2nthrt (problem 3.4.6)

Percentage Accurate: 53.6% → 84.9%

Time: 40.3s

Alternatives: 19

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: 53.6% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\left(2 \cdot \log \left(\sqrt{\frac{x}{1 + x}}\right)\right) \cdot \frac{-1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-26)
   (/ (/ (cbrt (pow x (/ 3.0 n))) n) x)
   (if (<= (/ 1.0 n) 5e-134)
     (* (* 2.0 (log (sqrt (/ x (+ 1.0 x))))) (/ -1.0 n))
     (if (<= (/ 1.0 n) 4e-79)
       (/ (/ 1.0 n) x)
       (if (<= (/ 1.0 n) 2e-22)
         (/ (log (/ (+ 1.0 x) x)) n)
         (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = (cbrt(pow(x, (3.0 / n))) / n) / x;
	} else if ((1.0 / n) <= 5e-134) {
		tmp = (2.0 * log(sqrt((x / (1.0 + x))))) * (-1.0 / n);
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e-22) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = (Math.cbrt(Math.pow(x, (3.0 / n))) / n) / x;
	} else if ((1.0 / n) <= 5e-134) {
		tmp = (2.0 * Math.log(Math.sqrt((x / (1.0 + x))))) * (-1.0 / n);
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e-22) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-26)
		tmp = Float64(Float64(cbrt((x ^ Float64(3.0 / n))) / n) / x);
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = Float64(Float64(2.0 * log(sqrt(Float64(x / Float64(1.0 + x))))) * Float64(-1.0 / n));
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-22)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-26], N[(N[(N[Power[N[Power[x, N[(3.0 / n), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], N[(N[(2.0 * N[Log[N[Sqrt[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-22], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-26

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. associate-/r* 97.0%

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

      2. mul-1-neg 97.0%

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

      3. log-rec 97.0%

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

      4. mul-1-neg 97.0%

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

      5. distribute-neg-frac 97.0%

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

      6. mul-1-neg 97.0%

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

      7. remove-double-neg 97.0%

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

      8. *-rgt-identity 97.0%

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

      9. associate-/l* 97.0%

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

      10. exp-to-pow 97.0%

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

   5. Simplified 97.0%

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

      1. add-cbrt-cube 97.0%

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

      2. pow3 97.0%

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

      3. pow-pow 97.0%

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

   7. Applied egg-rr 97.0%

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

      1. associate-*l/ 97.0%

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

      2. metadata-eval 97.0%

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

   9. Simplified 97.0%

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

   if -5.00000000000000019e-26 < (/.f64 1 n) < 5.0000000000000003e-134

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf 87.9%

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

      1. log1p-define 87.9%

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

   5. Simplified 87.9%

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

      1. log1p-undefine 87.9%

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

      2. diff-log 88.0%

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

   7. Applied egg-rr 88.0%

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

      1. +-commutative 88.0%

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

   9. Simplified 88.0%

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

       1. frac-2neg 88.0%

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

       2. div-inv 87.9%

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

       3. neg-log 87.9%

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

       4. clear-num 87.9%

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

   11. Applied egg-rr 87.9%

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

       1. distribute-frac-neg2 87.9%

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

       2. distribute-neg-frac 87.9%

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

       3. metadata-eval 87.9%

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

   13. Simplified 87.9%

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

       1. add-sqr-sqrt 87.9%

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

       2. log-prod 88.0%

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

   15. Applied egg-rr 88.0%

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

       1. count-2 88.0%

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

   17. Simplified 88.0%

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

   if 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. associate-/r* 76.3%

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

      2. mul-1-neg 76.3%

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

      3. log-rec 76.3%

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

      4. mul-1-neg 76.3%

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

      5. distribute-neg-frac 76.3%

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

      6. mul-1-neg 76.3%

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

      7. remove-double-neg 76.3%

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

      8. *-rgt-identity 76.3%

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

      9. associate-/l* 76.3%

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

      10. exp-to-pow 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in n around inf 76.3%

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

   if 4e-79 < (/.f64 1 n) < 2.0000000000000001e-22

   1. Initial program 19.1%

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

   3. Taylor expanded in n around inf 79.7%

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

      1. log1p-define 79.7%

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

   5. Simplified 79.7%

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

      1. log1p-undefine 79.7%

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

      2. diff-log 79.7%

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

   7. Applied egg-rr 79.7%

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

      1. +-commutative 79.7%

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

   9. Simplified 79.7%

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

   if 2.0000000000000001e-22 < (/.f64 1 n)

   1. Initial program 54.1%

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

   3. Taylor expanded in n around 0 54.1%

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

      1. log1p-define 95.1%

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

      2. *-rgt-identity 95.1%

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

      3. associate-*l/ 95.1%

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

      4. associate-/l* 95.1%

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

      5. exp-to-pow 95.1%

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

   5. Simplified 95.1%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\left(2 \cdot \log \left(\sqrt{\frac{x}{1 + x}}\right)\right) \cdot \frac{-1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\left(2 \cdot \log \left(\sqrt{\frac{x}{1 + x}}\right)\right) \cdot \frac{-1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-26)
   (/ (/ (cbrt (pow x (/ 3.0 n))) n) x)
   (if (<= (/ 1.0 n) 5e-134)
     (* (* 2.0 (log (sqrt (/ x (+ 1.0 x))))) (/ -1.0 n))
     (if (<= (/ 1.0 n) 4e-79)
       (/ (/ 1.0 n) x)
       (if (<= (/ 1.0 n) 4e-8)
         (/ (log (/ (+ 1.0 x) x)) n)
         (-
          (+ 1.0 (* x (+ (/ 1.0 n) (/ (+ (* x -0.5) (* 0.5 (/ x n))) n))))
          (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = (cbrt(pow(x, (3.0 / n))) / n) / x;
	} else if ((1.0 / n) <= 5e-134) {
		tmp = (2.0 * log(sqrt((x / (1.0 + x))))) * (-1.0 / n);
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = (Math.cbrt(Math.pow(x, (3.0 / n))) / n) / x;
	} else if ((1.0 / n) <= 5e-134) {
		tmp = (2.0 * Math.log(Math.sqrt((x / (1.0 + x))))) * (-1.0 / n);
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-26)
		tmp = Float64(Float64(cbrt((x ^ Float64(3.0 / n))) / n) / x);
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = Float64(Float64(2.0 * log(sqrt(Float64(x / Float64(1.0 + x))))) * Float64(-1.0 / n));
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 4e-8)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(Float64(x * -0.5) + Float64(0.5 * Float64(x / n))) / n)))) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-26], N[(N[(N[Power[N[Power[x, N[(3.0 / n), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], N[(N[(2.0 * N[Log[N[Sqrt[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-8], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(x * -0.5), $MachinePrecision] + N[(0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-26

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. associate-/r* 97.0%

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

      2. mul-1-neg 97.0%

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

      3. log-rec 97.0%

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

      4. mul-1-neg 97.0%

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

      5. distribute-neg-frac 97.0%

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

      6. mul-1-neg 97.0%

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

      7. remove-double-neg 97.0%

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

      8. *-rgt-identity 97.0%

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

      9. associate-/l* 97.0%

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

      10. exp-to-pow 97.0%

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

   5. Simplified 97.0%

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

      1. add-cbrt-cube 97.0%

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

      2. pow3 97.0%

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

      3. pow-pow 97.0%

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

   7. Applied egg-rr 97.0%

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

      1. associate-*l/ 97.0%

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

      2. metadata-eval 97.0%

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

   9. Simplified 97.0%

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

   if -5.00000000000000019e-26 < (/.f64 1 n) < 5.0000000000000003e-134

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf 87.9%

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

      1. log1p-define 87.9%

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

   5. Simplified 87.9%

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

      1. log1p-undefine 87.9%

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

      2. diff-log 88.0%

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

   7. Applied egg-rr 88.0%

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

      1. +-commutative 88.0%

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

   9. Simplified 88.0%

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

       1. frac-2neg 88.0%

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

       2. div-inv 87.9%

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

       3. neg-log 87.9%

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

       4. clear-num 87.9%

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

   11. Applied egg-rr 87.9%

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

       1. distribute-frac-neg2 87.9%

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

       2. distribute-neg-frac 87.9%

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

       3. metadata-eval 87.9%

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

   13. Simplified 87.9%

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

       1. add-sqr-sqrt 87.9%

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

       2. log-prod 88.0%

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

   15. Applied egg-rr 88.0%

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

       1. count-2 88.0%

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

   17. Simplified 88.0%

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

   if 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. associate-/r* 76.3%

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

      2. mul-1-neg 76.3%

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

      3. log-rec 76.3%

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

      4. mul-1-neg 76.3%

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

      5. distribute-neg-frac 76.3%

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

      6. mul-1-neg 76.3%

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

      7. remove-double-neg 76.3%

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

      8. *-rgt-identity 76.3%

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

      9. associate-/l* 76.3%

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

      10. exp-to-pow 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in n around inf 76.3%

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

   if 4e-79 < (/.f64 1 n) < 4.0000000000000001e-8

   1. Initial program 18.4%

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

   3. Taylor expanded in n around inf 76.0%

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

      1. log1p-define 76.0%

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

   5. Simplified 76.0%

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

      1. log1p-undefine 76.0%

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

      2. diff-log 76.0%

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

   7. Applied egg-rr 76.0%

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

      1. +-commutative 76.0%

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

   9. Simplified 76.0%

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

   if 4.0000000000000001e-8 < (/.f64 1 n)

   1. Initial program 55.4%

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

   3. Taylor expanded in x around 0 80.0%

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

   4. Taylor expanded in n around inf 87.7%

      \[\leadsto \left(1 + x \cdot \left(\color{blue}{\frac{-0.5 \cdot x + 0.5 \cdot \frac{x}{n}}{n}} + \frac{1}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\left(2 \cdot \log \left(\sqrt{\frac{x}{1 + x}}\right)\right) \cdot \frac{-1}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -5e-26)
     (/ (/ (cbrt (pow x (/ 3.0 n))) n) x)
     (if (<= (/ 1.0 n) 5e-134)
       t_0
       (if (<= (/ 1.0 n) 4e-79)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 4e-8)
           t_0
           (-
            (+ 1.0 (* x (+ (/ 1.0 n) (/ (+ (* x -0.5) (* 0.5 (/ x n))) n))))
            (pow x (/ 1.0 n)))))))))

C

double code(double x, double n) {
	double t_0 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = (cbrt(pow(x, (3.0 / n))) / n) / x;
	} else if ((1.0 / n) <= 5e-134) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = (Math.cbrt(Math.pow(x, (3.0 / n))) / n) / x;
	} else if ((1.0 / n) <= 5e-134) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-26)
		tmp = Float64(Float64(cbrt((x ^ Float64(3.0 / n))) / n) / x);
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 4e-8)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(Float64(x * -0.5) + Float64(0.5 * Float64(x / n))) / n)))) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-26], N[(N[(N[Power[N[Power[x, N[(3.0 / n), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-8], t$95$0, N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(x * -0.5), $MachinePrecision] + N[(0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-26

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. associate-/r* 97.0%

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

      2. mul-1-neg 97.0%

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

      3. log-rec 97.0%

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

      4. mul-1-neg 97.0%

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

      5. distribute-neg-frac 97.0%

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

      6. mul-1-neg 97.0%

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

      7. remove-double-neg 97.0%

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

      8. *-rgt-identity 97.0%

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

      9. associate-/l* 97.0%

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

      10. exp-to-pow 97.0%

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

   5. Simplified 97.0%

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

      1. add-cbrt-cube 97.0%

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

      2. pow3 97.0%

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

      3. pow-pow 97.0%

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

   7. Applied egg-rr 97.0%

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

      1. associate-*l/ 97.0%

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

      2. metadata-eval 97.0%

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

   9. Simplified 97.0%

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

   if -5.00000000000000019e-26 < (/.f64 1 n) < 5.0000000000000003e-134 or 4e-79 < (/.f64 1 n) < 4.0000000000000001e-8

   1. Initial program 33.2%

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

   3. Taylor expanded in n around inf 86.1%

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

      1. log1p-define 86.1%

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

   5. Simplified 86.1%

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

      1. log1p-undefine 86.1%

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

      2. diff-log 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. +-commutative 86.1%

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

   9. Simplified 86.1%

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

   if 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. associate-/r* 76.3%

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

      2. mul-1-neg 76.3%

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

      3. log-rec 76.3%

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

      4. mul-1-neg 76.3%

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

      5. distribute-neg-frac 76.3%

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

      6. mul-1-neg 76.3%

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

      7. remove-double-neg 76.3%

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

      8. *-rgt-identity 76.3%

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

      9. associate-/l* 76.3%

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

      10. exp-to-pow 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in n around inf 76.3%

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

   if 4.0000000000000001e-8 < (/.f64 1 n)

   1. Initial program 55.4%

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

   3. Taylor expanded in x around 0 80.0%

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

   4. Taylor expanded in n around inf 87.7%

      \[\leadsto \left(1 + x \cdot \left(\color{blue}{\frac{-0.5 \cdot x + 0.5 \cdot \frac{x}{n}}{n}} + \frac{1}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{x}^{\left(\frac{3}{n}\right)}}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{t\_1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ (+ 1.0 x) x)) n)) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-26)
     (* (/ t_1 n) (/ 1.0 x))
     (if (<= (/ 1.0 n) 5e-134)
       t_0
       (if (<= (/ 1.0 n) 4e-79)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 4e-8)
           t_0
           (-
            (+ 1.0 (* x (+ (/ 1.0 n) (/ (+ (* x -0.5) (* 0.5 (/ x n))) n))))
            t_1)))))))

C

double code(double x, double n) {
	double t_0 = log(((1.0 + x) / x)) / n;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = (t_1 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_1;
	}
	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 = log(((1.0d0 + x) / x)) / n
    t_1 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-26)) then
        tmp = (t_1 / n) * (1.0d0 / x)
    else if ((1.0d0 / n) <= 5d-134) then
        tmp = t_0
    else if ((1.0d0 / n) <= 4d-79) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 4d-8) then
        tmp = t_0
    else
        tmp = (1.0d0 + (x * ((1.0d0 / n) + (((x * (-0.5d0)) + (0.5d0 * (x / n))) / n)))) - t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(((1.0 + x) / x)) / n;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = (t_1 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(((1.0 + x) / x)) / n
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-26:
		tmp = (t_1 / n) * (1.0 / x)
	elif (1.0 / n) <= 5e-134:
		tmp = t_0
	elif (1.0 / n) <= 4e-79:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 4e-8:
		tmp = t_0
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_1
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-26)
		tmp = Float64(Float64(t_1 / n) * Float64(1.0 / x));
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 4e-8)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(Float64(x * -0.5) + Float64(0.5 * Float64(x / n))) / n)))) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(((1.0 + x) / x)) / n;
	t_1 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-26)
		tmp = (t_1 / n) * (1.0 / x);
	elseif ((1.0 / n) <= 5e-134)
		tmp = t_0;
	elseif ((1.0 / n) <= 4e-79)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 4e-8)
		tmp = t_0;
	else
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-26], N[(N[(t$95$1 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-8], t$95$0, N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(x * -0.5), $MachinePrecision] + N[(0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-26

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. associate-/r* 97.0%

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

      2. mul-1-neg 97.0%

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

      3. log-rec 97.0%

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

      4. mul-1-neg 97.0%

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

      5. distribute-neg-frac 97.0%

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

      6. mul-1-neg 97.0%

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

      7. remove-double-neg 97.0%

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

      8. *-rgt-identity 97.0%

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

      9. associate-/l* 97.0%

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

      10. exp-to-pow 97.0%

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

   5. Simplified 97.0%

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

      1. div-inv 97.0%

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

   7. Applied egg-rr 97.0%

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

   if -5.00000000000000019e-26 < (/.f64 1 n) < 5.0000000000000003e-134 or 4e-79 < (/.f64 1 n) < 4.0000000000000001e-8

   1. Initial program 33.2%

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

   3. Taylor expanded in n around inf 86.1%

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

      1. log1p-define 86.1%

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

   5. Simplified 86.1%

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

      1. log1p-undefine 86.1%

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

      2. diff-log 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. +-commutative 86.1%

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

   9. Simplified 86.1%

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

   if 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. associate-/r* 76.3%

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

      2. mul-1-neg 76.3%

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

      3. log-rec 76.3%

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

      4. mul-1-neg 76.3%

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

      5. distribute-neg-frac 76.3%

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

      6. mul-1-neg 76.3%

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

      7. remove-double-neg 76.3%

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

      8. *-rgt-identity 76.3%

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

      9. associate-/l* 76.3%

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

      10. exp-to-pow 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in n around inf 76.3%

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

   if 4.0000000000000001e-8 < (/.f64 1 n)

   1. Initial program 55.4%

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

   3. Taylor expanded in x around 0 80.0%

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

   4. Taylor expanded in n around inf 87.7%

      \[\leadsto \left(1 + x \cdot \left(\color{blue}{\frac{-0.5 \cdot x + 0.5 \cdot \frac{x}{n}}{n}} + \frac{1}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{t\_1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ (+ 1.0 x) x)) n)) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-26)
     (* (/ t_1 n) (/ 1.0 x))
     (if (<= (/ 1.0 n) 5e-134)
       t_0
       (if (<= (/ 1.0 n) 4e-79)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 4e-8)
           t_0
           (if (<= (/ 1.0 n) 5e+157)
             (- (+ 1.0 (/ x n)) t_1)
             (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n))))))))

C

double code(double x, double n) {
	double t_0 = log(((1.0 + x) / x)) / n;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = (t_1 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / 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) :: t_1
    real(8) :: tmp
    t_0 = log(((1.0d0 + x) / x)) / n
    t_1 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-26)) then
        tmp = (t_1 / n) * (1.0d0 / x)
    else if ((1.0d0 / n) <= 5d-134) then
        tmp = t_0
    else if ((1.0d0 / n) <= 4d-79) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 4d-8) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d+157) then
        tmp = (1.0d0 + (x / n)) - t_1
    else
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(((1.0 + x) / x)) / n;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = (t_1 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(((1.0 + x) / x)) / n
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-26:
		tmp = (t_1 / n) * (1.0 / x)
	elif (1.0 / n) <= 5e-134:
		tmp = t_0
	elif (1.0 / n) <= 4e-79:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 4e-8:
		tmp = t_0
	elif (1.0 / n) <= 5e+157:
		tmp = (1.0 + (x / n)) - t_1
	else:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-26)
		tmp = Float64(Float64(t_1 / n) * Float64(1.0 / x));
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 4e-8)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e+157)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(((1.0 + x) / x)) / n;
	t_1 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-26)
		tmp = (t_1 / n) * (1.0 / x);
	elseif ((1.0 / n) <= 5e-134)
		tmp = t_0;
	elseif ((1.0 / n) <= 4e-79)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 4e-8)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e+157)
		tmp = (1.0 + (x / n)) - t_1;
	else
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-26], N[(N[(t$95$1 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-8], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+157], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-26

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. associate-/r* 97.0%

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

      2. mul-1-neg 97.0%

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

      3. log-rec 97.0%

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

      4. mul-1-neg 97.0%

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

      5. distribute-neg-frac 97.0%

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

      6. mul-1-neg 97.0%

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

      7. remove-double-neg 97.0%

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

      8. *-rgt-identity 97.0%

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

      9. associate-/l* 97.0%

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

      10. exp-to-pow 97.0%

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

   5. Simplified 97.0%

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

      1. div-inv 97.0%

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

   7. Applied egg-rr 97.0%

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

   if -5.00000000000000019e-26 < (/.f64 1 n) < 5.0000000000000003e-134 or 4e-79 < (/.f64 1 n) < 4.0000000000000001e-8

   1. Initial program 33.2%

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

   3. Taylor expanded in n around inf 86.1%

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

      1. log1p-define 86.1%

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

   5. Simplified 86.1%

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

      1. log1p-undefine 86.1%

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

      2. diff-log 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. +-commutative 86.1%

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

   9. Simplified 86.1%

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

   if 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. associate-/r* 76.3%

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

      2. mul-1-neg 76.3%

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

      3. log-rec 76.3%

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

      4. mul-1-neg 76.3%

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

      5. distribute-neg-frac 76.3%

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

      6. mul-1-neg 76.3%

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

      7. remove-double-neg 76.3%

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

      8. *-rgt-identity 76.3%

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

      9. associate-/l* 76.3%

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

      10. exp-to-pow 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in n around inf 76.3%

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

   if 4.0000000000000001e-8 < (/.f64 1 n) < 4.99999999999999976e157

   1. Initial program 92.4%

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

   3. Taylor expanded in x around 0 93.8%

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

   if 4.99999999999999976e157 < (/.f64 1 n)

   1. Initial program 29.6%

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

   3. Taylor expanded in n around inf 7.3%

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

      1. log1p-define 7.3%

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

   5. Simplified 7.3%

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

      1. log1p-undefine 7.3%

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

      2. diff-log 7.3%

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

   7. Applied egg-rr 7.3%

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

      1. +-commutative 7.3%

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

   9. Simplified 7.3%

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

   10. Taylor expanded in x around inf 70.7%

       \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{0.3333333333333333}{{x}^{2}}\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   12. Simplified 70.7%

       \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -5e+48)
     (/ (/ 0.3333333333333333 n) (pow x 3.0))
     (if (<= (/ 1.0 n) 5e-134)
       t_0
       (if (<= (/ 1.0 n) 4e-79)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 4e-8)
           t_0
           (if (<= (/ 1.0 n) 5e+157)
             (- 1.0 (pow x (/ 1.0 n)))
             (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n))))))))

C

double code(double x, double n) {
	double t_0 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e+48) {
		tmp = (0.3333333333333333 / n) / pow(x, 3.0);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / 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(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-5d+48)) then
        tmp = (0.3333333333333333d0 / n) / (x ** 3.0d0)
    else if ((1.0d0 / n) <= 5d-134) then
        tmp = t_0
    else if ((1.0d0 / n) <= 4d-79) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 4d-8) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d+157) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e+48) {
		tmp = (0.3333333333333333 / n) / Math.pow(x, 3.0);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -5e+48:
		tmp = (0.3333333333333333 / n) / math.pow(x, 3.0)
	elif (1.0 / n) <= 5e-134:
		tmp = t_0
	elif (1.0 / n) <= 4e-79:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 4e-8:
		tmp = t_0
	elif (1.0 / n) <= 5e+157:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+48)
		tmp = Float64(Float64(0.3333333333333333 / n) / (x ^ 3.0));
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 4e-8)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e+157)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e+48)
		tmp = (0.3333333333333333 / n) / (x ^ 3.0);
	elseif ((1.0 / n) <= 5e-134)
		tmp = t_0;
	elseif ((1.0 / n) <= 4e-79)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 4e-8)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e+157)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+48], N[(N[(0.3333333333333333 / n), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-8], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+157], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.99999999999999973e48

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 58.2%

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

      1. log1p-define 58.2%

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

   5. Simplified 58.2%

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

   6. Taylor expanded in x around -inf 34.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in x around 0 75.1%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 75.1%

         \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]

   9. Simplified 75.1%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]

   if -4.99999999999999973e48 < (/.f64 1 n) < 5.0000000000000003e-134 or 4e-79 < (/.f64 1 n) < 4.0000000000000001e-8

   1. Initial program 35.7%

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

   3. Taylor expanded in n around inf 83.5%

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

      1. log1p-define 83.5%

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

   5. Simplified 83.5%

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

      1. log1p-undefine 83.5%

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

      2. diff-log 83.6%

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

   7. Applied egg-rr 83.6%

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

      1. +-commutative 83.6%

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

   9. Simplified 83.6%

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

   if 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. associate-/r* 76.3%

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

      2. mul-1-neg 76.3%

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

      3. log-rec 76.3%

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

      4. mul-1-neg 76.3%

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

      5. distribute-neg-frac 76.3%

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

      6. mul-1-neg 76.3%

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

      7. remove-double-neg 76.3%

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

      8. *-rgt-identity 76.3%

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

      9. associate-/l* 76.3%

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

      10. exp-to-pow 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in n around inf 76.3%

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

   if 4.0000000000000001e-8 < (/.f64 1 n) < 4.99999999999999976e157

   1. Initial program 92.4%

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

   3. Taylor expanded in x around 0 92.4%

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

      1. *-rgt-identity 92.4%

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

      2. associate-*l/ 92.4%

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

      3. associate-/l* 92.4%

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

      4. exp-to-pow 92.4%

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

   5. Simplified 92.4%

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

   if 4.99999999999999976e157 < (/.f64 1 n)

   1. Initial program 29.6%

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

   3. Taylor expanded in n around inf 7.3%

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

      1. log1p-define 7.3%

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

   5. Simplified 7.3%

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

      1. log1p-undefine 7.3%

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

      2. diff-log 7.3%

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

   7. Applied egg-rr 7.3%

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

      1. +-commutative 7.3%

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

   9. Simplified 7.3%

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

   10. Taylor expanded in x around inf 70.7%

       \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{0.3333333333333333}{{x}^{2}}\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   12. Simplified 70.7%

       \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -5e-26)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-134)
       t_1
       (if (<= (/ 1.0 n) 4e-79)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 4e-8)
           t_1
           (if (<= (/ 1.0 n) 5e+157)
             (- 1.0 t_0)
             (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-134) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / 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) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-5d-26)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-134) then
        tmp = t_1
    else if ((1.0d0 / n) <= 4d-79) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 4d-8) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d+157) then
        tmp = 1.0d0 - t_0
    else
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    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 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-134) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -5e-26:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-134:
		tmp = t_1
	elif (1.0 / n) <= 4e-79:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 4e-8:
		tmp = t_1
	elif (1.0 / n) <= 5e+157:
		tmp = 1.0 - t_0
	else:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-26)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 4e-8)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+157)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-26)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-134)
		tmp = t_1;
	elseif ((1.0 / n) <= 4e-79)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 4e-8)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e+157)
		tmp = 1.0 - t_0;
	else
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	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[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-26], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-8], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+157], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-26

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. associate-/r* 97.0%

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

      2. mul-1-neg 97.0%

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

      3. log-rec 97.0%

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

      4. mul-1-neg 97.0%

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

      5. distribute-neg-frac 97.0%

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

      6. mul-1-neg 97.0%

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

      7. remove-double-neg 97.0%

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

      8. *-rgt-identity 97.0%

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

      9. associate-/l* 97.0%

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

      10. exp-to-pow 97.0%

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

   5. Simplified 97.0%

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

   if -5.00000000000000019e-26 < (/.f64 1 n) < 5.0000000000000003e-134 or 4e-79 < (/.f64 1 n) < 4.0000000000000001e-8

   1. Initial program 33.2%

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

   3. Taylor expanded in n around inf 86.1%

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

      1. log1p-define 86.1%

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

   5. Simplified 86.1%

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

      1. log1p-undefine 86.1%

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

      2. diff-log 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. +-commutative 86.1%

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

   9. Simplified 86.1%

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

   if 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. associate-/r* 76.3%

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

      2. mul-1-neg 76.3%

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

      3. log-rec 76.3%

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

      4. mul-1-neg 76.3%

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

      5. distribute-neg-frac 76.3%

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

      6. mul-1-neg 76.3%

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

      7. remove-double-neg 76.3%

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

      8. *-rgt-identity 76.3%

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

      9. associate-/l* 76.3%

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

      10. exp-to-pow 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in n around inf 76.3%

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

   if 4.0000000000000001e-8 < (/.f64 1 n) < 4.99999999999999976e157

   1. Initial program 92.4%

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

   3. Taylor expanded in x around 0 92.4%

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

      1. *-rgt-identity 92.4%

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

      2. associate-*l/ 92.4%

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

      3. associate-/l* 92.4%

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

      4. exp-to-pow 92.4%

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

   5. Simplified 92.4%

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

   if 4.99999999999999976e157 < (/.f64 1 n)

   1. Initial program 29.6%

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

   3. Taylor expanded in n around inf 7.3%

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

      1. log1p-define 7.3%

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

   5. Simplified 7.3%

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

      1. log1p-undefine 7.3%

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

      2. diff-log 7.3%

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

   7. Applied egg-rr 7.3%

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

      1. +-commutative 7.3%

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

   9. Simplified 7.3%

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

   10. Taylor expanded in x around inf 70.7%

       \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{0.3333333333333333}{{x}^{2}}\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   12. Simplified 70.7%

       \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{t\_1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;1 - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ (+ 1.0 x) x)) n)) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-26)
     (* (/ t_1 n) (/ 1.0 x))
     (if (<= (/ 1.0 n) 5e-134)
       t_0
       (if (<= (/ 1.0 n) 4e-79)
         (/ (/ 1.0 n) x)
         (if (<= (/ 1.0 n) 4e-8)
           t_0
           (if (<= (/ 1.0 n) 5e+157)
             (- 1.0 t_1)
             (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n))))))))

C

double code(double x, double n) {
	double t_0 = log(((1.0 + x) / x)) / n;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = (t_1 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = 1.0 - t_1;
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / 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) :: t_1
    real(8) :: tmp
    t_0 = log(((1.0d0 + x) / x)) / n
    t_1 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-26)) then
        tmp = (t_1 / n) * (1.0d0 / x)
    else if ((1.0d0 / n) <= 5d-134) then
        tmp = t_0
    else if ((1.0d0 / n) <= 4d-79) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 4d-8) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d+157) then
        tmp = 1.0d0 - t_1
    else
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(((1.0 + x) / x)) / n;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = (t_1 / n) * (1.0 / x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = 1.0 - t_1;
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(((1.0 + x) / x)) / n
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-26:
		tmp = (t_1 / n) * (1.0 / x)
	elif (1.0 / n) <= 5e-134:
		tmp = t_0
	elif (1.0 / n) <= 4e-79:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 4e-8:
		tmp = t_0
	elif (1.0 / n) <= 5e+157:
		tmp = 1.0 - t_1
	else:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-26)
		tmp = Float64(Float64(t_1 / n) * Float64(1.0 / x));
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 4e-8)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e+157)
		tmp = Float64(1.0 - t_1);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(((1.0 + x) / x)) / n;
	t_1 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-26)
		tmp = (t_1 / n) * (1.0 / x);
	elseif ((1.0 / n) <= 5e-134)
		tmp = t_0;
	elseif ((1.0 / n) <= 4e-79)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 4e-8)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e+157)
		tmp = 1.0 - t_1;
	else
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-26], N[(N[(t$95$1 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-8], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+157], N[(1.0 - t$95$1), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-26

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. associate-/r* 97.0%

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

      2. mul-1-neg 97.0%

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

      3. log-rec 97.0%

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

      4. mul-1-neg 97.0%

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

      5. distribute-neg-frac 97.0%

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

      6. mul-1-neg 97.0%

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

      7. remove-double-neg 97.0%

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

      8. *-rgt-identity 97.0%

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

      9. associate-/l* 97.0%

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

      10. exp-to-pow 97.0%

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

   5. Simplified 97.0%

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

      1. div-inv 97.0%

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

   7. Applied egg-rr 97.0%

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

   if -5.00000000000000019e-26 < (/.f64 1 n) < 5.0000000000000003e-134 or 4e-79 < (/.f64 1 n) < 4.0000000000000001e-8

   1. Initial program 33.2%

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

   3. Taylor expanded in n around inf 86.1%

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

      1. log1p-define 86.1%

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

   5. Simplified 86.1%

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

      1. log1p-undefine 86.1%

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

      2. diff-log 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. +-commutative 86.1%

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

   9. Simplified 86.1%

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

   if 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in x around inf 73.5%

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

      1. associate-/r* 76.3%

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

      2. mul-1-neg 76.3%

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

      3. log-rec 76.3%

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

      4. mul-1-neg 76.3%

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

      5. distribute-neg-frac 76.3%

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

      6. mul-1-neg 76.3%

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

      7. remove-double-neg 76.3%

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

      8. *-rgt-identity 76.3%

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

      9. associate-/l* 76.3%

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

      10. exp-to-pow 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in n around inf 76.3%

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

   if 4.0000000000000001e-8 < (/.f64 1 n) < 4.99999999999999976e157

   1. Initial program 92.4%

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

   3. Taylor expanded in x around 0 92.4%

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

      1. *-rgt-identity 92.4%

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

      2. associate-*l/ 92.4%

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

      3. associate-/l* 92.4%

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

      4. exp-to-pow 92.4%

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

   5. Simplified 92.4%

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

   if 4.99999999999999976e157 < (/.f64 1 n)

   1. Initial program 29.6%

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

   3. Taylor expanded in n around inf 7.3%

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

      1. log1p-define 7.3%

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

   5. Simplified 7.3%

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

      1. log1p-undefine 7.3%

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

      2. diff-log 7.3%

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

   7. Applied egg-rr 7.3%

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

      1. +-commutative 7.3%

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

   9. Simplified 7.3%

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

   10. Taylor expanded in x around inf 70.7%

       \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{0.3333333333333333}{{x}^{2}}\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   12. Simplified 70.7%

       \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 21000:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+72} \lor \neg \left(x \leq 2.55 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 21000.0)
   (/ (- x (log x)) n)
   (if (or (<= x 1.05e+72) (not (<= x 2.55e+173)))
     (/ 0.0 n)
     (/ (- (/ 1.0 n) (/ (/ 0.5 n) x)) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 21000.0) {
		tmp = (x - log(x)) / n;
	} else if ((x <= 1.05e+72) || !(x <= 2.55e+173)) {
		tmp = 0.0 / n;
	} else {
		tmp = ((1.0 / n) - ((0.5 / n) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 21000.0d0) then
        tmp = (x - log(x)) / n
    else if ((x <= 1.05d+72) .or. (.not. (x <= 2.55d+173))) then
        tmp = 0.0d0 / n
    else
        tmp = ((1.0d0 / n) - ((0.5d0 / n) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 21000.0) {
		tmp = (x - Math.log(x)) / n;
	} else if ((x <= 1.05e+72) || !(x <= 2.55e+173)) {
		tmp = 0.0 / n;
	} else {
		tmp = ((1.0 / n) - ((0.5 / n) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 21000.0:
		tmp = (x - math.log(x)) / n
	elif (x <= 1.05e+72) or not (x <= 2.55e+173):
		tmp = 0.0 / n
	else:
		tmp = ((1.0 / n) - ((0.5 / n) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 21000.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif ((x <= 1.05e+72) || !(x <= 2.55e+173))
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(0.5 / n) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 21000.0)
		tmp = (x - log(x)) / n;
	elseif ((x <= 1.05e+72) || ~((x <= 2.55e+173)))
		tmp = 0.0 / n;
	else
		tmp = ((1.0 / n) - ((0.5 / n) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 21000.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[Or[LessEqual[x, 1.05e+72], N[Not[LessEqual[x, 2.55e+173]], $MachinePrecision]], N[(0.0 / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(0.5 / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 21000

   1. Initial program 33.0%

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

   3. Taylor expanded in n around inf 60.5%

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

      1. log1p-define 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in x around 0 58.9%

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

   if 21000 < x < 1.0500000000000001e72 or 2.54999999999999975e173 < x

   1. Initial program 83.1%

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

   3. Taylor expanded in n around inf 79.6%

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

      1. log1p-define 79.6%

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

   5. Simplified 79.6%

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

      1. log1p-undefine 79.6%

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

      2. diff-log 79.6%

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

   7. Applied egg-rr 79.6%

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

      1. +-commutative 79.6%

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

   9. Simplified 79.6%

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

   10. Taylor expanded in x around inf 83.2%

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

   if 1.0500000000000001e72 < x < 2.54999999999999975e173

   1. Initial program 42.6%

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

   3. Taylor expanded in n around inf 42.6%

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

      1. log1p-define 42.6%

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

   5. Simplified 42.6%

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

   6. Taylor expanded in x around inf 82.8%

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

      1. associate-*r/ 82.8%

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

      2. metadata-eval 82.8%

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

      3. associate-/r* 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 21000:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+72} \lor \neg \left(x \leq 2.55 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+72} \lor \neg \left(x \leq 5.5 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (log x) (- n))
   (if (or (<= x 1.05e+72) (not (<= x 5.5e+173)))
     (/ 0.0 n)
     (/ (- (/ 1.0 n) (/ (/ 0.5 n) x)) x))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = log(x) / -n;
	} else if ((x <= 1.05e+72) || !(x <= 5.5e+173)) {
		tmp = 0.0 / n;
	} else {
		tmp = ((1.0 / n) - ((0.5 / n) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = log(x) / -n
    else if ((x <= 1.05d+72) .or. (.not. (x <= 5.5d+173))) then
        tmp = 0.0d0 / n
    else
        tmp = ((1.0d0 / n) - ((0.5d0 / n) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = Math.log(x) / -n;
	} else if ((x <= 1.05e+72) || !(x <= 5.5e+173)) {
		tmp = 0.0 / n;
	} else {
		tmp = ((1.0 / n) - ((0.5 / n) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = math.log(x) / -n
	elif (x <= 1.05e+72) or not (x <= 5.5e+173):
		tmp = 0.0 / n
	else:
		tmp = ((1.0 / n) - ((0.5 / n) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(log(x) / Float64(-n));
	elseif ((x <= 1.05e+72) || !(x <= 5.5e+173))
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(0.5 / n) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = log(x) / -n;
	elseif ((x <= 1.05e+72) || ~((x <= 5.5e+173)))
		tmp = 0.0 / n;
	else
		tmp = ((1.0 / n) - ((0.5 / n) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[Or[LessEqual[x, 1.05e+72], N[Not[LessEqual[x, 5.5e+173]], $MachinePrecision]], N[(0.0 / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(0.5 / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 33.4%

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

   3. Taylor expanded in n around inf 60.3%

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

      1. log1p-define 60.3%

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

   5. Simplified 60.3%

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

   6. Taylor expanded in x around 0 58.9%

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

      1. neg-mul-1 58.9%

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

   8. Simplified 58.9%

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

   if 1 < x < 1.0500000000000001e72 or 5.50000000000000049e173 < x

   1. Initial program 81.3%

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

   3. Taylor expanded in n around inf 79.6%

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

      1. log1p-define 79.6%

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

   5. Simplified 79.6%

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

      1. log1p-undefine 79.6%

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

      2. diff-log 79.7%

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

   7. Applied egg-rr 79.7%

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

      1. +-commutative 79.7%

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

   9. Simplified 79.7%

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

   10. Taylor expanded in x around inf 81.3%

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

   if 1.0500000000000001e72 < x < 5.50000000000000049e173

   1. Initial program 42.6%

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

   3. Taylor expanded in n around inf 42.6%

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

      1. log1p-define 42.6%

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

   5. Simplified 42.6%

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

   6. Taylor expanded in x around inf 82.8%

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

      1. associate-*r/ 82.8%

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

      2. metadata-eval 82.8%

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

      3. associate-/r* 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+72} \lor \neg \left(x \leq 5.5 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -500000.0)
   (/ 0.0 n)
   (/ (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) x)) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500000.0) {
		tmp = 0.0 / n;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-500000.0d0)) then
        tmp = 0.0d0 / n
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500000.0) {
		tmp = 0.0 / n;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -500000.0:
		tmp = 0.0 / n
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500000.0)
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -500000.0)
		tmp = 0.0 / n;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -500000.0], N[(0.0 / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 63.3%

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

      1. log1p-define 63.3%

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

   5. Simplified 63.3%

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

      1. log1p-undefine 63.3%

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

      2. diff-log 63.3%

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

   7. Applied egg-rr 63.3%

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

      1. +-commutative 63.3%

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

   9. Simplified 63.3%

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

   10. Taylor expanded in x around inf 67.2%

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

   if -5e5 < (/.f64 1 n)

   1. Initial program 36.3%

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

   3. Taylor expanded in n around inf 64.6%

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

      1. log1p-define 64.6%

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

   5. Simplified 64.6%

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

      1. log1p-undefine 64.6%

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

      2. diff-log 64.7%

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

   7. Applied egg-rr 64.7%

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

      1. +-commutative 64.7%

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

   9. Simplified 64.7%

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

   10. Taylor expanded in x around inf 42.3%

       \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]
   11. Step-by-step derivation

   12. Simplified 49.0%

       \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}\\ \mathbf{if}\;x \leq 4.9 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{1 + t\_0}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{-1 + t\_0}{n \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)))
   (if (<= x 4.9e+173)
     (/ (/ (+ 1.0 t_0) x) n)
     (+ -1.0 (- 1.0 (/ (+ -1.0 t_0) (* n x)))))))

C

double code(double x, double n) {
	double t_0 = (-0.5 + (0.3333333333333333 / x)) / x;
	double tmp;
	if (x <= 4.9e+173) {
		tmp = ((1.0 + t_0) / x) / n;
	} else {
		tmp = -1.0 + (1.0 - ((-1.0 + 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 = ((-0.5d0) + (0.3333333333333333d0 / x)) / x
    if (x <= 4.9d+173) then
        tmp = ((1.0d0 + t_0) / x) / n
    else
        tmp = (-1.0d0) + (1.0d0 - (((-1.0d0) + t_0) / (n * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (-0.5 + (0.3333333333333333 / x)) / x;
	double tmp;
	if (x <= 4.9e+173) {
		tmp = ((1.0 + t_0) / x) / n;
	} else {
		tmp = -1.0 + (1.0 - ((-1.0 + t_0) / (n * x)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (-0.5 + (0.3333333333333333 / x)) / x
	tmp = 0
	if x <= 4.9e+173:
		tmp = ((1.0 + t_0) / x) / n
	else:
		tmp = -1.0 + (1.0 - ((-1.0 + t_0) / (n * x)))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)
	tmp = 0.0
	if (x <= 4.9e+173)
		tmp = Float64(Float64(Float64(1.0 + t_0) / x) / n);
	else
		tmp = Float64(-1.0 + Float64(1.0 - Float64(Float64(-1.0 + t_0) / Float64(n * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (-0.5 + (0.3333333333333333 / x)) / x;
	tmp = 0.0;
	if (x <= 4.9e+173)
		tmp = ((1.0 + t_0) / x) / n;
	else
		tmp = -1.0 + (1.0 - ((-1.0 + t_0) / (n * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, 4.9e+173], N[(N[(N[(1.0 + t$95$0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(-1.0 + N[(1.0 - N[(N[(-1.0 + t$95$0), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9000000000000001e173

   1. Initial program 39.4%

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

   3. Taylor expanded in n around inf 57.4%

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

      1. log1p-define 57.4%

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

   5. Simplified 57.4%

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

      1. log1p-undefine 57.4%

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

      2. diff-log 57.5%

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

   7. Applied egg-rr 57.5%

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

      1. +-commutative 57.5%

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

   9. Simplified 57.5%

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

   10. Taylor expanded in x around inf 39.1%

       \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{0.3333333333333333}{{x}^{2}}\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   12. Simplified 39.1%

       \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

   if 4.9000000000000001e173 < x

   1. Initial program 87.4%

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

   3. Taylor expanded in n around inf 87.5%

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

      1. log1p-define 87.5%

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

   5. Simplified 87.5%

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

   6. Taylor expanded in x around -inf 64.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]
   7. Step-by-step derivation

      1. div-sub 64.4%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x} - \frac{1}{x}\right)}}{n} \]

      2. add-sqr-sqrt 64.4%

         \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{\sqrt{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}} \cdot \sqrt{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}}{x} - \frac{1}{x}\right)}{n} \]

      3. sqrt-unprod 64.4%

         \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{\sqrt{\left(-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right) \cdot \left(-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right)}}}{x} - \frac{1}{x}\right)}{n} \]

      4. mul-1-neg 64.4%

         \[\leadsto \frac{-1 \cdot \left(\frac{\sqrt{\color{blue}{\left(-\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right)} \cdot \left(-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right)}}{x} - \frac{1}{x}\right)}{n} \]

      5. mul-1-neg 64.4%

         \[\leadsto \frac{-1 \cdot \left(\frac{\sqrt{\left(-\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right) \cdot \color{blue}{\left(-\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right)}}}{x} - \frac{1}{x}\right)}{n} \]

      6. sqr-neg 64.4%

         \[\leadsto \frac{-1 \cdot \left(\frac{\sqrt{\color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}}{x} - \frac{1}{x}\right)}{n} \]

      7. sqrt-unprod 0.0%

         \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{\sqrt{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}} \cdot \sqrt{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}}{x} - \frac{1}{x}\right)}{n} \]

      8. add-sqr-sqrt 64.4%

         \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}{x} - \frac{1}{x}\right)}{n} \]

      9. sub-neg 64.4%

         \[\leadsto \frac{-1 \cdot \left(\frac{\frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{x} + \left(-0.5\right)}}{x}}{x} - \frac{1}{x}\right)}{n} \]

      10. un-div-inv 64.4%

          \[\leadsto \frac{-1 \cdot \left(\frac{\frac{\color{blue}{\frac{0.3333333333333333}{x}} + \left(-0.5\right)}{x}}{x} - \frac{1}{x}\right)}{n} \]

      11. metadata-eval 64.4%

          \[\leadsto \frac{-1 \cdot \left(\frac{\frac{\frac{0.3333333333333333}{x} + \color{blue}{-0.5}}{x}}{x} - \frac{1}{x}\right)}{n} \]

   8. Applied egg-rr 64.4%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{x} - \frac{1}{x}\right)}}{n} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 64.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1 \cdot \left(\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{x} - \frac{1}{x}\right)}{n}\right)\right)} \]

      2. expm1-undefine 77.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1 \cdot \left(\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{x} - \frac{1}{x}\right)}{n}\right)} - 1} \]

      3. mul-1-neg 77.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{-\left(\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{x} - \frac{1}{x}\right)}}{n}\right)} - 1 \]

      4. sub-div 77.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-\color{blue}{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}}{n}\right)} - 1 \]

   10. Applied egg-rr 77.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)} - 1} \]
   11. Step-by-step derivation

       1. sub-neg 77.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)} + \left(-1\right)} \]

       2. metadata-eval 77.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)} + \color{blue}{-1} \]

       3. +-commutative 77.3%

          \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{-\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)}} \]

       4. log1p-undefine 77.3%

          \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \frac{-\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)}} \]

       5. rem-exp-log 77.6%

          \[\leadsto -1 + \color{blue}{\left(1 + \frac{-\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)} \]

       6. distribute-frac-neg 77.6%

          \[\leadsto -1 + \left(1 + \color{blue}{\left(-\frac{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)}\right) \]

       7. unsub-neg 77.6%

          \[\leadsto -1 + \color{blue}{\left(1 - \frac{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)} \]

       8. associate-/r* 77.6%

          \[\leadsto -1 + \left(1 - \color{blue}{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x \cdot n}}\right) \]

       9. sub-neg 77.6%

          \[\leadsto -1 + \left(1 - \frac{\color{blue}{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + \left(-1\right)}}{x \cdot n}\right) \]

       10. metadata-eval 77.6%

           \[\leadsto -1 + \left(1 - \frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + \color{blue}{-1}}{x \cdot n}\right) \]

       11. +-commutative 77.6%

           \[\leadsto -1 + \left(1 - \frac{\color{blue}{-1 + \frac{\frac{0.3333333333333333}{x} + -0.5}{x}}}{x \cdot n}\right) \]

       12. +-commutative 77.6%

           \[\leadsto -1 + \left(1 - \frac{-1 + \frac{\color{blue}{-0.5 + \frac{0.3333333333333333}{x}}}{x}}{x \cdot n}\right) \]

   12. Simplified 77.6%

       \[\leadsto \color{blue}{-1 + \left(1 - \frac{-1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.9 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{-1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.72 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{-1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.72e+179)
   (/ (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) x)) x)
   (+
    -1.0
    (- 1.0 (/ (+ -1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) (* n x))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.72e+179) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	} else {
		tmp = -1.0 + (1.0 - ((-1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (n * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.72d+179) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / x)) / x
    else
        tmp = (-1.0d0) + (1.0d0 - (((-1.0d0) + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / (n * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.72e+179) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	} else {
		tmp = -1.0 + (1.0 - ((-1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (n * x)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.72e+179:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x
	else:
		tmp = -1.0 + (1.0 - ((-1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (n * x)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.72e+179)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / x)) / x);
	else
		tmp = Float64(-1.0 + Float64(1.0 - Float64(Float64(-1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / Float64(n * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.72e+179)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	else
		tmp = -1.0 + (1.0 - ((-1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (n * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.72e+179], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(-1.0 + N[(1.0 - N[(N[(-1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7200000000000001e179

   1. Initial program 40.4%

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

   3. Taylor expanded in n around inf 58.1%

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

      1. log1p-define 58.1%

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

   5. Simplified 58.1%

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

      1. log1p-undefine 58.1%

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

      2. diff-log 58.1%

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

   7. Applied egg-rr 58.1%

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

      1. +-commutative 58.1%

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

   9. Simplified 58.1%

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

   10. Taylor expanded in x around inf 27.9%

       \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]
   11. Step-by-step derivation

   12. Simplified 39.0%

       \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}} \]

   if 1.7200000000000001e179 < x

   1. Initial program 86.8%

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

   3. Taylor expanded in n around inf 86.8%

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

      1. log1p-define 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in x around -inf 66.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]
   7. Step-by-step derivation

      1. div-sub 66.0%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x} - \frac{1}{x}\right)}}{n} \]

      2. add-sqr-sqrt 66.0%

         \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{\sqrt{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}} \cdot \sqrt{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}}{x} - \frac{1}{x}\right)}{n} \]

      3. sqrt-unprod 66.0%

         \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{\sqrt{\left(-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right) \cdot \left(-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right)}}}{x} - \frac{1}{x}\right)}{n} \]

      4. mul-1-neg 66.0%

         \[\leadsto \frac{-1 \cdot \left(\frac{\sqrt{\color{blue}{\left(-\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right)} \cdot \left(-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right)}}{x} - \frac{1}{x}\right)}{n} \]

      5. mul-1-neg 66.0%

         \[\leadsto \frac{-1 \cdot \left(\frac{\sqrt{\left(-\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right) \cdot \color{blue}{\left(-\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right)}}}{x} - \frac{1}{x}\right)}{n} \]

      6. sqr-neg 66.0%

         \[\leadsto \frac{-1 \cdot \left(\frac{\sqrt{\color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}}{x} - \frac{1}{x}\right)}{n} \]

      7. sqrt-unprod 0.0%

         \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{\sqrt{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}} \cdot \sqrt{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}}{x} - \frac{1}{x}\right)}{n} \]

      8. add-sqr-sqrt 66.0%

         \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}{x} - \frac{1}{x}\right)}{n} \]

      9. sub-neg 66.0%

         \[\leadsto \frac{-1 \cdot \left(\frac{\frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{x} + \left(-0.5\right)}}{x}}{x} - \frac{1}{x}\right)}{n} \]

      10. un-div-inv 66.0%

          \[\leadsto \frac{-1 \cdot \left(\frac{\frac{\color{blue}{\frac{0.3333333333333333}{x}} + \left(-0.5\right)}{x}}{x} - \frac{1}{x}\right)}{n} \]

      11. metadata-eval 66.0%

          \[\leadsto \frac{-1 \cdot \left(\frac{\frac{\frac{0.3333333333333333}{x} + \color{blue}{-0.5}}{x}}{x} - \frac{1}{x}\right)}{n} \]

   8. Applied egg-rr 66.0%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{x} - \frac{1}{x}\right)}}{n} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 65.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1 \cdot \left(\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{x} - \frac{1}{x}\right)}{n}\right)\right)} \]

      2. expm1-undefine 79.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1 \cdot \left(\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{x} - \frac{1}{x}\right)}{n}\right)} - 1} \]

      3. mul-1-neg 79.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{-\left(\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{x} - \frac{1}{x}\right)}}{n}\right)} - 1 \]

      4. sub-div 79.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-\color{blue}{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}}{n}\right)} - 1 \]

   10. Applied egg-rr 79.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)} - 1} \]
   11. Step-by-step derivation

       1. sub-neg 79.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)} + \left(-1\right)} \]

       2. metadata-eval 79.7%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{-\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)} + \color{blue}{-1} \]

       3. +-commutative 79.7%

          \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{-\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)}} \]

       4. log1p-undefine 79.7%

          \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \frac{-\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)}} \]

       5. rem-exp-log 79.8%

          \[\leadsto -1 + \color{blue}{\left(1 + \frac{-\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)} \]

       6. distribute-frac-neg 79.8%

          \[\leadsto -1 + \left(1 + \color{blue}{\left(-\frac{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)}\right) \]

       7. unsub-neg 79.8%

          \[\leadsto -1 + \color{blue}{\left(1 - \frac{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x}}{n}\right)} \]

       8. associate-/r* 79.8%

          \[\leadsto -1 + \left(1 - \color{blue}{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} - 1}{x \cdot n}}\right) \]

       9. sub-neg 79.8%

          \[\leadsto -1 + \left(1 - \frac{\color{blue}{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + \left(-1\right)}}{x \cdot n}\right) \]

       10. metadata-eval 79.8%

           \[\leadsto -1 + \left(1 - \frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + \color{blue}{-1}}{x \cdot n}\right) \]

       11. +-commutative 79.8%

           \[\leadsto -1 + \left(1 - \frac{\color{blue}{-1 + \frac{\frac{0.3333333333333333}{x} + -0.5}{x}}}{x \cdot n}\right) \]

       12. +-commutative 79.8%

           \[\leadsto -1 + \left(1 - \frac{-1 + \frac{\color{blue}{-0.5 + \frac{0.3333333333333333}{x}}}{x}}{x \cdot n}\right) \]

   12. Simplified 79.8%

       \[\leadsto \color{blue}{-1 + \left(1 - \frac{-1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.72 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{-1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.9% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{n} \cdot \frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (* (/ 1.0 n) (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x)))

C

double code(double x, double n) {
	return (1.0 / n) * ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / n) * ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x)
end function

Java

public static double code(double x, double n) {
	return (1.0 / n) * ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x);
}

Python

def code(x, n):
	return (1.0 / n) * ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x)

Julia

function code(x, n)
	return Float64(Float64(1.0 / n) * Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x))
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / n) * ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x);
end

Wolfram

code[x_, n_] := N[(N[(1.0 / n), $MachinePrecision] * N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in n around inf 64.3%

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

   1. log1p-define 64.3%

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

5. Simplified 64.3%

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

6. Taylor expanded in x around -inf 44.9%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]
7. Step-by-step derivation

   1. div-inv 44.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}\right) \cdot \frac{1}{n}} \]

8. Applied egg-rr 44.9%

   \[\leadsto \color{blue}{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + 1}{x} \cdot \frac{1}{n}} \]

9. Final simplification 44.9%

   \[\leadsto \frac{1}{n} \cdot \frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x} \]
10. Add Preprocessing

Alternative 15: 45.9% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n))

C

double code(double x, double n) {
	return ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
end function

Java

public static double code(double x, double n) {
	return ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
}

Python

def code(x, n):
	return ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in n around inf 64.3%

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

   1. log1p-define 64.3%

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

5. Simplified 64.3%

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

   1. log1p-undefine 64.3%

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

   2. diff-log 64.4%

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

7. Applied egg-rr 64.4%

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

   1. +-commutative 64.4%

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

9. Simplified 64.4%

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

10. Taylor expanded in x around inf 44.9%

    \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{0.3333333333333333}{{x}^{2}}\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

12. Simplified 44.9%

    \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}}{n} \]

13. Final simplification 44.9%

    \[\leadsto \frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n} \]
14. Add Preprocessing

Alternative 16: 44.0% accurate, 19.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n} + \frac{0.5}{n \cdot x}}{x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (+ (/ 1.0 n) (/ 0.5 (* n x))) x))

C

double code(double x, double n) {
	return ((1.0 / n) + (0.5 / (n * x))) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 / n) + (0.5d0 / (n * x))) / x
end function

Java

public static double code(double x, double n) {
	return ((1.0 / n) + (0.5 / (n * x))) / x;
}

Python

def code(x, n):
	return ((1.0 / n) + (0.5 / (n * x))) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 / n) + Float64(0.5 / Float64(n * x))) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 / n) + (0.5 / (n * x))) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 / n), $MachinePrecision] + N[(0.5 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in n around inf 64.3%

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

   1. log1p-define 64.3%

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

5. Simplified 64.3%

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

6. Taylor expanded in x around -inf 44.9%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]
7. Step-by-step derivation

   1. div-sub 44.9%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x} - \frac{1}{x}\right)}}{n} \]

   2. add-sqr-sqrt 30.9%

      \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{\sqrt{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}} \cdot \sqrt{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}}{x} - \frac{1}{x}\right)}{n} \]

   3. sqrt-unprod 31.5%

      \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{\sqrt{\left(-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right) \cdot \left(-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right)}}}{x} - \frac{1}{x}\right)}{n} \]

   4. mul-1-neg 31.5%

      \[\leadsto \frac{-1 \cdot \left(\frac{\sqrt{\color{blue}{\left(-\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right)} \cdot \left(-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right)}}{x} - \frac{1}{x}\right)}{n} \]

   5. mul-1-neg 31.5%

      \[\leadsto \frac{-1 \cdot \left(\frac{\sqrt{\left(-\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right) \cdot \color{blue}{\left(-\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}\right)}}}{x} - \frac{1}{x}\right)}{n} \]

   6. sqr-neg 31.5%

      \[\leadsto \frac{-1 \cdot \left(\frac{\sqrt{\color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}}{x} - \frac{1}{x}\right)}{n} \]

   7. sqrt-unprod 0.6%

      \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{\sqrt{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}} \cdot \sqrt{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}}{x} - \frac{1}{x}\right)}{n} \]

   8. add-sqr-sqrt 31.2%

      \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}{x} - \frac{1}{x}\right)}{n} \]

   9. sub-neg 31.2%

      \[\leadsto \frac{-1 \cdot \left(\frac{\frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{x} + \left(-0.5\right)}}{x}}{x} - \frac{1}{x}\right)}{n} \]

   10. un-div-inv 31.2%

       \[\leadsto \frac{-1 \cdot \left(\frac{\frac{\color{blue}{\frac{0.3333333333333333}{x}} + \left(-0.5\right)}{x}}{x} - \frac{1}{x}\right)}{n} \]

   11. metadata-eval 31.2%

       \[\leadsto \frac{-1 \cdot \left(\frac{\frac{\frac{0.3333333333333333}{x} + \color{blue}{-0.5}}{x}}{x} - \frac{1}{x}\right)}{n} \]

8. Applied egg-rr 31.2%

   \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{x} - \frac{1}{x}\right)}}{n} \]

9. Taylor expanded in x around inf 43.7%

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

    1. associate-*r/ 43.7%

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

    2. metadata-eval 43.7%

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

    3. *-commutative 43.7%

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

11. Simplified 43.7%

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

12. Final simplification 43.7%

    \[\leadsto \frac{\frac{1}{n} + \frac{0.5}{n \cdot x}}{x} \]
13. Add Preprocessing

Alternative 17: 39.5% 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 50.5%

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

3. Taylor expanded in n around inf 64.3%

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

   1. log1p-define 64.3%

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

5. Simplified 64.3%

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

6. Taylor expanded in x around inf 41.3%

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

   1. *-commutative 41.3%

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

8. Simplified 41.3%

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

9. Final simplification 41.3%

   \[\leadsto \frac{1}{n \cdot x} \]
10. Add Preprocessing

Alternative 18: 40.1% 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 50.5%

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

3. Taylor expanded in x around inf 55.3%

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

   1. associate-/r* 55.8%

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

   2. mul-1-neg 55.8%

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

   3. log-rec 55.8%

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

   4. mul-1-neg 55.8%

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

   5. distribute-neg-frac 55.8%

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

   6. mul-1-neg 55.8%

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

   7. remove-double-neg 55.8%

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

   8. *-rgt-identity 55.8%

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

   9. associate-/l* 55.8%

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

   10. exp-to-pow 55.8%

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

5. Simplified 55.8%

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

6. Taylor expanded in n around inf 41.6%

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

7. Final simplification 41.6%

   \[\leadsto \frac{\frac{1}{n}}{x} \]
8. Add Preprocessing

Alternative 19: 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 50.5%

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

3. Taylor expanded in x around 0 25.7%

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

4. Taylor expanded in x around inf 4.6%

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

5. Final simplification 4.6%

   \[\leadsto \frac{x}{n} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))