2. sqrt-unprod 81.6%

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

   3. distribute-lft1-in 81.6%

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

   4. metadata-eval 81.6%

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

   5. distribute-lft1-in 81.6%

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

   6. metadata-eval 81.6%

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

   7. swap-sqr 81.6%

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

   8. metadata-eval 81.6%

      \[\leadsto \varepsilon \cdot \sqrt{\color{blue}{25} \cdot \left({x}^{4} \cdot {x}^{4}\right)} \]

   9. pow-prod-up 81.6%

      \[\leadsto \varepsilon \cdot \sqrt{25 \cdot \color{blue}{{x}^{\left(4 + 4\right)}}} \]

   10. metadata-eval 81.6%

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

4. Applied egg-rr 81.6%

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

   1. sqrt-prod 81.6%

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

   2. metadata-eval 81.6%

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

   3. sqrt-pow1 83.4%

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

   4. metadata-eval 83.4%

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

   5. metadata-eval 83.4%

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

   6. pow-sqr 83.3%

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

   7. pow2 83.3%

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

   8. pow2 83.3%

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

   9. associate-*l* 83.4%

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

   10. *-commutative 83.4%

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

   11. associate-*r* 83.4%

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

   12. associate-*r* 83.4%

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

6. Applied egg-rr 83.4%

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

7. Final simplification 83.4%

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

Alternative 7: 82.4% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

2. Taylor expanded in x around inf 83.4%

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

   1. distribute-lft1-in 83.4%

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

   2. metadata-eval 83.4%

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

   3. associate-*l* 83.4%

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

4. Simplified 83.4%

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

   1. associate-*r* 83.4%

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

   2. *-commutative 83.4%

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

   3. add-sqr-sqrt 77.1%

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

   4. pow2 77.1%

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

   5. sqrt-prod 43.1%

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

   6. *-commutative 43.1%

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

   7. sqrt-pow1 43.1%

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

   8. metadata-eval 43.1%

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

   9. pow2 43.1%

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

6. Applied egg-rr 43.1%

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

   1. unpow2 43.1%

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

   2. swap-sqr 43.1%

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

   3. add-sqr-sqrt 83.3%

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

   4. *-commutative 83.3%

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

   5. pow2 83.3%

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

   6. pow2 83.3%

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

   7. pow-prod-up 83.4%

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

   8. metadata-eval 83.4%

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

   9. associate-*r* 83.4%

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

   10. metadata-eval 83.4%

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

   11. pow-sqr 83.3%

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

   12. pow-prod-down 83.3%

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

   13. pow2 83.3%

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

   14. associate-*l* 83.4%

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

   15. *-commutative 83.4%

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

   16. associate-*r* 83.4%

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

8. Applied egg-rr 83.4%

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

9. Final simplification 83.4%

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

Alternative 8: 82.4% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

2. Taylor expanded in x around inf 83.4%

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

   1. distribute-lft1-in 83.4%

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

   2. metadata-eval 83.4%

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

   3. associate-*l* 83.4%

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

4. Simplified 83.4%

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

   1. associate-*r* 83.4%

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

   2. *-commutative 83.4%

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

   3. add-sqr-sqrt 77.1%

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

   4. pow2 77.1%

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

   5. sqrt-prod 43.1%

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

   6. *-commutative 43.1%

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

   7. sqrt-pow1 43.1%

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

   8. metadata-eval 43.1%

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

   9. pow2 43.1%

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

6. Applied egg-rr 43.1%

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

   1. unpow2 43.1%

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

   2. swap-sqr 43.1%

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

   3. add-sqr-sqrt 83.3%

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

   4. *-commutative 83.3%

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

   5. pow2 83.3%

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

   6. pow2 83.3%

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

   7. pow-prod-up 83.4%

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

   8. metadata-eval 83.4%

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

   9. associate-*r* 83.4%

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

   10. metadata-eval 83.4%

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

   11. pow-sqr 83.3%

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

   12. pow-prod-down 83.3%

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

   13. pow2 83.3%

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

   14. associate-*l* 83.4%

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

   15. associate-*r* 83.4%

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

   16. associate-*r* 83.4%

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

8. Applied egg-rr 83.4%

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

9. Final simplification 83.4%

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

Reproduce

herbie shell --seed 2023195 
(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)))