2nthrt (problem 3.4.6)

Percentage Accurate: 54.4% → 84.9%

Time: 19.1s

Alternatives: 14

Speedup: 1.9×

• could not determine a ground truth

  1. x = -2
  2. n = 4.94066e-324

Specification

Math

\[\begin{array}{l} \\ {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))

C

double code(double x, double n) {
	return pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
end function

Java

public static double code(double x, double n) {
	return Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
}

Python

def code(x, n):
	return math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))

Julia

function code(x, n)
	return Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
end

MATLAB

function tmp = code(x, n)
	tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
end

Wolfram

code[x_, n_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 54.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))

C

double code(double x, double n) {
	return pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((x + 1.0d0) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
end function

Java

public static double code(double x, double n) {
	return Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
}

Python

def code(x, n):
	return math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))

Julia

function code(x, n)
	return Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
end

MATLAB

function tmp = code(x, n)
	tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
end

Wolfram

code[x_, n_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-100}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{log1p}\left(x\right) - \log x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 20000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -5e-80)
     t_1
     (if (<= (/ 1.0 n) 2e-100)
       (/ (log (exp (- (log1p x) (log x)))) n)
       (if (<= (/ 1.0 n) 20000000.0) t_1 (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -5e-80) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-100) {
		tmp = log(exp((log1p(x) - log(x)))) / n;
	} else if ((1.0 / n) <= 20000000.0) {
		tmp = t_1;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -5e-80) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-100) {
		tmp = Math.log(Math.exp((Math.log1p(x) - Math.log(x)))) / n;
	} else if ((1.0 / n) <= 20000000.0) {
		tmp = t_1;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -5e-80:
		tmp = t_1
	elif (1.0 / n) <= 2e-100:
		tmp = math.log(math.exp((math.log1p(x) - math.log(x)))) / n
	elif (1.0 / n) <= 20000000.0:
		tmp = t_1
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-80)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-100)
		tmp = Float64(log(exp(Float64(log1p(x) - log(x)))) / n);
	elseif (Float64(1.0 / n) <= 20000000.0)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-80], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-100], N[(N[Log[N[Exp[N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 20000000.0], t$95$1, N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -5e-80 or 2e-100 < (/.f64 1 n) < 2e7

   1. Initial program 68.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 87.2%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. log-rec 87.2%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      2. mul-1-neg 87.2%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 87.2%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      4. distribute-frac-neg 87.2%

         \[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      5. neg-mul-1 87.2%

         \[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      6. remove-double-neg 87.2%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      7. *-rgt-identity 87.2%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-*r/ 87.2%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. unpow-1 87.2%

         \[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]

      10. exp-to-pow 87.2%

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

      11. unpow-1 87.2%

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative 87.2%

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   4. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 87.2%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}}{x \cdot n} \]

      2. pow3 87.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}}{x \cdot n} \]

      3. pow-to-exp 87.2%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot 3}}}{x \cdot n} \]

      4. pow1/3 87.2%

         \[\leadsto \frac{e^{\log \color{blue}{\left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333}\right)} \cdot 3}}{x \cdot n} \]

      5. log-pow 87.2%

         \[\leadsto \frac{e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot 3}}{x \cdot n} \]

      6. pow-to-exp 87.2%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \log \color{blue}{\left(e^{\log x \cdot \frac{1}{n}}\right)}\right) \cdot 3}}{x \cdot n} \]

      7. add-log-exp 87.2%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log x \cdot \frac{1}{n}\right)}\right) \cdot 3}}{x \cdot n} \]

      8. un-div-inv 87.2%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\frac{\log x}{n}}\right) \cdot 3}}{x \cdot n} \]

   6. Applied egg-rr 87.2%

      \[\leadsto \frac{\color{blue}{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x \cdot n} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 87.2%

         \[\leadsto \frac{\color{blue}{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}} \cdot \sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}}{x \cdot n} \]

      2. times-frac 87.7%

         \[\leadsto \color{blue}{\frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x} \cdot \frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{n}} \]

   8. Applied egg-rr 87.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{x} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}} \]
   9. Step-by-step derivation

      1. associate-*l/ 87.7%

         \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x}} \]

      2. associate-*r/ 87.7%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}}{x} \]

      3. add-sqr-sqrt 87.7%

         \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x} \]

   10. Applied egg-rr 87.7%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]

   if -5e-80 < (/.f64 1 n) < 2e-100

   1. Initial program 37.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 84.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 84.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. add-log-exp 84.4%

         \[\leadsto \frac{\color{blue}{\log \left(e^{\mathsf{log1p}\left(x\right) - \log x}\right)}}{n} \]

   6. Applied egg-rr 84.4%

      \[\leadsto \frac{\color{blue}{\log \left(e^{\mathsf{log1p}\left(x\right) - \log x}\right)}}{n} \]

   if 2e7 < (/.f64 1 n)

   1. Initial program 59.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around 0 59.2%

      \[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}} - e^{\frac{\log x}{n}}} \]
   3. Step-by-step derivation

      1. log1p-def 100.0%

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - e^{\frac{\log x}{n}} \]

      2. *-rgt-identity 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      3. associate-*r/ 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      4. unpow-1 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\log x \cdot \color{blue}{{n}^{-1}}} \]

      5. exp-to-pow 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - \color{blue}{{x}^{\left({n}^{-1}\right)}} \]

      6. /-rgt-identity 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{1}\right)}} \]

      7. metadata-eval 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{\frac{2}{2}}}\right)} \]

      8. associate-/l* 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1} \cdot 2}{2}\right)}} \]

      9. *-commutative 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{2 \cdot {n}^{-1}}}{2}\right)} \]

      10. *-commutative 100.0%

          \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{{n}^{-1} \cdot 2}}{2}\right)} \]

      11. associate-/l* 100.0%

          \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{\frac{2}{2}}\right)}} \]

      12. metadata-eval 100.0%

          \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{1}}\right)} \]

      13. /-rgt-identity 100.0%

          \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left({n}^{-1}\right)}} \]

      14. unpow-1 100.0%

          \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-100}:\\ \;\;\;\;\frac{\log \left(e^{\mathsf{log1p}\left(x\right) - \log x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 20000000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-100}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 20000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -5e-80)
     t_1
     (if (<= (/ 1.0 n) 2e-100)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 20000000.0) t_1 (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -5e-80) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-100) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 20000000.0) {
		tmp = t_1;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -5e-80) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-100) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 20000000.0) {
		tmp = t_1;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -5e-80:
		tmp = t_1
	elif (1.0 / n) <= 2e-100:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 20000000.0:
		tmp = t_1
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-80)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-100)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 20000000.0)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-80], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-100], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 20000000.0], t$95$1, N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -5e-80 or 2e-100 < (/.f64 1 n) < 2e7

   1. Initial program 68.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 87.2%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. log-rec 87.2%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      2. mul-1-neg 87.2%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 87.2%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      4. distribute-frac-neg 87.2%

         \[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      5. neg-mul-1 87.2%

         \[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      6. remove-double-neg 87.2%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      7. *-rgt-identity 87.2%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-*r/ 87.2%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. unpow-1 87.2%

         \[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]

      10. exp-to-pow 87.2%

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

      11. unpow-1 87.2%

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative 87.2%

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   4. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 87.2%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}}{x \cdot n} \]

      2. pow3 87.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}}{x \cdot n} \]

      3. pow-to-exp 87.2%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot 3}}}{x \cdot n} \]

      4. pow1/3 87.2%

         \[\leadsto \frac{e^{\log \color{blue}{\left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333}\right)} \cdot 3}}{x \cdot n} \]

      5. log-pow 87.2%

         \[\leadsto \frac{e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot 3}}{x \cdot n} \]

      6. pow-to-exp 87.2%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \log \color{blue}{\left(e^{\log x \cdot \frac{1}{n}}\right)}\right) \cdot 3}}{x \cdot n} \]

      7. add-log-exp 87.2%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log x \cdot \frac{1}{n}\right)}\right) \cdot 3}}{x \cdot n} \]

      8. un-div-inv 87.2%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\frac{\log x}{n}}\right) \cdot 3}}{x \cdot n} \]

   6. Applied egg-rr 87.2%

      \[\leadsto \frac{\color{blue}{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x \cdot n} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 87.2%

         \[\leadsto \frac{\color{blue}{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}} \cdot \sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}}{x \cdot n} \]

      2. times-frac 87.7%

         \[\leadsto \color{blue}{\frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x} \cdot \frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{n}} \]

   8. Applied egg-rr 87.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{x} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}} \]
   9. Step-by-step derivation

      1. associate-*l/ 87.7%

         \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x}} \]

      2. associate-*r/ 87.7%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}}{x} \]

      3. add-sqr-sqrt 87.7%

         \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x} \]

   10. Applied egg-rr 87.7%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]

   if -5e-80 < (/.f64 1 n) < 2e-100

   1. Initial program 37.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 84.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 84.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 2e7 < (/.f64 1 n)

   1. Initial program 59.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around 0 59.2%

      \[\leadsto \color{blue}{e^{\frac{\log \left(1 + x\right)}{n}} - e^{\frac{\log x}{n}}} \]
   3. Step-by-step derivation

      1. log1p-def 100.0%

         \[\leadsto e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - e^{\frac{\log x}{n}} \]

      2. *-rgt-identity 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      3. associate-*r/ 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      4. unpow-1 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - e^{\log x \cdot \color{blue}{{n}^{-1}}} \]

      5. exp-to-pow 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - \color{blue}{{x}^{\left({n}^{-1}\right)}} \]

      6. /-rgt-identity 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{1}\right)}} \]

      7. metadata-eval 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{\frac{2}{2}}}\right)} \]

      8. associate-/l* 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1} \cdot 2}{2}\right)}} \]

      9. *-commutative 100.0%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{2 \cdot {n}^{-1}}}{2}\right)} \]

      10. *-commutative 100.0%

          \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{{n}^{-1} \cdot 2}}{2}\right)} \]

      11. associate-/l* 100.0%

          \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{\frac{2}{2}}\right)}} \]

      12. metadata-eval 100.0%

          \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{1}}\right)} \]

      13. /-rgt-identity 100.0%

          \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left({n}^{-1}\right)}} \]

      14. unpow-1 100.0%

          \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-100}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 20000000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 3: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-100}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 20000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+152}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{n}}{x}}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -5e-80)
     t_1
     (if (<= (/ 1.0 n) 2e-100)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 20000000.0)
         t_1
         (if (<= (/ 1.0 n) 4e+152)
           (- (pow (+ 1.0 x) (/ 1.0 n)) t_0)
           (/ (/ (/ 1.0 n) x) t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -5e-80) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-100) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 20000000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e+152) {
		tmp = pow((1.0 + x), (1.0 / n)) - t_0;
	} else {
		tmp = ((1.0 / n) / x) / t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -5e-80) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-100) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 20000000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e+152) {
		tmp = Math.pow((1.0 + x), (1.0 / n)) - t_0;
	} else {
		tmp = ((1.0 / n) / x) / t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -5e-80:
		tmp = t_1
	elif (1.0 / n) <= 2e-100:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 20000000.0:
		tmp = t_1
	elif (1.0 / n) <= 4e+152:
		tmp = math.pow((1.0 + x), (1.0 / n)) - t_0
	else:
		tmp = ((1.0 / n) / x) / t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-80)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-100)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 20000000.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 4e+152)
		tmp = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) / x) / t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-80], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-100], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 20000000.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+152], N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5e-80 or 2e-100 < (/.f64 1 n) < 2e7

   1. Initial program 68.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 87.2%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. log-rec 87.2%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      2. mul-1-neg 87.2%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 87.2%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      4. distribute-frac-neg 87.2%

         \[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      5. neg-mul-1 87.2%

         \[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      6. remove-double-neg 87.2%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      7. *-rgt-identity 87.2%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-*r/ 87.2%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. unpow-1 87.2%

         \[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]

      10. exp-to-pow 87.2%

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

      11. unpow-1 87.2%

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative 87.2%

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   4. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 87.2%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}}{x \cdot n} \]

      2. pow3 87.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}}{x \cdot n} \]

      3. pow-to-exp 87.2%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot 3}}}{x \cdot n} \]

      4. pow1/3 87.2%

         \[\leadsto \frac{e^{\log \color{blue}{\left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333}\right)} \cdot 3}}{x \cdot n} \]

      5. log-pow 87.2%

         \[\leadsto \frac{e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot 3}}{x \cdot n} \]

      6. pow-to-exp 87.2%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \log \color{blue}{\left(e^{\log x \cdot \frac{1}{n}}\right)}\right) \cdot 3}}{x \cdot n} \]

      7. add-log-exp 87.2%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log x \cdot \frac{1}{n}\right)}\right) \cdot 3}}{x \cdot n} \]

      8. un-div-inv 87.2%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\frac{\log x}{n}}\right) \cdot 3}}{x \cdot n} \]

   6. Applied egg-rr 87.2%

      \[\leadsto \frac{\color{blue}{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x \cdot n} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 87.2%

         \[\leadsto \frac{\color{blue}{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}} \cdot \sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}}{x \cdot n} \]

      2. times-frac 87.7%

         \[\leadsto \color{blue}{\frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x} \cdot \frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{n}} \]

   8. Applied egg-rr 87.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{x} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}} \]
   9. Step-by-step derivation

      1. associate-*l/ 87.7%

         \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x}} \]

      2. associate-*r/ 87.7%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}}{x} \]

      3. add-sqr-sqrt 87.7%

         \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x} \]

   10. Applied egg-rr 87.7%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]

   if -5e-80 < (/.f64 1 n) < 2e-100

   1. Initial program 37.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 84.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 84.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 2e7 < (/.f64 1 n) < 4.0000000000000002e152

   1. Initial program 91.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   if 4.0000000000000002e152 < (/.f64 1 n)

   1. Initial program 22.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 0.7%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. log-rec 0.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      2. mul-1-neg 0.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 0.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      4. distribute-frac-neg 0.7%

         \[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      5. neg-mul-1 0.7%

         \[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      6. remove-double-neg 0.7%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      7. *-rgt-identity 0.7%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-*r/ 0.7%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. unpow-1 0.7%

         \[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]

      10. exp-to-pow 0.7%

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

      11. unpow-1 0.7%

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative 0.7%

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   4. Simplified 0.7%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 0.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}}{x \cdot n} \]

      2. pow3 0.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}}{x \cdot n} \]

      3. pow-to-exp 0.7%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot 3}}}{x \cdot n} \]

      4. pow1/3 0.7%

         \[\leadsto \frac{e^{\log \color{blue}{\left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333}\right)} \cdot 3}}{x \cdot n} \]

      5. log-pow 0.7%

         \[\leadsto \frac{e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot 3}}{x \cdot n} \]

      6. pow-to-exp 0.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \log \color{blue}{\left(e^{\log x \cdot \frac{1}{n}}\right)}\right) \cdot 3}}{x \cdot n} \]

      7. add-log-exp 0.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log x \cdot \frac{1}{n}\right)}\right) \cdot 3}}{x \cdot n} \]

      8. un-div-inv 0.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\frac{\log x}{n}}\right) \cdot 3}}{x \cdot n} \]

   6. Applied egg-rr 0.7%

      \[\leadsto \frac{\color{blue}{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x \cdot n} \]
   7. Step-by-step derivation

      1. clear-num 0.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot n}{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}} \]

      2. inv-pow 0.7%

         \[\leadsto \color{blue}{{\left(\frac{x \cdot n}{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}\right)}^{-1}} \]

   8. Applied egg-rr 87.1%

      \[\leadsto \color{blue}{{\left(\left(x \cdot n\right) \cdot {x}^{\left(\frac{1}{n}\right)}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 87.1%

         \[\leadsto \color{blue}{\frac{1}{\left(x \cdot n\right) \cdot {x}^{\left(\frac{1}{n}\right)}}} \]

      2. associate-/r* 87.1%

         \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot n}}{{x}^{\left(\frac{1}{n}\right)}}} \]

      3. *-commutative 87.1%

         \[\leadsto \frac{\frac{1}{\color{blue}{n \cdot x}}}{{x}^{\left(\frac{1}{n}\right)}} \]

      4. associate-/r* 87.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{n}}{x}}}{{x}^{\left(\frac{1}{n}\right)}} \]

   10. Simplified 87.1%

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{n}}{x}}{{x}^{\left(\frac{1}{n}\right)}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-100}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 20000000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+152}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{n}}{x}}{{x}^{\left(\frac{1}{n}\right)}}\\ \end{array} \]

Alternative 4: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-100}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 20000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right) + \left(1 + \frac{x}{n}\right)\right) - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -5e-80)
     t_1
     (if (<= (/ 1.0 n) 2e-100)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 20000000.0)
         t_1
         (-
          (+ (* (- (/ 0.5 (* n n)) (/ 0.5 n)) (* x x)) (+ 1.0 (/ x n)))
          t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -5e-80) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-100) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 20000000.0) {
		tmp = t_1;
	} else {
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (x / n))) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -5e-80) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-100) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 20000000.0) {
		tmp = t_1;
	} else {
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (x / n))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -5e-80:
		tmp = t_1
	elif (1.0 / n) <= 2e-100:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 20000000.0:
		tmp = t_1
	else:
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (x / n))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-80)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-100)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 20000000.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)) * Float64(x * x)) + Float64(1.0 + Float64(x / n))) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-80], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-100], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 20000000.0], t$95$1, N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -5e-80 or 2e-100 < (/.f64 1 n) < 2e7

   1. Initial program 68.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 87.2%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. log-rec 87.2%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      2. mul-1-neg 87.2%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 87.2%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      4. distribute-frac-neg 87.2%

         \[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      5. neg-mul-1 87.2%

         \[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      6. remove-double-neg 87.2%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      7. *-rgt-identity 87.2%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-*r/ 87.2%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. unpow-1 87.2%

         \[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]

      10. exp-to-pow 87.2%

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

      11. unpow-1 87.2%

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative 87.2%

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   4. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 87.2%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}}{x \cdot n} \]

      2. pow3 87.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}}{x \cdot n} \]

      3. pow-to-exp 87.2%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot 3}}}{x \cdot n} \]

      4. pow1/3 87.2%

         \[\leadsto \frac{e^{\log \color{blue}{\left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333}\right)} \cdot 3}}{x \cdot n} \]

      5. log-pow 87.2%

         \[\leadsto \frac{e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot 3}}{x \cdot n} \]

      6. pow-to-exp 87.2%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \log \color{blue}{\left(e^{\log x \cdot \frac{1}{n}}\right)}\right) \cdot 3}}{x \cdot n} \]

      7. add-log-exp 87.2%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log x \cdot \frac{1}{n}\right)}\right) \cdot 3}}{x \cdot n} \]

      8. un-div-inv 87.2%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\frac{\log x}{n}}\right) \cdot 3}}{x \cdot n} \]

   6. Applied egg-rr 87.2%

      \[\leadsto \frac{\color{blue}{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x \cdot n} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 87.2%

         \[\leadsto \frac{\color{blue}{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}} \cdot \sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}}{x \cdot n} \]

      2. times-frac 87.7%

         \[\leadsto \color{blue}{\frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x} \cdot \frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{n}} \]

   8. Applied egg-rr 87.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{x} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}} \]
   9. Step-by-step derivation

      1. associate-*l/ 87.7%

         \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x}} \]

      2. associate-*r/ 87.7%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}}{x} \]

      3. add-sqr-sqrt 87.7%

         \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x} \]

   10. Applied egg-rr 87.7%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]

   if -5e-80 < (/.f64 1 n) < 2e-100

   1. Initial program 37.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 84.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 84.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   if 2e7 < (/.f64 1 n)

   1. Initial program 59.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 69.4%

      \[\leadsto \color{blue}{\left(\frac{x}{n} + \left(1 + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 69.4%

         \[\leadsto \color{blue}{\left(\left(\frac{x}{n} + 1\right) + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. +-commutative 69.4%

         \[\leadsto \left(\color{blue}{\left(1 + \frac{x}{n}\right)} + \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      3. associate-*r/ 69.4%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\color{blue}{\frac{0.5 \cdot 1}{{n}^{2}}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      4. metadata-eval 69.4%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{\color{blue}{0.5}}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      5. unpow2 69.4%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{\color{blue}{n \cdot n}} - 0.5 \cdot \frac{1}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      6. associate-*r/ 69.4%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \color{blue}{\frac{0.5 \cdot 1}{n}}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      7. metadata-eval 69.4%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{\color{blue}{0.5}}{n}\right) \cdot {x}^{2}\right) - {x}^{\left(\frac{1}{n}\right)} \]

      8. unpow2 69.4%

         \[\leadsto \left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   4. Simplified 69.4%

      \[\leadsto \color{blue}{\left(\left(1 + \frac{x}{n}\right) + \left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-100}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 20000000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right) + \left(1 + \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 5: 70.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.4 \cdot 10^{-236}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-118}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{elif}\;x \leq 0.00043:\\ \;\;\;\;-0.5 \cdot \frac{x \cdot x}{n} + \left(\frac{x}{n} - \frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 1.4e-236)
     (- 1.0 t_0)
     (if (<= x 3.05e-186)
       (/ (- (log x)) n)
       (if (<= x 2.2e-118)
         (- (+ 1.0 (/ x n)) t_0)
         (if (<= x 0.00043)
           (+ (* -0.5 (/ (* x x) n)) (- (/ x n) (/ (log x) n)))
           (/ (/ t_0 n) x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.4e-236) {
		tmp = 1.0 - t_0;
	} else if (x <= 3.05e-186) {
		tmp = -log(x) / n;
	} else if (x <= 2.2e-118) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 0.00043) {
		tmp = (-0.5 * ((x * x) / n)) + ((x / n) - (log(x) / n));
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 1.4d-236) then
        tmp = 1.0d0 - t_0
    else if (x <= 3.05d-186) then
        tmp = -log(x) / n
    else if (x <= 2.2d-118) then
        tmp = (1.0d0 + (x / n)) - t_0
    else if (x <= 0.00043d0) then
        tmp = ((-0.5d0) * ((x * x) / n)) + ((x / n) - (log(x) / n))
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.4e-236) {
		tmp = 1.0 - t_0;
	} else if (x <= 3.05e-186) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.2e-118) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 0.00043) {
		tmp = (-0.5 * ((x * x) / n)) + ((x / n) - (Math.log(x) / n));
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.4e-236:
		tmp = 1.0 - t_0
	elif x <= 3.05e-186:
		tmp = -math.log(x) / n
	elif x <= 2.2e-118:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 0.00043:
		tmp = (-0.5 * ((x * x) / n)) + ((x / n) - (math.log(x) / n))
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 1.4e-236)
		tmp = Float64(1.0 - t_0);
	elseif (x <= 3.05e-186)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.2e-118)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 0.00043)
		tmp = Float64(Float64(-0.5 * Float64(Float64(x * x) / n)) + Float64(Float64(x / n) - Float64(log(x) / n)));
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 1.4e-236)
		tmp = 1.0 - t_0;
	elseif (x <= 3.05e-186)
		tmp = -log(x) / n;
	elseif (x <= 2.2e-118)
		tmp = (1.0 + (x / n)) - t_0;
	elseif (x <= 0.00043)
		tmp = (-0.5 * ((x * x) / n)) + ((x / n) - (log(x) / n));
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.4e-236], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[x, 3.05e-186], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.2e-118], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 0.00043], N[(N[(-0.5 * N[(N[(x * x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.39999999999999993e-236

   1. Initial program 64.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 64.4%

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   3. Step-by-step derivation

      1. *-rgt-identity 64.4%

         \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      2. associate-*r/ 64.4%

         \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      3. unpow-1 64.4%

         \[\leadsto 1 - e^{\log x \cdot \color{blue}{{n}^{-1}}} \]

      4. exp-to-pow 64.4%

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

      5. unpow-1 64.4%

         \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   4. Simplified 64.4%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   if 1.39999999999999993e-236 < x < 3.04999999999999991e-186

   1. Initial program 30.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 61.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 61.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 61.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 61.6%

         \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   7. Simplified 61.6%

      \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   if 3.04999999999999991e-186 < x < 2.19999999999999984e-118

   1. Initial program 73.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 75.1%

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

   if 2.19999999999999984e-118 < x < 4.29999999999999989e-4

   1. Initial program 32.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 59.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 59.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 59.3%

      \[\leadsto \color{blue}{\frac{x}{n} + \left(-1 \cdot \frac{\log x}{n} + -0.5 \cdot \frac{{x}^{2}}{n}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 59.3%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\log x}{n} + -0.5 \cdot \frac{{x}^{2}}{n}\right) + \frac{x}{n}} \]

      2. neg-mul-1 59.3%

         \[\leadsto \left(\color{blue}{\left(-\frac{\log x}{n}\right)} + -0.5 \cdot \frac{{x}^{2}}{n}\right) + \frac{x}{n} \]

      3. +-commutative 59.3%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{{x}^{2}}{n} + \left(-\frac{\log x}{n}\right)\right)} + \frac{x}{n} \]

      4. associate-+l+ 59.3%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{x}^{2}}{n} + \left(\left(-\frac{\log x}{n}\right) + \frac{x}{n}\right)} \]

      5. unpow2 59.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{x \cdot x}}{n} + \left(\left(-\frac{\log x}{n}\right) + \frac{x}{n}\right) \]

      6. +-commutative 59.3%

         \[\leadsto -0.5 \cdot \frac{x \cdot x}{n} + \color{blue}{\left(\frac{x}{n} + \left(-\frac{\log x}{n}\right)\right)} \]

      7. unsub-neg 59.3%

         \[\leadsto -0.5 \cdot \frac{x \cdot x}{n} + \color{blue}{\left(\frac{x}{n} - \frac{\log x}{n}\right)} \]

   7. Simplified 59.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{x \cdot x}{n} + \left(\frac{x}{n} - \frac{\log x}{n}\right)} \]

   if 4.29999999999999989e-4 < x

   1. Initial program 64.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 96.7%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. log-rec 96.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      2. mul-1-neg 96.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 96.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      4. distribute-frac-neg 96.7%

         \[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      5. neg-mul-1 96.7%

         \[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      6. remove-double-neg 96.7%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      7. *-rgt-identity 96.7%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-*r/ 96.7%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. unpow-1 96.7%

         \[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]

      10. exp-to-pow 96.7%

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

      11. unpow-1 96.7%

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative 96.7%

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   4. Simplified 96.7%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 96.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}}{x \cdot n} \]

      2. pow3 96.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}}{x \cdot n} \]

      3. pow-to-exp 96.7%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot 3}}}{x \cdot n} \]

      4. pow1/3 96.7%

         \[\leadsto \frac{e^{\log \color{blue}{\left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333}\right)} \cdot 3}}{x \cdot n} \]

      5. log-pow 96.7%

         \[\leadsto \frac{e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot 3}}{x \cdot n} \]

      6. pow-to-exp 96.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \log \color{blue}{\left(e^{\log x \cdot \frac{1}{n}}\right)}\right) \cdot 3}}{x \cdot n} \]

      7. add-log-exp 96.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log x \cdot \frac{1}{n}\right)}\right) \cdot 3}}{x \cdot n} \]

      8. un-div-inv 96.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\frac{\log x}{n}}\right) \cdot 3}}{x \cdot n} \]

   6. Applied egg-rr 96.7%

      \[\leadsto \frac{\color{blue}{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x \cdot n} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 96.7%

         \[\leadsto \frac{\color{blue}{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}} \cdot \sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}}{x \cdot n} \]

      2. times-frac 98.4%

         \[\leadsto \color{blue}{\frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x} \cdot \frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{n}} \]

   8. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{x} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}} \]
   9. Step-by-step derivation

      1. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x}} \]

      2. associate-*r/ 98.4%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}}{x} \]

      3. add-sqr-sqrt 98.4%

         \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x} \]

   10. Applied egg-rr 98.4%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-236}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-118}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.00043:\\ \;\;\;\;-0.5 \cdot \frac{x \cdot x}{n} + \left(\frac{x}{n} - \frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]

Alternative 6: 70.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 3.6 \cdot 10^{-236}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{-187}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-118}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{elif}\;x \leq 0.0035:\\ \;\;\;\;\left(\frac{x}{n} + -0.5 \cdot \frac{x \cdot x}{n}\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 3.6e-236)
     (- 1.0 t_0)
     (if (<= x 6.3e-187)
       (/ (- (log x)) n)
       (if (<= x 2.5e-118)
         (- (+ 1.0 (/ x n)) t_0)
         (if (<= x 0.0035)
           (- (+ (/ x n) (* -0.5 (/ (* x x) n))) (/ (log x) n))
           (/ (/ t_0 n) x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 3.6e-236) {
		tmp = 1.0 - t_0;
	} else if (x <= 6.3e-187) {
		tmp = -log(x) / n;
	} else if (x <= 2.5e-118) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 0.0035) {
		tmp = ((x / n) + (-0.5 * ((x * x) / n))) - (log(x) / n);
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 3.6d-236) then
        tmp = 1.0d0 - t_0
    else if (x <= 6.3d-187) then
        tmp = -log(x) / n
    else if (x <= 2.5d-118) then
        tmp = (1.0d0 + (x / n)) - t_0
    else if (x <= 0.0035d0) then
        tmp = ((x / n) + ((-0.5d0) * ((x * x) / n))) - (log(x) / n)
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 3.6e-236) {
		tmp = 1.0 - t_0;
	} else if (x <= 6.3e-187) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.5e-118) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 0.0035) {
		tmp = ((x / n) + (-0.5 * ((x * x) / n))) - (Math.log(x) / n);
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 3.6e-236:
		tmp = 1.0 - t_0
	elif x <= 6.3e-187:
		tmp = -math.log(x) / n
	elif x <= 2.5e-118:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 0.0035:
		tmp = ((x / n) + (-0.5 * ((x * x) / n))) - (math.log(x) / n)
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 3.6e-236)
		tmp = Float64(1.0 - t_0);
	elseif (x <= 6.3e-187)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.5e-118)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 0.0035)
		tmp = Float64(Float64(Float64(x / n) + Float64(-0.5 * Float64(Float64(x * x) / n))) - Float64(log(x) / n));
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 3.6e-236)
		tmp = 1.0 - t_0;
	elseif (x <= 6.3e-187)
		tmp = -log(x) / n;
	elseif (x <= 2.5e-118)
		tmp = (1.0 + (x / n)) - t_0;
	elseif (x <= 0.0035)
		tmp = ((x / n) + (-0.5 * ((x * x) / n))) - (log(x) / n);
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 3.6e-236], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[x, 6.3e-187], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.5e-118], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 0.0035], N[(N[(N[(x / n), $MachinePrecision] + N[(-0.5 * N[(N[(x * x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 3.60000000000000008e-236

   1. Initial program 64.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 64.4%

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   3. Step-by-step derivation

      1. *-rgt-identity 64.4%

         \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      2. associate-*r/ 64.4%

         \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      3. unpow-1 64.4%

         \[\leadsto 1 - e^{\log x \cdot \color{blue}{{n}^{-1}}} \]

      4. exp-to-pow 64.4%

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

      5. unpow-1 64.4%

         \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   4. Simplified 64.4%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   if 3.60000000000000008e-236 < x < 6.29999999999999952e-187

   1. Initial program 30.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 61.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 61.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 61.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 61.6%

         \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   7. Simplified 61.6%

      \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   if 6.29999999999999952e-187 < x < 2.50000000000000007e-118

   1. Initial program 73.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 75.1%

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

   if 2.50000000000000007e-118 < x < 0.00350000000000000007

   1. Initial program 32.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 59.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 59.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. div-sub 59.3%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]

   6. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}} \]

   7. Taylor expanded in x around 0 59.3%

      \[\leadsto \color{blue}{\left(\frac{x}{n} + -0.5 \cdot \frac{{x}^{2}}{n}\right)} - \frac{\log x}{n} \]
   8. Step-by-step derivation

      1. unpow2 59.3%

         \[\leadsto \left(\frac{x}{n} + -0.5 \cdot \frac{\color{blue}{x \cdot x}}{n}\right) - \frac{\log x}{n} \]

   9. Simplified 59.3%

      \[\leadsto \color{blue}{\left(\frac{x}{n} + -0.5 \cdot \frac{x \cdot x}{n}\right)} - \frac{\log x}{n} \]

   if 0.00350000000000000007 < x

   1. Initial program 64.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 96.7%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. log-rec 96.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      2. mul-1-neg 96.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 96.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      4. distribute-frac-neg 96.7%

         \[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      5. neg-mul-1 96.7%

         \[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      6. remove-double-neg 96.7%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      7. *-rgt-identity 96.7%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-*r/ 96.7%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. unpow-1 96.7%

         \[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]

      10. exp-to-pow 96.7%

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

      11. unpow-1 96.7%

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative 96.7%

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   4. Simplified 96.7%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 96.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}}{x \cdot n} \]

      2. pow3 96.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}}{x \cdot n} \]

      3. pow-to-exp 96.7%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot 3}}}{x \cdot n} \]

      4. pow1/3 96.7%

         \[\leadsto \frac{e^{\log \color{blue}{\left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333}\right)} \cdot 3}}{x \cdot n} \]

      5. log-pow 96.7%

         \[\leadsto \frac{e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot 3}}{x \cdot n} \]

      6. pow-to-exp 96.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \log \color{blue}{\left(e^{\log x \cdot \frac{1}{n}}\right)}\right) \cdot 3}}{x \cdot n} \]

      7. add-log-exp 96.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log x \cdot \frac{1}{n}\right)}\right) \cdot 3}}{x \cdot n} \]

      8. un-div-inv 96.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\frac{\log x}{n}}\right) \cdot 3}}{x \cdot n} \]

   6. Applied egg-rr 96.7%

      \[\leadsto \frac{\color{blue}{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x \cdot n} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 96.7%

         \[\leadsto \frac{\color{blue}{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}} \cdot \sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}}{x \cdot n} \]

      2. times-frac 98.4%

         \[\leadsto \color{blue}{\frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x} \cdot \frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{n}} \]

   8. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{x} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}} \]
   9. Step-by-step derivation

      1. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x}} \]

      2. associate-*r/ 98.4%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}}{x} \]

      3. add-sqr-sqrt 98.4%

         \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x} \]

   10. Applied egg-rr 98.4%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-236}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{-187}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-118}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.0035:\\ \;\;\;\;\left(\frac{x}{n} + -0.5 \cdot \frac{x \cdot x}{n}\right) - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]

Alternative 7: 70.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := 1 - t_0\\ \mathbf{if}\;x \leq 2.8 \cdot 10^{-236}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.00012:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- 1.0 t_0)))
   (if (<= x 2.8e-236)
     t_1
     (if (<= x 2.25e-186)
       (/ (- (log x)) n)
       (if (<= x 2.4e-118)
         t_1
         (if (<= x 0.00012) (/ (- x (log x)) n) (/ (/ t_0 n) x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = 1.0 - t_0;
	double tmp;
	if (x <= 2.8e-236) {
		tmp = t_1;
	} else if (x <= 2.25e-186) {
		tmp = -log(x) / n;
	} else if (x <= 2.4e-118) {
		tmp = t_1;
	} else if (x <= 0.00012) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = 1.0d0 - t_0
    if (x <= 2.8d-236) then
        tmp = t_1
    else if (x <= 2.25d-186) then
        tmp = -log(x) / n
    else if (x <= 2.4d-118) then
        tmp = t_1
    else if (x <= 0.00012d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = 1.0 - t_0;
	double tmp;
	if (x <= 2.8e-236) {
		tmp = t_1;
	} else if (x <= 2.25e-186) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.4e-118) {
		tmp = t_1;
	} else if (x <= 0.00012) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = 1.0 - t_0
	tmp = 0
	if x <= 2.8e-236:
		tmp = t_1
	elif x <= 2.25e-186:
		tmp = -math.log(x) / n
	elif x <= 2.4e-118:
		tmp = t_1
	elif x <= 0.00012:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(1.0 - t_0)
	tmp = 0.0
	if (x <= 2.8e-236)
		tmp = t_1;
	elseif (x <= 2.25e-186)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.4e-118)
		tmp = t_1;
	elseif (x <= 0.00012)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = 1.0 - t_0;
	tmp = 0.0;
	if (x <= 2.8e-236)
		tmp = t_1;
	elseif (x <= 2.25e-186)
		tmp = -log(x) / n;
	elseif (x <= 2.4e-118)
		tmp = t_1;
	elseif (x <= 0.00012)
		tmp = (x - log(x)) / n;
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[x, 2.8e-236], t$95$1, If[LessEqual[x, 2.25e-186], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.4e-118], t$95$1, If[LessEqual[x, 0.00012], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.79999999999999986e-236 or 2.2499999999999999e-186 < x < 2.4000000000000001e-118

   1. Initial program 69.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 69.4%

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   3. Step-by-step derivation

      1. *-rgt-identity 69.4%

         \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      2. associate-*r/ 69.4%

         \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      3. unpow-1 69.4%

         \[\leadsto 1 - e^{\log x \cdot \color{blue}{{n}^{-1}}} \]

      4. exp-to-pow 69.4%

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

      5. unpow-1 69.4%

         \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   4. Simplified 69.4%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   if 2.79999999999999986e-236 < x < 2.2499999999999999e-186

   1. Initial program 30.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 61.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 61.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 61.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 61.6%

         \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   7. Simplified 61.6%

      \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   if 2.4000000000000001e-118 < x < 1.20000000000000003e-4

   1. Initial program 32.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 59.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 59.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 59.2%

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 59.2%

         \[\leadsto \frac{x + \color{blue}{\left(-\log x\right)}}{n} \]

      2. unsub-neg 59.2%

         \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   7. Simplified 59.2%

      \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   if 1.20000000000000003e-4 < x

   1. Initial program 64.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 96.7%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. log-rec 96.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      2. mul-1-neg 96.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 96.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      4. distribute-frac-neg 96.7%

         \[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      5. neg-mul-1 96.7%

         \[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      6. remove-double-neg 96.7%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      7. *-rgt-identity 96.7%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-*r/ 96.7%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. unpow-1 96.7%

         \[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]

      10. exp-to-pow 96.7%

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

      11. unpow-1 96.7%

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative 96.7%

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   4. Simplified 96.7%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 96.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}}{x \cdot n} \]

      2. pow3 96.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}}{x \cdot n} \]

      3. pow-to-exp 96.7%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot 3}}}{x \cdot n} \]

      4. pow1/3 96.7%

         \[\leadsto \frac{e^{\log \color{blue}{\left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333}\right)} \cdot 3}}{x \cdot n} \]

      5. log-pow 96.7%

         \[\leadsto \frac{e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot 3}}{x \cdot n} \]

      6. pow-to-exp 96.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \log \color{blue}{\left(e^{\log x \cdot \frac{1}{n}}\right)}\right) \cdot 3}}{x \cdot n} \]

      7. add-log-exp 96.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log x \cdot \frac{1}{n}\right)}\right) \cdot 3}}{x \cdot n} \]

      8. un-div-inv 96.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\frac{\log x}{n}}\right) \cdot 3}}{x \cdot n} \]

   6. Applied egg-rr 96.7%

      \[\leadsto \frac{\color{blue}{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x \cdot n} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 96.7%

         \[\leadsto \frac{\color{blue}{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}} \cdot \sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}}{x \cdot n} \]

      2. times-frac 98.4%

         \[\leadsto \color{blue}{\frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x} \cdot \frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{n}} \]

   8. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{x} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}} \]
   9. Step-by-step derivation

      1. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x}} \]

      2. associate-*r/ 98.4%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}}{x} \]

      3. add-sqr-sqrt 98.4%

         \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x} \]

   10. Applied egg-rr 98.4%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-236}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-118}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.00012:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]

Alternative 8: 70.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.9 \cdot 10^{-236}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-118}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{elif}\;x \leq 0.00023:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 1.9e-236)
     (- 1.0 t_0)
     (if (<= x 4.8e-186)
       (/ (- (log x)) n)
       (if (<= x 2.2e-118)
         (- (+ 1.0 (/ x n)) t_0)
         (if (<= x 0.00023) (/ (- x (log x)) n) (/ (/ t_0 n) x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.9e-236) {
		tmp = 1.0 - t_0;
	} else if (x <= 4.8e-186) {
		tmp = -log(x) / n;
	} else if (x <= 2.2e-118) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 0.00023) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 1.9d-236) then
        tmp = 1.0d0 - t_0
    else if (x <= 4.8d-186) then
        tmp = -log(x) / n
    else if (x <= 2.2d-118) then
        tmp = (1.0d0 + (x / n)) - t_0
    else if (x <= 0.00023d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.9e-236) {
		tmp = 1.0 - t_0;
	} else if (x <= 4.8e-186) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.2e-118) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 0.00023) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.9e-236:
		tmp = 1.0 - t_0
	elif x <= 4.8e-186:
		tmp = -math.log(x) / n
	elif x <= 2.2e-118:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 0.00023:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 1.9e-236)
		tmp = Float64(1.0 - t_0);
	elseif (x <= 4.8e-186)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.2e-118)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 0.00023)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 1.9e-236)
		tmp = 1.0 - t_0;
	elseif (x <= 4.8e-186)
		tmp = -log(x) / n;
	elseif (x <= 2.2e-118)
		tmp = (1.0 + (x / n)) - t_0;
	elseif (x <= 0.00023)
		tmp = (x - log(x)) / n;
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.9e-236], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[x, 4.8e-186], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.2e-118], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 0.00023], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.9e-236

   1. Initial program 64.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 64.4%

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   3. Step-by-step derivation

      1. *-rgt-identity 64.4%

         \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      2. associate-*r/ 64.4%

         \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      3. unpow-1 64.4%

         \[\leadsto 1 - e^{\log x \cdot \color{blue}{{n}^{-1}}} \]

      4. exp-to-pow 64.4%

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

      5. unpow-1 64.4%

         \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   4. Simplified 64.4%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   if 1.9e-236 < x < 4.80000000000000006e-186

   1. Initial program 30.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 61.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 61.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 61.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 61.6%

         \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   7. Simplified 61.6%

      \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   if 4.80000000000000006e-186 < x < 2.19999999999999984e-118

   1. Initial program 73.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 75.1%

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

   if 2.19999999999999984e-118 < x < 2.3000000000000001e-4

   1. Initial program 32.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 59.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 59.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 59.2%

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 59.2%

         \[\leadsto \frac{x + \color{blue}{\left(-\log x\right)}}{n} \]

      2. unsub-neg 59.2%

         \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   7. Simplified 59.2%

      \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   if 2.3000000000000001e-4 < x

   1. Initial program 64.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 96.7%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. log-rec 96.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      2. mul-1-neg 96.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 96.7%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      4. distribute-frac-neg 96.7%

         \[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      5. neg-mul-1 96.7%

         \[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      6. remove-double-neg 96.7%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      7. *-rgt-identity 96.7%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-*r/ 96.7%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. unpow-1 96.7%

         \[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]

      10. exp-to-pow 96.7%

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

      11. unpow-1 96.7%

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative 96.7%

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   4. Simplified 96.7%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 96.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}}{x \cdot n} \]

      2. pow3 96.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}}{x \cdot n} \]

      3. pow-to-exp 96.7%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot 3}}}{x \cdot n} \]

      4. pow1/3 96.7%

         \[\leadsto \frac{e^{\log \color{blue}{\left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333}\right)} \cdot 3}}{x \cdot n} \]

      5. log-pow 96.7%

         \[\leadsto \frac{e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot 3}}{x \cdot n} \]

      6. pow-to-exp 96.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \log \color{blue}{\left(e^{\log x \cdot \frac{1}{n}}\right)}\right) \cdot 3}}{x \cdot n} \]

      7. add-log-exp 96.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log x \cdot \frac{1}{n}\right)}\right) \cdot 3}}{x \cdot n} \]

      8. un-div-inv 96.7%

         \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\frac{\log x}{n}}\right) \cdot 3}}{x \cdot n} \]

   6. Applied egg-rr 96.7%

      \[\leadsto \frac{\color{blue}{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x \cdot n} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 96.7%

         \[\leadsto \frac{\color{blue}{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}} \cdot \sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}}{x \cdot n} \]

      2. times-frac 98.4%

         \[\leadsto \color{blue}{\frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x} \cdot \frac{\sqrt{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{n}} \]

   8. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{x} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}} \]
   9. Step-by-step derivation

      1. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x}} \]

      2. associate-*r/ 98.4%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}}{n}}}{x} \]

      3. add-sqr-sqrt 98.4%

         \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x} \]

   10. Applied egg-rr 98.4%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-236}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-118}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.00023:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]

Alternative 9: 55.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.25 \cdot 10^{-235}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 1.25e-235)
     t_0
     (if (<= x 4.2e-186)
       (/ (- (log x)) n)
       (if (<= x 2.5e-118)
         t_0
         (if (<= x 1.0)
           (/ (- x (log x)) n)
           (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.25e-235) {
		tmp = t_0;
	} else if (x <= 4.2e-186) {
		tmp = -log(x) / n;
	} else if (x <= 2.5e-118) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 1.25d-235) then
        tmp = t_0
    else if (x <= 4.2d-186) then
        tmp = -log(x) / n
    else if (x <= 2.5d-118) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.25e-235) {
		tmp = t_0;
	} else if (x <= 4.2e-186) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.5e-118) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.25e-235:
		tmp = t_0
	elif x <= 4.2e-186:
		tmp = -math.log(x) / n
	elif x <= 2.5e-118:
		tmp = t_0
	elif x <= 1.0:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 1.25e-235)
		tmp = t_0;
	elseif (x <= 4.2e-186)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.5e-118)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 1.25e-235)
		tmp = t_0;
	elseif (x <= 4.2e-186)
		tmp = -log(x) / n;
	elseif (x <= 2.5e-118)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.25e-235], t$95$0, If[LessEqual[x, 4.2e-186], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.5e-118], t$95$0, If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.2499999999999999e-235 or 4.2000000000000004e-186 < x < 2.50000000000000007e-118

   1. Initial program 69.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 69.4%

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   3. Step-by-step derivation

      1. *-rgt-identity 69.4%

         \[\leadsto 1 - e^{\frac{\color{blue}{\log x \cdot 1}}{n}} \]

      2. associate-*r/ 69.4%

         \[\leadsto 1 - e^{\color{blue}{\log x \cdot \frac{1}{n}}} \]

      3. unpow-1 69.4%

         \[\leadsto 1 - e^{\log x \cdot \color{blue}{{n}^{-1}}} \]

      4. exp-to-pow 69.4%

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

      5. unpow-1 69.4%

         \[\leadsto 1 - {x}^{\color{blue}{\left(\frac{1}{n}\right)}} \]

   4. Simplified 69.4%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]

   if 1.2499999999999999e-235 < x < 4.2000000000000004e-186

   1. Initial program 30.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 61.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 61.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 61.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 61.6%

         \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   7. Simplified 61.6%

      \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   if 2.50000000000000007e-118 < x < 1

   1. Initial program 34.9%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 57.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 57.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 57.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 57.4%

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 57.4%

         \[\leadsto \frac{x + \color{blue}{\left(-\log x\right)}}{n} \]

      2. unsub-neg 57.4%

         \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   7. Simplified 57.4%

      \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   if 1 < x

   1. Initial program 63.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 64.0%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 64.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 63.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate-*r/ 63.3%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 63.3%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

      3. unpow2 63.3%

         \[\leadsto \frac{\frac{1}{x} - \frac{0.5}{\color{blue}{x \cdot x}}}{n} \]

   7. Simplified 63.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{x \cdot x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-235}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-118}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \]

Alternative 10: 53.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-187}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.2e-187)
   (/ (- (log x)) n)
   (if (<= x 3.8e-137)
     (/ 1.0 (* n x))
     (if (<= x 0.98)
       (/ (- x (log x)) n)
       (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.2e-187) {
		tmp = -log(x) / n;
	} else if (x <= 3.8e-137) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.98) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.2d-187) then
        tmp = -log(x) / n
    else if (x <= 3.8d-137) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 0.98d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.2e-187) {
		tmp = -Math.log(x) / n;
	} else if (x <= 3.8e-137) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.98) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.2e-187:
		tmp = -math.log(x) / n
	elif x <= 3.8e-137:
		tmp = 1.0 / (n * x)
	elif x <= 0.98:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.2e-187)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 3.8e-137)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 0.98)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.2e-187)
		tmp = -log(x) / n;
	elseif (x <= 3.8e-137)
		tmp = 1.0 / (n * x);
	elseif (x <= 0.98)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.2e-187], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 3.8e-137], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.98], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.20000000000000007e-187

   1. Initial program 49.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 51.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 51.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 51.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 51.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 51.5%

         \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   7. Simplified 51.5%

      \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   if 1.20000000000000007e-187 < x < 3.79999999999999999e-137

   1. Initial program 79.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 69.1%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. log-rec 69.1%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      2. mul-1-neg 69.1%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 69.1%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      4. distribute-frac-neg 69.1%

         \[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      5. neg-mul-1 69.1%

         \[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      6. remove-double-neg 69.1%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      7. *-rgt-identity 69.1%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-*r/ 69.1%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. unpow-1 69.1%

         \[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]

      10. exp-to-pow 69.1%

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

      11. unpow-1 69.1%

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative 69.1%

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   4. Simplified 69.1%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   5. Taylor expanded in n around inf 44.1%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   6. Step-by-step derivation

      1. *-commutative 44.1%

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

   7. Simplified 44.1%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

   if 3.79999999999999999e-137 < x < 0.97999999999999998

   1. Initial program 36.5%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 55.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 55.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 55.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 55.3%

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 55.3%

         \[\leadsto \frac{x + \color{blue}{\left(-\log x\right)}}{n} \]

      2. unsub-neg 55.3%

         \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   7. Simplified 55.3%

      \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   if 0.97999999999999998 < x

   1. Initial program 63.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 64.0%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 64.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 63.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate-*r/ 63.3%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 63.3%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

      3. unpow2 63.3%

         \[\leadsto \frac{\frac{1}{x} - \frac{0.5}{\color{blue}{x \cdot x}}}{n} \]

   7. Simplified 63.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{x \cdot x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-187}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \]

Alternative 11: 53.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 2.3 \cdot 10^{-187}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)))
   (if (<= x 2.3e-187)
     t_0
     (if (<= x 3.7e-137)
       (/ 1.0 (* n x))
       (if (<= x 0.68) t_0 (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double tmp;
	if (x <= 2.3e-187) {
		tmp = t_0;
	} else if (x <= 3.7e-137) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.68) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -log(x) / n
    if (x <= 2.3d-187) then
        tmp = t_0
    else if (x <= 3.7d-137) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 0.68d0) then
        tmp = t_0
    else
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double tmp;
	if (x <= 2.3e-187) {
		tmp = t_0;
	} else if (x <= 3.7e-137) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.68) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	tmp = 0
	if x <= 2.3e-187:
		tmp = t_0
	elif x <= 3.7e-137:
		tmp = 1.0 / (n * x)
	elif x <= 0.68:
		tmp = t_0
	else:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 2.3e-187)
		tmp = t_0;
	elseif (x <= 3.7e-137)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 0.68)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log(x) / n;
	tmp = 0.0;
	if (x <= 2.3e-187)
		tmp = t_0;
	elseif (x <= 3.7e-137)
		tmp = 1.0 / (n * x);
	elseif (x <= 0.68)
		tmp = t_0;
	else
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 2.3e-187], t$95$0, If[LessEqual[x, 3.7e-137], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.68], t$95$0, N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.29999999999999998e-187 or 3.7e-137 < x < 0.680000000000000049

   1. Initial program 41.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 53.9%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 53.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 53.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 53.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 53.3%

         \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   7. Simplified 53.3%

      \[\leadsto \frac{\color{blue}{-\log x}}{n} \]

   if 2.29999999999999998e-187 < x < 3.7e-137

   1. Initial program 79.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 69.1%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. log-rec 69.1%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      2. mul-1-neg 69.1%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 69.1%

         \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      4. distribute-frac-neg 69.1%

         \[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      5. neg-mul-1 69.1%

         \[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

      6. remove-double-neg 69.1%

         \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

      7. *-rgt-identity 69.1%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

      8. associate-*r/ 69.1%

         \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

      9. unpow-1 69.1%

         \[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]

      10. exp-to-pow 69.1%

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

      11. unpow-1 69.1%

          \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

      12. *-commutative 69.1%

          \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

   4. Simplified 69.1%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

   5. Taylor expanded in n around inf 44.1%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   6. Step-by-step derivation

      1. *-commutative 44.1%

         \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

   7. Simplified 44.1%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

   if 0.680000000000000049 < x

   1. Initial program 63.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 64.0%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. log1p-def 64.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 63.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate-*r/ 63.3%

         \[\leadsto \frac{\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}}{n} \]

      2. metadata-eval 63.3%

         \[\leadsto \frac{\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}}{n} \]

      3. unpow2 63.3%

         \[\leadsto \frac{\frac{1}{x} - \frac{0.5}{\color{blue}{x \cdot x}}}{n} \]

   7. Simplified 63.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{x \cdot x}}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-187}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \]

Alternative 12: 40.1% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{n \cdot x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* n x)))

C

double code(double x, double n) {
	return 1.0 / (n * x);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 1.0d0 / (n * x)
end function

Java

public static double code(double x, double n) {
	return 1.0 / (n * x);
}

Python

def code(x, n):
	return 1.0 / (n * x)

Julia

function code(x, n)
	return Float64(1.0 / Float64(n * x))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (n * x);
end

Wolfram

code[x_, n_] := N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.4%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

2. Taylor expanded in x around inf 62.9%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
3. Step-by-step derivation

   1. log-rec 62.9%

      \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   2. mul-1-neg 62.9%

      \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 62.9%

      \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   4. distribute-frac-neg 62.9%

      \[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

   5. neg-mul-1 62.9%

      \[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

   6. remove-double-neg 62.9%

      \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

   7. *-rgt-identity 62.9%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

   8. associate-*r/ 62.9%

      \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

   9. unpow-1 62.9%

      \[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]

   10. exp-to-pow 62.9%

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

   11. unpow-1 62.9%

       \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

   12. *-commutative 62.9%

       \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

4. Simplified 62.9%

   \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

5. Taylor expanded in n around inf 41.1%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
6. Step-by-step derivation

   1. *-commutative 41.1%

      \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

7. Simplified 41.1%

   \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

8. Final simplification 41.1%

   \[\leadsto \frac{1}{n \cdot x} \]

Alternative 13: 40.6% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n}}{x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 n) x))

C

double code(double x, double n) {
	return (1.0 / n) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / n) / x
end function

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

function code(x, n)
	return Float64(Float64(1.0 / n) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / n) / x;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 55.4%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

2. Taylor expanded in x around inf 62.9%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
3. Step-by-step derivation

   1. log-rec 62.9%

      \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   2. mul-1-neg 62.9%

      \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 62.9%

      \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   4. distribute-frac-neg 62.9%

      \[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

   5. neg-mul-1 62.9%

      \[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

   6. remove-double-neg 62.9%

      \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

   7. *-rgt-identity 62.9%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

   8. associate-*r/ 62.9%

      \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

   9. unpow-1 62.9%

      \[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]

   10. exp-to-pow 62.9%

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

   11. unpow-1 62.9%

       \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

   12. *-commutative 62.9%

       \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

4. Simplified 62.9%

   \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
5. Step-by-step derivation

   1. add-cube-cbrt 62.9%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}}}{x \cdot n} \]

   2. pow3 62.9%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}}{x \cdot n} \]

   3. pow-to-exp 62.9%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot 3}}}{x \cdot n} \]

   4. pow1/3 62.9%

      \[\leadsto \frac{e^{\log \color{blue}{\left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^{0.3333333333333333}\right)} \cdot 3}}{x \cdot n} \]

   5. log-pow 62.9%

      \[\leadsto \frac{e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left({x}^{\left(\frac{1}{n}\right)}\right)\right)} \cdot 3}}{x \cdot n} \]

   6. pow-to-exp 62.9%

      \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \log \color{blue}{\left(e^{\log x \cdot \frac{1}{n}}\right)}\right) \cdot 3}}{x \cdot n} \]

   7. add-log-exp 62.9%

      \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log x \cdot \frac{1}{n}\right)}\right) \cdot 3}}{x \cdot n} \]

   8. un-div-inv 62.9%

      \[\leadsto \frac{e^{\left(0.3333333333333333 \cdot \color{blue}{\frac{\log x}{n}}\right) \cdot 3}}{x \cdot n} \]

6. Applied egg-rr 62.9%

   \[\leadsto \frac{\color{blue}{e^{\left(0.3333333333333333 \cdot \frac{\log x}{n}\right) \cdot 3}}}{x \cdot n} \]

7. Taylor expanded in n around inf 41.1%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
8. Step-by-step derivation

   1. associate-/r* 42.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

9. Simplified 42.0%

   \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

10. Final simplification 42.0%

    \[\leadsto \frac{\frac{1}{n}}{x} \]

Alternative 14: 4.5% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ x n))

C

double code(double x, double n) {
	return x / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = x / n
end function

Java

public static double code(double x, double n) {
	return x / n;
}

Python

def code(x, n):
	return x / n

Julia

function code(x, n)
	return Float64(x / n)
end

MATLAB

function tmp = code(x, n)
	tmp = x / n;
end

Wolfram

code[x_, n_] := N[(x / n), $MachinePrecision]

Derivation

1. Initial program 55.4%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

2. Taylor expanded in x around inf 62.9%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
3. Step-by-step derivation

   1. log-rec 62.9%

      \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   2. mul-1-neg 62.9%

      \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 62.9%

      \[\leadsto \frac{e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   4. distribute-frac-neg 62.9%

      \[\leadsto \frac{e^{-1 \cdot \color{blue}{\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

   5. neg-mul-1 62.9%

      \[\leadsto \frac{e^{\color{blue}{-\left(-\frac{\log x}{n}\right)}}}{n \cdot x} \]

   6. remove-double-neg 62.9%

      \[\leadsto \frac{e^{\color{blue}{\frac{\log x}{n}}}}{n \cdot x} \]

   7. *-rgt-identity 62.9%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} \]

   8. associate-*r/ 62.9%

      \[\leadsto \frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} \]

   9. unpow-1 62.9%

      \[\leadsto \frac{e^{\log x \cdot \color{blue}{{n}^{-1}}}}{n \cdot x} \]

   10. exp-to-pow 62.9%

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

   11. unpow-1 62.9%

       \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n}\right)}}}{n \cdot x} \]

   12. *-commutative 62.9%

       \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} \]

4. Simplified 62.9%

   \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

5. Taylor expanded in n around inf 41.1%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
6. Step-by-step derivation

   1. *-commutative 41.1%

      \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

7. Simplified 41.1%

   \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 32.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x \cdot n}\right)\right)} \]

   2. expm1-udef 24.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{x \cdot n}\right)} - 1} \]

   3. associate-/r* 24.1%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{x}}{n}}\right)} - 1 \]

   4. add-exp-log 24.1%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{e^{\log \left(\frac{1}{x}\right)}}}{n}\right)} - 1 \]

   5. neg-log 24.1%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{-\log x}}}{n}\right)} - 1 \]

   6. add-sqr-sqrt 3.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{\sqrt{-\log x} \cdot \sqrt{-\log x}}}}{n}\right)} - 1 \]

   7. sqrt-unprod 10.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{\sqrt{\left(-\log x\right) \cdot \left(-\log x\right)}}}}{n}\right)} - 1 \]

   8. sqr-neg 10.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\sqrt{\color{blue}{\log x \cdot \log x}}}}{n}\right)} - 1 \]

   9. sqrt-unprod 6.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{\sqrt{\log x} \cdot \sqrt{\log x}}}}{n}\right)} - 1 \]

   10. add-sqr-sqrt 8.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{e^{\color{blue}{\log x}}}{n}\right)} - 1 \]

   11. add-exp-log 8.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{n}\right)} - 1 \]

9. Applied egg-rr 8.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{n}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 3.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{n}\right)\right)} \]

    2. expm1-log1p 4.3%

       \[\leadsto \color{blue}{\frac{x}{n}} \]

11. Simplified 4.3%

    \[\leadsto \color{blue}{\frac{x}{n}} \]

12. Final simplification 4.3%

    \[\leadsto \frac{x}{n} \]

Reproduce

herbie shell --seed 2023238 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))