2nthrt (problem 3.4.6)

Percentage Accurate: 54.5% → 84.4%

Time: 20.0s

Alternatives: 16

Speedup: 2.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% 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: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{{x}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n} + -1\right)\right)\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-56)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-142)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 10.0)
         (/ (pow x (log1p (expm1 (+ (/ 1.0 n) -1.0)))) n)
         (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = pow(x, log1p(expm1(((1.0 / n) + -1.0)))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = Math.pow(x, Math.log1p(Math.expm1(((1.0 / n) + -1.0)))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-56:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-142:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 10.0:
		tmp = math.pow(x, math.log1p(math.expm1(((1.0 / n) + -1.0)))) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-56)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-142)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64((x ^ log1p(expm1(Float64(Float64(1.0 / n) + -1.0)))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-56], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-142], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[Power[x, N[Log[1 + N[(Exp[N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -4.99999999999999997e-56

   1. Initial program 85.7%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. mul-1-neg 95.7%

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

      2. log-rec 95.7%

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

      3. mul-1-neg 95.7%

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

      4. distribute-neg-frac 95.7%

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

      5. mul-1-neg 95.7%

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

      6. remove-double-neg 95.7%

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

      7. *-commutative 95.7%

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

   4. Simplified 95.7%

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

      1. expm1-log1p-u 54.6%

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

      2. expm1-udef 44.0%

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

      3. associate-/r* 44.0%

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

      4. div-inv 44.0%

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

      5. pow-to-exp 44.0%

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

      6. pow1 44.0%

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

      7. pow-div 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. expm1-def 54.0%

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

      2. expm1-log1p 95.1%

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

      3. sub-neg 95.1%

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

      4. metadata-eval 95.1%

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

   8. Simplified 95.1%

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

      1. unpow-prod-up 95.7%

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

      2. inv-pow 95.7%

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

      3. *-un-lft-identity 95.7%

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

      4. times-frac 95.8%

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

   10. Applied egg-rr 95.8%

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

       1. /-rgt-identity 95.8%

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

       2. associate-/l/ 95.7%

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

       3. associate-/r* 95.7%

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

       4. associate-*r/ 95.7%

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

       5. associate-*r/ 95.8%

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

       6. *-rgt-identity 95.8%

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

   12. Simplified 95.8%

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

   if -4.99999999999999997e-56 < (/.f64 1 n) < 5.0000000000000002e-142

   1. Initial program 44.4%

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

   2. Taylor expanded in n around inf 91.6%

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

      1. +-rgt-identity 91.6%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 91.6%

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

      3. log1p-def 91.6%

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

   4. Simplified 91.6%

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

      1. log1p-udef 91.6%

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

      2. diff-log 91.8%

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

   6. Applied egg-rr 91.8%

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

   if 5.0000000000000002e-142 < (/.f64 1 n) < 10

   1. Initial program 8.7%

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

   2. Taylor expanded in x around inf 68.3%

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

      1. mul-1-neg 68.3%

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

      2. log-rec 68.3%

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

      3. mul-1-neg 68.3%

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

      4. distribute-neg-frac 68.3%

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

      5. mul-1-neg 68.3%

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

      6. remove-double-neg 68.3%

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

      7. *-commutative 68.3%

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

   4. Simplified 68.3%

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

      1. expm1-log1p-u 68.3%

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

      2. expm1-udef 11.6%

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

      3. associate-/r* 11.6%

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

      4. div-inv 11.6%

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

      5. pow-to-exp 11.7%

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

      6. pow1 11.7%

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

      7. pow-div 14.3%

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

   6. Applied egg-rr 14.3%

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

      1. expm1-def 72.3%

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

      2. expm1-log1p 72.3%

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

      3. sub-neg 72.3%

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

      4. metadata-eval 72.3%

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

   8. Simplified 72.3%

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

      1. log1p-expm1-u 72.3%

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

   10. Applied egg-rr 72.3%

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

   if 10 < (/.f64 1 n)

   1. Initial program 54.3%

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

   2. Taylor expanded in n around 0 54.3%

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

      1. log1p-def 96.7%

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

   4. Simplified 96.7%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{{x}^{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{n} + -1\right)\right)\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{{x}^{\left(\frac{1 - {n}^{-2}}{-1 + \frac{-1}{n}}\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-56)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-142)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 10.0)
         (/ (pow x (/ (- 1.0 (pow n -2.0)) (+ -1.0 (/ -1.0 n)))) n)
         (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = pow(x, ((1.0 - pow(n, -2.0)) / (-1.0 + (-1.0 / n)))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = Math.pow(x, ((1.0 - Math.pow(n, -2.0)) / (-1.0 + (-1.0 / n)))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-56:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-142:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 10.0:
		tmp = math.pow(x, ((1.0 - math.pow(n, -2.0)) / (-1.0 + (-1.0 / n)))) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-56)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-142)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64((x ^ Float64(Float64(1.0 - (n ^ -2.0)) / Float64(-1.0 + Float64(-1.0 / n)))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-56], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-142], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[Power[x, N[(N[(1.0 - N[Power[n, -2.0], $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -4.99999999999999997e-56

   1. Initial program 85.7%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. mul-1-neg 95.7%

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

      2. log-rec 95.7%

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

      3. mul-1-neg 95.7%

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

      4. distribute-neg-frac 95.7%

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

      5. mul-1-neg 95.7%

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

      6. remove-double-neg 95.7%

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

      7. *-commutative 95.7%

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

   4. Simplified 95.7%

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

      1. expm1-log1p-u 54.6%

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

      2. expm1-udef 44.0%

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

      3. associate-/r* 44.0%

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

      4. div-inv 44.0%

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

      5. pow-to-exp 44.0%

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

      6. pow1 44.0%

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

      7. pow-div 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. expm1-def 54.0%

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

      2. expm1-log1p 95.1%

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

      3. sub-neg 95.1%

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

      4. metadata-eval 95.1%

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

   8. Simplified 95.1%

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

      1. unpow-prod-up 95.7%

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

      2. inv-pow 95.7%

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

      3. *-un-lft-identity 95.7%

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

      4. times-frac 95.8%

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

   10. Applied egg-rr 95.8%

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

       1. /-rgt-identity 95.8%

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

       2. associate-/l/ 95.7%

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

       3. associate-/r* 95.7%

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

       4. associate-*r/ 95.7%

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

       5. associate-*r/ 95.8%

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

       6. *-rgt-identity 95.8%

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

   12. Simplified 95.8%

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

   if -4.99999999999999997e-56 < (/.f64 1 n) < 5.0000000000000002e-142

   1. Initial program 44.4%

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

   2. Taylor expanded in n around inf 91.6%

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

      1. +-rgt-identity 91.6%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 91.6%

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

      3. log1p-def 91.6%

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

   4. Simplified 91.6%

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

      1. log1p-udef 91.6%

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

      2. diff-log 91.8%

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

   6. Applied egg-rr 91.8%

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

   if 5.0000000000000002e-142 < (/.f64 1 n) < 10

   1. Initial program 8.7%

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

   2. Taylor expanded in x around inf 68.3%

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

      1. mul-1-neg 68.3%

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

      2. log-rec 68.3%

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

      3. mul-1-neg 68.3%

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

      4. distribute-neg-frac 68.3%

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

      5. mul-1-neg 68.3%

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

      6. remove-double-neg 68.3%

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

      7. *-commutative 68.3%

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

   4. Simplified 68.3%

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

      1. expm1-log1p-u 68.3%

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

      2. expm1-udef 11.6%

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

      3. associate-/r* 11.6%

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

      4. div-inv 11.6%

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

      5. pow-to-exp 11.7%

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

      6. pow1 11.7%

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

      7. pow-div 14.3%

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

   6. Applied egg-rr 14.3%

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

      1. expm1-def 72.3%

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

      2. expm1-log1p 72.3%

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

      3. sub-neg 72.3%

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

      4. metadata-eval 72.3%

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

   8. Simplified 72.3%

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

      1. flip-+ 72.2%

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

      2. frac-2neg 72.2%

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

      3. metadata-eval 72.2%

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

      4. sub-neg 72.2%

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

      5. inv-pow 72.2%

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

      6. inv-pow 72.2%

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

      7. pow-prod-up 72.3%

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

      8. metadata-eval 72.3%

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

      9. metadata-eval 72.3%

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

      10. sub-neg 72.3%

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

      11. metadata-eval 72.3%

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

   10. Applied egg-rr 72.3%

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

       1. neg-sub0 72.3%

          \[\leadsto \frac{{x}^{\left(\frac{\color{blue}{0 - \left({n}^{-2} + -1\right)}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       2. +-commutative 72.3%

          \[\leadsto \frac{{x}^{\left(\frac{0 - \color{blue}{\left(-1 + {n}^{-2}\right)}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       3. associate--r+ 72.3%

          \[\leadsto \frac{{x}^{\left(\frac{\color{blue}{\left(0 - -1\right) - {n}^{-2}}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       4. metadata-eval 72.3%

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

       5. +-commutative 72.3%

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

       6. distribute-neg-in 72.3%

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

       7. metadata-eval 72.3%

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

       8. distribute-neg-frac 72.3%

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

       9. metadata-eval 72.3%

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

   12. Simplified 72.3%

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

   if 10 < (/.f64 1 n)

   1. Initial program 54.3%

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

   2. Taylor expanded in n around 0 54.3%

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

      1. log1p-def 96.7%

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

   4. Simplified 96.7%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{{x}^{\left(\frac{1 - {n}^{-2}}{-1 + \frac{-1}{n}}\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 3: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(-1 + \frac{-1}{n}\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-56)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-142)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 10.0)
         (/ (pow x (+ (/ 1.0 n) -1.0)) n)
         (if (<= (/ 1.0 n) 2e+123)
           (- (pow (+ 1.0 x) (/ 1.0 n)) t_0)
           (/ (pow x (+ -1.0 (/ -1.0 n))) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e+123) {
		tmp = pow((1.0 + x), (1.0 / n)) - t_0;
	} else {
		tmp = pow(x, (-1.0 + (-1.0 / n))) / 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 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-56)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-142) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 10.0d0) then
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    else if ((1.0d0 / n) <= 2d+123) then
        tmp = ((1.0d0 + x) ** (1.0d0 / n)) - t_0
    else
        tmp = (x ** ((-1.0d0) + ((-1.0d0) / n))) / 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 tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e+123) {
		tmp = Math.pow((1.0 + x), (1.0 / n)) - t_0;
	} else {
		tmp = Math.pow(x, (-1.0 + (-1.0 / n))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-56:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-142:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 10.0:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 2e+123:
		tmp = math.pow((1.0 + x), (1.0 / n)) - t_0
	else:
		tmp = math.pow(x, (-1.0 + (-1.0 / n))) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-56)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-142)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 2e+123)
		tmp = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0);
	else
		tmp = Float64((x ^ Float64(-1.0 + Float64(-1.0 / n))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-56)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-142)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 10.0)
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	elseif ((1.0 / n) <= 2e+123)
		tmp = ((1.0 + x) ^ (1.0 / n)) - t_0;
	else
		tmp = (x ^ (-1.0 + (-1.0 / n))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-56], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-142], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+123], N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Power[x, N[(-1.0 + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.99999999999999997e-56

   1. Initial program 85.7%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. mul-1-neg 95.7%

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

      2. log-rec 95.7%

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

      3. mul-1-neg 95.7%

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

      4. distribute-neg-frac 95.7%

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

      5. mul-1-neg 95.7%

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

      6. remove-double-neg 95.7%

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

      7. *-commutative 95.7%

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

   4. Simplified 95.7%

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

      1. expm1-log1p-u 54.6%

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

      2. expm1-udef 44.0%

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

      3. associate-/r* 44.0%

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

      4. div-inv 44.0%

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

      5. pow-to-exp 44.0%

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

      6. pow1 44.0%

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

      7. pow-div 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. expm1-def 54.0%

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

      2. expm1-log1p 95.1%

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

      3. sub-neg 95.1%

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

      4. metadata-eval 95.1%

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

   8. Simplified 95.1%

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

      1. unpow-prod-up 95.7%

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

      2. inv-pow 95.7%

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

      3. *-un-lft-identity 95.7%

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

      4. times-frac 95.8%

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

   10. Applied egg-rr 95.8%

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

       1. /-rgt-identity 95.8%

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

       2. associate-/l/ 95.7%

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

       3. associate-/r* 95.7%

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

       4. associate-*r/ 95.7%

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

       5. associate-*r/ 95.8%

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

       6. *-rgt-identity 95.8%

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

   12. Simplified 95.8%

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

   if -4.99999999999999997e-56 < (/.f64 1 n) < 5.0000000000000002e-142

   1. Initial program 44.4%

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

   2. Taylor expanded in n around inf 91.6%

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

      1. +-rgt-identity 91.6%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 91.6%

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

      3. log1p-def 91.6%

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

   4. Simplified 91.6%

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

      1. log1p-udef 91.6%

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

      2. diff-log 91.8%

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

   6. Applied egg-rr 91.8%

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

   if 5.0000000000000002e-142 < (/.f64 1 n) < 10

   1. Initial program 8.7%

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

   2. Taylor expanded in x around inf 68.3%

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

      1. mul-1-neg 68.3%

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

      2. log-rec 68.3%

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

      3. mul-1-neg 68.3%

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

      4. distribute-neg-frac 68.3%

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

      5. mul-1-neg 68.3%

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

      6. remove-double-neg 68.3%

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

      7. *-commutative 68.3%

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

   4. Simplified 68.3%

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

      1. expm1-log1p-u 68.3%

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

      2. expm1-udef 11.6%

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

      3. associate-/r* 11.6%

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

      4. div-inv 11.6%

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

      5. pow-to-exp 11.7%

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

      6. pow1 11.7%

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

      7. pow-div 14.3%

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

   6. Applied egg-rr 14.3%

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

      1. expm1-def 72.3%

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

      2. expm1-log1p 72.3%

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

      3. sub-neg 72.3%

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

      4. metadata-eval 72.3%

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

   8. Simplified 72.3%

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

   if 10 < (/.f64 1 n) < 1.99999999999999996e123

   1. Initial program 67.5%

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

   if 1.99999999999999996e123 < (/.f64 1 n)

   1. Initial program 43.9%

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

   2. Taylor expanded in x around inf 0.8%

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

      1. mul-1-neg 0.8%

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

      2. log-rec 0.8%

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

      3. mul-1-neg 0.8%

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

      4. distribute-neg-frac 0.8%

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

      5. mul-1-neg 0.8%

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

      6. remove-double-neg 0.8%

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

      7. *-commutative 0.8%

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

   4. Simplified 0.8%

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

      1. expm1-log1p-u 0.8%

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

      2. expm1-udef 0.8%

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

      3. associate-/r* 2.1%

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

      4. div-inv 2.1%

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

      5. pow-to-exp 2.1%

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

      6. pow1 2.1%

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

      7. pow-div 2.1%

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

   6. Applied egg-rr 2.1%

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

      1. expm1-def 2.1%

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

      2. expm1-log1p 2.1%

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

      3. sub-neg 2.1%

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

      4. metadata-eval 2.1%

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

   8. Simplified 2.1%

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

      1. flip-+ 2.1%

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

      2. frac-2neg 2.1%

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

      3. metadata-eval 2.1%

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

      4. sub-neg 2.1%

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

      5. inv-pow 2.1%

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

      6. inv-pow 2.1%

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

      7. pow-prod-up 2.1%

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

      8. metadata-eval 2.1%

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

      9. metadata-eval 2.1%

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

      10. sub-neg 2.1%

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

      11. metadata-eval 2.1%

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

   10. Applied egg-rr 2.1%

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

       1. neg-sub0 2.1%

          \[\leadsto \frac{{x}^{\left(\frac{\color{blue}{0 - \left({n}^{-2} + -1\right)}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       2. +-commutative 2.1%

          \[\leadsto \frac{{x}^{\left(\frac{0 - \color{blue}{\left(-1 + {n}^{-2}\right)}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       3. associate--r+ 2.1%

          \[\leadsto \frac{{x}^{\left(\frac{\color{blue}{\left(0 - -1\right) - {n}^{-2}}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       4. metadata-eval 2.1%

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

       5. +-commutative 2.1%

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

       6. distribute-neg-in 2.1%

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

       7. metadata-eval 2.1%

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

       8. distribute-neg-frac 2.1%

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

       9. metadata-eval 2.1%

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

   12. Simplified 2.1%

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

       1. sub-neg 2.1%

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

       2. metadata-eval 2.1%

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

       3. pow-prod-up 2.1%

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

       4. inv-pow 2.1%

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

       5. inv-pow 2.1%

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

       6. distribute-rgt-neg-in 2.1%

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

       7. metadata-eval 2.1%

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

       8. associate-*r/ 2.1%

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

       9. mul-1-neg 2.1%

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

       10. mul-1-neg 2.1%

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

       11. div-inv 2.1%

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

       12. cancel-sign-sub-inv 2.1%

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

       13. metadata-eval 2.1%

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

       14. flip-- 2.1%

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

       15. add-sqr-sqrt 0.0%

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

       16. sqrt-unprod 64.3%

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

       17. frac-times 64.3%

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

       18. metadata-eval 64.3%

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

       19. metadata-eval 64.3%

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

       20. frac-times 64.3%

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

       21. sqrt-prod 64.3%

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

       22. add-sqr-sqrt 64.3%

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

   14. Applied egg-rr 64.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(-1 + \frac{-1}{n}\right)}}{n}\\ \end{array} \]

Alternative 4: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := -1 + \frac{-1}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{{x}^{\left(\frac{1 - {n}^{-2}}{t_1}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{t_1}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (+ -1.0 (/ -1.0 n))))
   (if (<= (/ 1.0 n) -5e-56)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-142)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 10.0)
         (/ (pow x (/ (- 1.0 (pow n -2.0)) t_1)) n)
         (if (<= (/ 1.0 n) 2e+123)
           (- (pow (+ 1.0 x) (/ 1.0 n)) t_0)
           (/ (pow x t_1) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = -1.0 + (-1.0 / n);
	double tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = pow(x, ((1.0 - pow(n, -2.0)) / t_1)) / n;
	} else if ((1.0 / n) <= 2e+123) {
		tmp = pow((1.0 + x), (1.0 / n)) - t_0;
	} else {
		tmp = pow(x, t_1) / 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 = (-1.0d0) + ((-1.0d0) / n)
    if ((1.0d0 / n) <= (-5d-56)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-142) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 10.0d0) then
        tmp = (x ** ((1.0d0 - (n ** (-2.0d0))) / t_1)) / n
    else if ((1.0d0 / n) <= 2d+123) then
        tmp = ((1.0d0 + x) ** (1.0d0 / n)) - t_0
    else
        tmp = (x ** t_1) / 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 = -1.0 + (-1.0 / n);
	double tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = Math.pow(x, ((1.0 - Math.pow(n, -2.0)) / t_1)) / n;
	} else if ((1.0 / n) <= 2e+123) {
		tmp = Math.pow((1.0 + x), (1.0 / n)) - t_0;
	} else {
		tmp = Math.pow(x, t_1) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = -1.0 + (-1.0 / n)
	tmp = 0
	if (1.0 / n) <= -5e-56:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-142:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 10.0:
		tmp = math.pow(x, ((1.0 - math.pow(n, -2.0)) / t_1)) / n
	elif (1.0 / n) <= 2e+123:
		tmp = math.pow((1.0 + x), (1.0 / n)) - t_0
	else:
		tmp = math.pow(x, t_1) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(-1.0 + Float64(-1.0 / n))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-56)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-142)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64((x ^ Float64(Float64(1.0 - (n ^ -2.0)) / t_1)) / n);
	elseif (Float64(1.0 / n) <= 2e+123)
		tmp = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0);
	else
		tmp = Float64((x ^ t_1) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = -1.0 + (-1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-56)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-142)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 10.0)
		tmp = (x ^ ((1.0 - (n ^ -2.0)) / t_1)) / n;
	elseif ((1.0 / n) <= 2e+123)
		tmp = ((1.0 + x) ^ (1.0 / n)) - t_0;
	else
		tmp = (x ^ t_1) / 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[(-1.0 + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-56], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-142], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[Power[x, N[(N[(1.0 - N[Power[n, -2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+123], N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Power[x, t$95$1], $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.99999999999999997e-56

   1. Initial program 85.7%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. mul-1-neg 95.7%

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

      2. log-rec 95.7%

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

      3. mul-1-neg 95.7%

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

      4. distribute-neg-frac 95.7%

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

      5. mul-1-neg 95.7%

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

      6. remove-double-neg 95.7%

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

      7. *-commutative 95.7%

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

   4. Simplified 95.7%

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

      1. expm1-log1p-u 54.6%

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

      2. expm1-udef 44.0%

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

      3. associate-/r* 44.0%

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

      4. div-inv 44.0%

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

      5. pow-to-exp 44.0%

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

      6. pow1 44.0%

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

      7. pow-div 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. expm1-def 54.0%

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

      2. expm1-log1p 95.1%

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

      3. sub-neg 95.1%

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

      4. metadata-eval 95.1%

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

   8. Simplified 95.1%

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

      1. unpow-prod-up 95.7%

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

      2. inv-pow 95.7%

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

      3. *-un-lft-identity 95.7%

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

      4. times-frac 95.8%

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

   10. Applied egg-rr 95.8%

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

       1. /-rgt-identity 95.8%

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

       2. associate-/l/ 95.7%

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

       3. associate-/r* 95.7%

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

       4. associate-*r/ 95.7%

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

       5. associate-*r/ 95.8%

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

       6. *-rgt-identity 95.8%

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

   12. Simplified 95.8%

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

   if -4.99999999999999997e-56 < (/.f64 1 n) < 5.0000000000000002e-142

   1. Initial program 44.4%

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

   2. Taylor expanded in n around inf 91.6%

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

      1. +-rgt-identity 91.6%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 91.6%

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

      3. log1p-def 91.6%

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

   4. Simplified 91.6%

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

      1. log1p-udef 91.6%

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

      2. diff-log 91.8%

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

   6. Applied egg-rr 91.8%

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

   if 5.0000000000000002e-142 < (/.f64 1 n) < 10

   1. Initial program 8.7%

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

   2. Taylor expanded in x around inf 68.3%

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

      1. mul-1-neg 68.3%

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

      2. log-rec 68.3%

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

      3. mul-1-neg 68.3%

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

      4. distribute-neg-frac 68.3%

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

      5. mul-1-neg 68.3%

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

      6. remove-double-neg 68.3%

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

      7. *-commutative 68.3%

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

   4. Simplified 68.3%

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

      1. expm1-log1p-u 68.3%

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

      2. expm1-udef 11.6%

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

      3. associate-/r* 11.6%

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

      4. div-inv 11.6%

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

      5. pow-to-exp 11.7%

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

      6. pow1 11.7%

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

      7. pow-div 14.3%

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

   6. Applied egg-rr 14.3%

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

      1. expm1-def 72.3%

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

      2. expm1-log1p 72.3%

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

      3. sub-neg 72.3%

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

      4. metadata-eval 72.3%

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

   8. Simplified 72.3%

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

      1. flip-+ 72.2%

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

      2. frac-2neg 72.2%

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

      3. metadata-eval 72.2%

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

      4. sub-neg 72.2%

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

      5. inv-pow 72.2%

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

      6. inv-pow 72.2%

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

      7. pow-prod-up 72.3%

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

      8. metadata-eval 72.3%

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

      9. metadata-eval 72.3%

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

      10. sub-neg 72.3%

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

      11. metadata-eval 72.3%

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

   10. Applied egg-rr 72.3%

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

       1. neg-sub0 72.3%

          \[\leadsto \frac{{x}^{\left(\frac{\color{blue}{0 - \left({n}^{-2} + -1\right)}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       2. +-commutative 72.3%

          \[\leadsto \frac{{x}^{\left(\frac{0 - \color{blue}{\left(-1 + {n}^{-2}\right)}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       3. associate--r+ 72.3%

          \[\leadsto \frac{{x}^{\left(\frac{\color{blue}{\left(0 - -1\right) - {n}^{-2}}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       4. metadata-eval 72.3%

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

       5. +-commutative 72.3%

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

       6. distribute-neg-in 72.3%

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

       7. metadata-eval 72.3%

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

       8. distribute-neg-frac 72.3%

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

       9. metadata-eval 72.3%

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

   12. Simplified 72.3%

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

   if 10 < (/.f64 1 n) < 1.99999999999999996e123

   1. Initial program 67.5%

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

   if 1.99999999999999996e123 < (/.f64 1 n)

   1. Initial program 43.9%

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

   2. Taylor expanded in x around inf 0.8%

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

      1. mul-1-neg 0.8%

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

      2. log-rec 0.8%

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

      3. mul-1-neg 0.8%

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

      4. distribute-neg-frac 0.8%

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

      5. mul-1-neg 0.8%

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

      6. remove-double-neg 0.8%

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

      7. *-commutative 0.8%

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

   4. Simplified 0.8%

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

      1. expm1-log1p-u 0.8%

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

      2. expm1-udef 0.8%

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

      3. associate-/r* 2.1%

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

      4. div-inv 2.1%

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

      5. pow-to-exp 2.1%

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

      6. pow1 2.1%

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

      7. pow-div 2.1%

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

   6. Applied egg-rr 2.1%

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

      1. expm1-def 2.1%

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

      2. expm1-log1p 2.1%

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

      3. sub-neg 2.1%

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

      4. metadata-eval 2.1%

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

   8. Simplified 2.1%

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

      1. flip-+ 2.1%

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

      2. frac-2neg 2.1%

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

      3. metadata-eval 2.1%

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

      4. sub-neg 2.1%

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

      5. inv-pow 2.1%

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

      6. inv-pow 2.1%

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

      7. pow-prod-up 2.1%

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

      8. metadata-eval 2.1%

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

      9. metadata-eval 2.1%

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

      10. sub-neg 2.1%

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

      11. metadata-eval 2.1%

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

   10. Applied egg-rr 2.1%

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

       1. neg-sub0 2.1%

          \[\leadsto \frac{{x}^{\left(\frac{\color{blue}{0 - \left({n}^{-2} + -1\right)}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       2. +-commutative 2.1%

          \[\leadsto \frac{{x}^{\left(\frac{0 - \color{blue}{\left(-1 + {n}^{-2}\right)}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       3. associate--r+ 2.1%

          \[\leadsto \frac{{x}^{\left(\frac{\color{blue}{\left(0 - -1\right) - {n}^{-2}}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       4. metadata-eval 2.1%

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

       5. +-commutative 2.1%

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

       6. distribute-neg-in 2.1%

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

       7. metadata-eval 2.1%

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

       8. distribute-neg-frac 2.1%

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

       9. metadata-eval 2.1%

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

   12. Simplified 2.1%

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

       1. sub-neg 2.1%

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

       2. metadata-eval 2.1%

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

       3. pow-prod-up 2.1%

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

       4. inv-pow 2.1%

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

       5. inv-pow 2.1%

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

       6. distribute-rgt-neg-in 2.1%

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

       7. metadata-eval 2.1%

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

       8. associate-*r/ 2.1%

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

       9. mul-1-neg 2.1%

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

       10. mul-1-neg 2.1%

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

       11. div-inv 2.1%

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

       12. cancel-sign-sub-inv 2.1%

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

       13. metadata-eval 2.1%

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

       14. flip-- 2.1%

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

       15. add-sqr-sqrt 0.0%

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

       16. sqrt-unprod 64.3%

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

       17. frac-times 64.3%

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

       18. metadata-eval 64.3%

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

       19. metadata-eval 64.3%

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

       20. frac-times 64.3%

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

       21. sqrt-prod 64.3%

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

       22. add-sqr-sqrt 64.3%

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

   14. Applied egg-rr 64.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{{x}^{\left(\frac{1 - {n}^{-2}}{-1 + \frac{-1}{n}}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(-1 + \frac{-1}{n}\right)}}{n}\\ \end{array} \]

Alternative 5: 65.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;\frac{t_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-147}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ 1.0 (* n (+ x 0.5)))))
   (if (<= (/ 1.0 n) -20000000.0)
     (/ t_0 n)
     (if (<= (/ 1.0 n) -5e-56)
       t_1
       (if (<= (/ 1.0 n) -2e-138)
         (/ (- x (log x)) n)
         (if (<= (/ 1.0 n) 1e-253)
           t_1
           (if (<= (/ 1.0 n) 1e-147)
             (/ (- (log x)) n)
             (if (<= (/ 1.0 n) 10.0) t_1 (- 1.0 t_0)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = 1.0 / (n * (x + 0.5));
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= -5e-56) {
		tmp = t_1;
	} else if ((1.0 / n) <= -2e-138) {
		tmp = (x - log(x)) / n;
	} else if ((1.0 / n) <= 1e-253) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-147) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = t_1;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = 1.0d0 / (n * (x + 0.5d0))
    if ((1.0d0 / n) <= (-20000000.0d0)) then
        tmp = t_0 / n
    else if ((1.0d0 / n) <= (-5d-56)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-2d-138)) then
        tmp = (x - log(x)) / n
    else if ((1.0d0 / n) <= 1d-253) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d-147) then
        tmp = -log(x) / n
    else if ((1.0d0 / n) <= 10.0d0) then
        tmp = t_1
    else
        tmp = 1.0d0 - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = 1.0 / (n * (x + 0.5));
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= -5e-56) {
		tmp = t_1;
	} else if ((1.0 / n) <= -2e-138) {
		tmp = (x - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1e-253) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e-147) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = t_1;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = 1.0 / (n * (x + 0.5))
	tmp = 0
	if (1.0 / n) <= -20000000.0:
		tmp = t_0 / n
	elif (1.0 / n) <= -5e-56:
		tmp = t_1
	elif (1.0 / n) <= -2e-138:
		tmp = (x - math.log(x)) / n
	elif (1.0 / n) <= 1e-253:
		tmp = t_1
	elif (1.0 / n) <= 1e-147:
		tmp = -math.log(x) / n
	elif (1.0 / n) <= 10.0:
		tmp = t_1
	else:
		tmp = 1.0 - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(1.0 / Float64(n * Float64(x + 0.5)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -20000000.0)
		tmp = Float64(t_0 / n);
	elseif (Float64(1.0 / n) <= -5e-56)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -2e-138)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e-253)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e-147)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = t_1;
	else
		tmp = Float64(1.0 - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = 1.0 / (n * (x + 0.5));
	tmp = 0.0;
	if ((1.0 / n) <= -20000000.0)
		tmp = t_0 / n;
	elseif ((1.0 / n) <= -5e-56)
		tmp = t_1;
	elseif ((1.0 / n) <= -2e-138)
		tmp = (x - log(x)) / n;
	elseif ((1.0 / n) <= 1e-253)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e-147)
		tmp = -log(x) / n;
	elseif ((1.0 / n) <= 10.0)
		tmp = t_1;
	else
		tmp = 1.0 - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(n * N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000000.0], N[(t$95$0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-56], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-138], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-253], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-147], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], t$95$1, N[(1.0 - t$95$0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2e7

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. expm1-log1p-u 50.7%

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

      2. expm1-udef 50.7%

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

      3. associate-/r* 50.7%

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

      4. div-inv 50.7%

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

      5. pow-to-exp 50.7%

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

      6. pow1 50.7%

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

      7. pow-div 50.7%

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

   6. Applied egg-rr 50.7%

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

      1. expm1-def 50.7%

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

      2. expm1-log1p 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in n around 0 100.0%

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

   if -2e7 < (/.f64 1 n) < -4.99999999999999997e-56 or -2.00000000000000013e-138 < (/.f64 1 n) < 1.0000000000000001e-253 or 9.9999999999999997e-148 < (/.f64 1 n) < 10

   1. Initial program 37.3%

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

   2. Taylor expanded in n around inf 69.1%

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

      1. +-rgt-identity 69.1%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 69.1%

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

      3. log1p-def 69.1%

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

   4. Simplified 69.1%

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

      1. clear-num 69.1%

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

      2. inv-pow 69.1%

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

   6. Applied egg-rr 69.1%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 69.1%

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

   8. Simplified 69.1%

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

   9. Taylor expanded in x around inf 66.0%

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

       1. +-commutative 66.0%

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

       2. *-commutative 66.0%

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

       3. distribute-lft-out 66.0%

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

   11. Simplified 66.0%

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

   if -4.99999999999999997e-56 < (/.f64 1 n) < -2.00000000000000013e-138

   1. Initial program 11.6%

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

   2. Taylor expanded in n around inf 85.1%

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

      1. +-rgt-identity 85.1%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 85.1%

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

      3. log1p-def 85.1%

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

   4. Simplified 85.1%

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

   5. Taylor expanded in x around 0 77.5%

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

      1. neg-mul-1 77.5%

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

      2. unsub-neg 77.5%

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

   7. Simplified 77.5%

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

   if 1.0000000000000001e-253 < (/.f64 1 n) < 9.9999999999999997e-148

   1. Initial program 27.5%

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

   2. Taylor expanded in x around 0 27.5%

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

   3. Taylor expanded in n around inf 68.0%

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

      1. neg-mul-1 68.0%

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

      2. distribute-neg-frac 68.0%

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

   5. Simplified 68.0%

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

   if 10 < (/.f64 1 n)

   1. Initial program 54.3%

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

   2. Taylor expanded in x around 0 48.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-253}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-147}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 6: 80.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+182}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(-1 + \frac{-1}{n}\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-56)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-142)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 10.0)
         (/ (pow x (+ (/ 1.0 n) -1.0)) n)
         (if (<= (/ 1.0 n) 5e+182)
           (- (+ 1.0 (/ x n)) t_0)
           (/ (pow x (+ -1.0 (/ -1.0 n))) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 5e+182) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = pow(x, (-1.0 + (-1.0 / n))) / 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 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-56)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-142) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 10.0d0) then
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    else if ((1.0d0 / n) <= 5d+182) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (x ** ((-1.0d0) + ((-1.0d0) / n))) / 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 tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 5e+182) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.pow(x, (-1.0 + (-1.0 / n))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-56:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-142:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 10.0:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 5e+182:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.pow(x, (-1.0 + (-1.0 / n))) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-56)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-142)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 5e+182)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64((x ^ Float64(-1.0 + Float64(-1.0 / n))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-56)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-142)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 10.0)
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	elseif ((1.0 / n) <= 5e+182)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = (x ^ (-1.0 + (-1.0 / n))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-56], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-142], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+182], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Power[x, N[(-1.0 + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.99999999999999997e-56

   1. Initial program 85.7%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. mul-1-neg 95.7%

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

      2. log-rec 95.7%

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

      3. mul-1-neg 95.7%

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

      4. distribute-neg-frac 95.7%

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

      5. mul-1-neg 95.7%

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

      6. remove-double-neg 95.7%

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

      7. *-commutative 95.7%

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

   4. Simplified 95.7%

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

      1. expm1-log1p-u 54.6%

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

      2. expm1-udef 44.0%

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

      3. associate-/r* 44.0%

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

      4. div-inv 44.0%

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

      5. pow-to-exp 44.0%

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

      6. pow1 44.0%

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

      7. pow-div 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. expm1-def 54.0%

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

      2. expm1-log1p 95.1%

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

      3. sub-neg 95.1%

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

      4. metadata-eval 95.1%

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

   8. Simplified 95.1%

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

      1. unpow-prod-up 95.7%

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

      2. inv-pow 95.7%

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

      3. *-un-lft-identity 95.7%

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

      4. times-frac 95.8%

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

   10. Applied egg-rr 95.8%

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

       1. /-rgt-identity 95.8%

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

       2. associate-/l/ 95.7%

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

       3. associate-/r* 95.7%

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

       4. associate-*r/ 95.7%

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

       5. associate-*r/ 95.8%

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

       6. *-rgt-identity 95.8%

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

   12. Simplified 95.8%

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

   if -4.99999999999999997e-56 < (/.f64 1 n) < 5.0000000000000002e-142

   1. Initial program 44.4%

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

   2. Taylor expanded in n around inf 91.6%

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

      1. +-rgt-identity 91.6%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 91.6%

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

      3. log1p-def 91.6%

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

   4. Simplified 91.6%

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

      1. log1p-udef 91.6%

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

      2. diff-log 91.8%

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

   6. Applied egg-rr 91.8%

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

   if 5.0000000000000002e-142 < (/.f64 1 n) < 10

   1. Initial program 8.7%

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

   2. Taylor expanded in x around inf 68.3%

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

      1. mul-1-neg 68.3%

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

      2. log-rec 68.3%

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

      3. mul-1-neg 68.3%

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

      4. distribute-neg-frac 68.3%

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

      5. mul-1-neg 68.3%

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

      6. remove-double-neg 68.3%

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

      7. *-commutative 68.3%

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

   4. Simplified 68.3%

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

      1. expm1-log1p-u 68.3%

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

      2. expm1-udef 11.6%

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

      3. associate-/r* 11.6%

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

      4. div-inv 11.6%

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

      5. pow-to-exp 11.7%

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

      6. pow1 11.7%

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

      7. pow-div 14.3%

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

   6. Applied egg-rr 14.3%

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

      1. expm1-def 72.3%

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

      2. expm1-log1p 72.3%

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

      3. sub-neg 72.3%

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

      4. metadata-eval 72.3%

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

   8. Simplified 72.3%

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

   if 10 < (/.f64 1 n) < 4.99999999999999973e182

   1. Initial program 61.1%

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

   2. Taylor expanded in x around 0 57.4%

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

   if 4.99999999999999973e182 < (/.f64 1 n)

   1. Initial program 35.4%

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

   2. Taylor expanded in x around inf 1.0%

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

      1. mul-1-neg 1.0%

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

      2. log-rec 1.0%

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

      3. mul-1-neg 1.0%

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

      4. distribute-neg-frac 1.0%

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

      5. mul-1-neg 1.0%

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

      6. remove-double-neg 1.0%

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

      7. *-commutative 1.0%

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

   4. Simplified 1.0%

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

      1. expm1-log1p-u 1.0%

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

      2. expm1-udef 1.0%

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

      3. associate-/r* 1.9%

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

      4. div-inv 1.9%

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

      5. pow-to-exp 1.9%

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

      6. pow1 1.9%

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

      7. pow-div 1.9%

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

   6. Applied egg-rr 1.9%

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

      1. expm1-def 1.9%

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

      2. expm1-log1p 1.9%

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

      3. sub-neg 1.9%

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

      4. metadata-eval 1.9%

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

   8. Simplified 1.9%

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

      1. flip-+ 1.9%

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

      2. frac-2neg 1.9%

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

      3. metadata-eval 1.9%

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

      4. sub-neg 1.9%

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

      5. inv-pow 1.9%

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

      6. inv-pow 1.9%

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

      7. pow-prod-up 1.9%

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

      8. metadata-eval 1.9%

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

      9. metadata-eval 1.9%

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

      10. sub-neg 1.9%

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

      11. metadata-eval 1.9%

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

   10. Applied egg-rr 1.9%

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

       1. neg-sub0 1.9%

          \[\leadsto \frac{{x}^{\left(\frac{\color{blue}{0 - \left({n}^{-2} + -1\right)}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       2. +-commutative 1.9%

          \[\leadsto \frac{{x}^{\left(\frac{0 - \color{blue}{\left(-1 + {n}^{-2}\right)}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       3. associate--r+ 1.9%

          \[\leadsto \frac{{x}^{\left(\frac{\color{blue}{\left(0 - -1\right) - {n}^{-2}}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       4. metadata-eval 1.9%

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

       5. +-commutative 1.9%

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

       6. distribute-neg-in 1.9%

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

       7. metadata-eval 1.9%

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

       8. distribute-neg-frac 1.9%

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

       9. metadata-eval 1.9%

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

   12. Simplified 1.9%

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

       1. sub-neg 1.9%

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

       2. metadata-eval 1.9%

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

       3. pow-prod-up 1.9%

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

       4. inv-pow 1.9%

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

       5. inv-pow 1.9%

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

       6. distribute-rgt-neg-in 1.9%

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

       7. metadata-eval 1.9%

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

       8. associate-*r/ 1.9%

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

       9. mul-1-neg 1.9%

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

       10. mul-1-neg 1.9%

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

       11. div-inv 1.9%

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

       12. cancel-sign-sub-inv 1.9%

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

       13. metadata-eval 1.9%

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

       14. flip-- 1.9%

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

       15. add-sqr-sqrt 0.0%

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

       16. sqrt-unprod 78.5%

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

       17. frac-times 78.5%

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

       18. metadata-eval 78.5%

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

       19. metadata-eval 78.5%

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

       20. frac-times 78.5%

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

       21. sqrt-prod 78.5%

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

       22. add-sqr-sqrt 78.5%

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

   14. Applied egg-rr 78.5%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+182}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(-1 + \frac{-1}{n}\right)}}{n}\\ \end{array} \]

Alternative 7: 80.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(-1 + \frac{-1}{n}\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-56)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-142)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 10.0)
         (/ (pow x (+ (/ 1.0 n) -1.0)) n)
         (if (<= (/ 1.0 n) 2e+123)
           (- 1.0 t_0)
           (/ (pow x (+ -1.0 (/ -1.0 n))) n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e+123) {
		tmp = 1.0 - t_0;
	} else {
		tmp = pow(x, (-1.0 + (-1.0 / n))) / 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 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-56)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-142) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 10.0d0) then
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    else if ((1.0d0 / n) <= 2d+123) then
        tmp = 1.0d0 - t_0
    else
        tmp = (x ** ((-1.0d0) + ((-1.0d0) / n))) / 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 tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 2e+123) {
		tmp = 1.0 - t_0;
	} else {
		tmp = Math.pow(x, (-1.0 + (-1.0 / n))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-56:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-142:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 10.0:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 2e+123:
		tmp = 1.0 - t_0
	else:
		tmp = math.pow(x, (-1.0 + (-1.0 / n))) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-56)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-142)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 2e+123)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64((x ^ Float64(-1.0 + Float64(-1.0 / n))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-56)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-142)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 10.0)
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	elseif ((1.0 / n) <= 2e+123)
		tmp = 1.0 - t_0;
	else
		tmp = (x ^ (-1.0 + (-1.0 / n))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-56], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-142], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+123], N[(1.0 - t$95$0), $MachinePrecision], N[(N[Power[x, N[(-1.0 + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.99999999999999997e-56

   1. Initial program 85.7%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. mul-1-neg 95.7%

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

      2. log-rec 95.7%

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

      3. mul-1-neg 95.7%

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

      4. distribute-neg-frac 95.7%

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

      5. mul-1-neg 95.7%

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

      6. remove-double-neg 95.7%

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

      7. *-commutative 95.7%

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

   4. Simplified 95.7%

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

      1. expm1-log1p-u 54.6%

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

      2. expm1-udef 44.0%

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

      3. associate-/r* 44.0%

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

      4. div-inv 44.0%

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

      5. pow-to-exp 44.0%

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

      6. pow1 44.0%

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

      7. pow-div 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. expm1-def 54.0%

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

      2. expm1-log1p 95.1%

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

      3. sub-neg 95.1%

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

      4. metadata-eval 95.1%

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

   8. Simplified 95.1%

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

      1. unpow-prod-up 95.7%

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

      2. inv-pow 95.7%

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

      3. *-un-lft-identity 95.7%

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

      4. times-frac 95.8%

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

   10. Applied egg-rr 95.8%

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

       1. /-rgt-identity 95.8%

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

       2. associate-/l/ 95.7%

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

       3. associate-/r* 95.7%

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

       4. associate-*r/ 95.7%

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

       5. associate-*r/ 95.8%

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

       6. *-rgt-identity 95.8%

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

   12. Simplified 95.8%

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

   if -4.99999999999999997e-56 < (/.f64 1 n) < 5.0000000000000002e-142

   1. Initial program 44.4%

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

   2. Taylor expanded in n around inf 91.6%

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

      1. +-rgt-identity 91.6%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 91.6%

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

      3. log1p-def 91.6%

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

   4. Simplified 91.6%

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

      1. log1p-udef 91.6%

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

      2. diff-log 91.8%

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

   6. Applied egg-rr 91.8%

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

   if 5.0000000000000002e-142 < (/.f64 1 n) < 10

   1. Initial program 8.7%

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

   2. Taylor expanded in x around inf 68.3%

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

      1. mul-1-neg 68.3%

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

      2. log-rec 68.3%

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

      3. mul-1-neg 68.3%

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

      4. distribute-neg-frac 68.3%

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

      5. mul-1-neg 68.3%

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

      6. remove-double-neg 68.3%

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

      7. *-commutative 68.3%

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

   4. Simplified 68.3%

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

      1. expm1-log1p-u 68.3%

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

      2. expm1-udef 11.6%

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

      3. associate-/r* 11.6%

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

      4. div-inv 11.6%

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

      5. pow-to-exp 11.7%

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

      6. pow1 11.7%

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

      7. pow-div 14.3%

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

   6. Applied egg-rr 14.3%

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

      1. expm1-def 72.3%

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

      2. expm1-log1p 72.3%

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

      3. sub-neg 72.3%

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

      4. metadata-eval 72.3%

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

   8. Simplified 72.3%

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

   if 10 < (/.f64 1 n) < 1.99999999999999996e123

   1. Initial program 67.5%

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

   2. Taylor expanded in x around 0 61.1%

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

   if 1.99999999999999996e123 < (/.f64 1 n)

   1. Initial program 43.9%

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

   2. Taylor expanded in x around inf 0.8%

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

      1. mul-1-neg 0.8%

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

      2. log-rec 0.8%

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

      3. mul-1-neg 0.8%

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

      4. distribute-neg-frac 0.8%

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

      5. mul-1-neg 0.8%

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

      6. remove-double-neg 0.8%

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

      7. *-commutative 0.8%

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

   4. Simplified 0.8%

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

      1. expm1-log1p-u 0.8%

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

      2. expm1-udef 0.8%

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

      3. associate-/r* 2.1%

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

      4. div-inv 2.1%

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

      5. pow-to-exp 2.1%

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

      6. pow1 2.1%

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

      7. pow-div 2.1%

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

   6. Applied egg-rr 2.1%

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

      1. expm1-def 2.1%

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

      2. expm1-log1p 2.1%

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

      3. sub-neg 2.1%

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

      4. metadata-eval 2.1%

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

   8. Simplified 2.1%

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

      1. flip-+ 2.1%

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

      2. frac-2neg 2.1%

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

      3. metadata-eval 2.1%

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

      4. sub-neg 2.1%

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

      5. inv-pow 2.1%

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

      6. inv-pow 2.1%

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

      7. pow-prod-up 2.1%

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

      8. metadata-eval 2.1%

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

      9. metadata-eval 2.1%

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

      10. sub-neg 2.1%

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

      11. metadata-eval 2.1%

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

   10. Applied egg-rr 2.1%

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

       1. neg-sub0 2.1%

          \[\leadsto \frac{{x}^{\left(\frac{\color{blue}{0 - \left({n}^{-2} + -1\right)}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       2. +-commutative 2.1%

          \[\leadsto \frac{{x}^{\left(\frac{0 - \color{blue}{\left(-1 + {n}^{-2}\right)}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       3. associate--r+ 2.1%

          \[\leadsto \frac{{x}^{\left(\frac{\color{blue}{\left(0 - -1\right) - {n}^{-2}}}{-\left(\frac{1}{n} + 1\right)}\right)}}{n} \]

       4. metadata-eval 2.1%

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

       5. +-commutative 2.1%

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

       6. distribute-neg-in 2.1%

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

       7. metadata-eval 2.1%

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

       8. distribute-neg-frac 2.1%

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

       9. metadata-eval 2.1%

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

   12. Simplified 2.1%

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

       1. sub-neg 2.1%

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

       2. metadata-eval 2.1%

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

       3. pow-prod-up 2.1%

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

       4. inv-pow 2.1%

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

       5. inv-pow 2.1%

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

       6. distribute-rgt-neg-in 2.1%

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

       7. metadata-eval 2.1%

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

       8. associate-*r/ 2.1%

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

       9. mul-1-neg 2.1%

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

       10. mul-1-neg 2.1%

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

       11. div-inv 2.1%

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

       12. cancel-sign-sub-inv 2.1%

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

       13. metadata-eval 2.1%

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

       14. flip-- 2.1%

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

       15. add-sqr-sqrt 0.0%

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

       16. sqrt-unprod 64.3%

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

       17. frac-times 64.3%

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

       18. metadata-eval 64.3%

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

       19. metadata-eval 64.3%

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

       20. frac-times 64.3%

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

       21. sqrt-prod 64.3%

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

       22. add-sqr-sqrt 64.3%

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

   14. Applied egg-rr 64.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(-1 + \frac{-1}{n}\right)}}{n}\\ \end{array} \]

Alternative 8: 77.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;\frac{t_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ 1.0 (* n (+ x 0.5)))))
   (if (<= (/ 1.0 n) -20000000.0)
     (/ t_0 n)
     (if (<= (/ 1.0 n) -5e-56)
       t_1
       (if (<= (/ 1.0 n) 5e-142)
         (/ (log (/ (+ 1.0 x) x)) n)
         (if (<= (/ 1.0 n) 10.0) t_1 (- 1.0 t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = 1.0 / (n * (x + 0.5));
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= -5e-56) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = t_1;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = 1.0d0 / (n * (x + 0.5d0))
    if ((1.0d0 / n) <= (-20000000.0d0)) then
        tmp = t_0 / n
    else if ((1.0d0 / n) <= (-5d-56)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-142) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 10.0d0) then
        tmp = t_1
    else
        tmp = 1.0d0 - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = 1.0 / (n * (x + 0.5));
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = t_0 / n;
	} else if ((1.0 / n) <= -5e-56) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = t_1;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = 1.0 / (n * (x + 0.5))
	tmp = 0
	if (1.0 / n) <= -20000000.0:
		tmp = t_0 / n
	elif (1.0 / n) <= -5e-56:
		tmp = t_1
	elif (1.0 / n) <= 5e-142:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 10.0:
		tmp = t_1
	else:
		tmp = 1.0 - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(1.0 / Float64(n * Float64(x + 0.5)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -20000000.0)
		tmp = Float64(t_0 / n);
	elseif (Float64(1.0 / n) <= -5e-56)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-142)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = t_1;
	else
		tmp = Float64(1.0 - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = 1.0 / (n * (x + 0.5));
	tmp = 0.0;
	if ((1.0 / n) <= -20000000.0)
		tmp = t_0 / n;
	elseif ((1.0 / n) <= -5e-56)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-142)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 10.0)
		tmp = t_1;
	else
		tmp = 1.0 - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(n * N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000000.0], N[(t$95$0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-56], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-142], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], t$95$1, N[(1.0 - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -2e7

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. expm1-log1p-u 50.7%

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

      2. expm1-udef 50.7%

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

      3. associate-/r* 50.7%

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

      4. div-inv 50.7%

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

      5. pow-to-exp 50.7%

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

      6. pow1 50.7%

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

      7. pow-div 50.7%

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

   6. Applied egg-rr 50.7%

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

      1. expm1-def 50.7%

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

      2. expm1-log1p 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in n around 0 100.0%

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

   if -2e7 < (/.f64 1 n) < -4.99999999999999997e-56 or 5.0000000000000002e-142 < (/.f64 1 n) < 10

   1. Initial program 11.1%

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

   2. Taylor expanded in n around inf 35.3%

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

      1. +-rgt-identity 35.3%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 35.3%

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

      3. log1p-def 35.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 35.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. clear-num 35.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

      2. inv-pow 35.4%

         \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]

   6. Applied egg-rr 35.4%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 35.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

   8. Simplified 35.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

   9. Taylor expanded in x around inf 66.6%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot n + n \cdot x}} \]
   10. Step-by-step derivation

       1. +-commutative 66.6%

          \[\leadsto \frac{1}{\color{blue}{n \cdot x + 0.5 \cdot n}} \]

       2. *-commutative 66.6%

          \[\leadsto \frac{1}{n \cdot x + \color{blue}{n \cdot 0.5}} \]

       3. distribute-lft-out 66.6%

          \[\leadsto \frac{1}{\color{blue}{n \cdot \left(x + 0.5\right)}} \]

   11. Simplified 66.6%

       \[\leadsto \frac{1}{\color{blue}{n \cdot \left(x + 0.5\right)}} \]

   if -4.99999999999999997e-56 < (/.f64 1 n) < 5.0000000000000002e-142

   1. Initial program 44.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 91.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 91.6%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 91.6%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 91.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 91.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. log1p-udef 91.6%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      2. diff-log 91.8%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   6. Applied egg-rr 91.8%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   if 10 < (/.f64 1 n)

   1. Initial program 54.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 48.6%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 9: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -5e-56)
     t_1
     (if (<= (/ 1.0 n) 5e-142)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 10.0) t_1 (- 1.0 t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = t_1;
	} else {
		tmp = 1.0 - t_0;
	}
	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 = (t_0 / n) / x
    if ((1.0d0 / n) <= (-5d-56)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-142) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 10.0d0) then
        tmp = t_1
    else
        tmp = 1.0d0 - t_0
    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 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = t_1;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -5e-56:
		tmp = t_1
	elif (1.0 / n) <= 5e-142:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 10.0:
		tmp = t_1
	else:
		tmp = 1.0 - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-56)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-142)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = t_1;
	else
		tmp = Float64(1.0 - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / n) / x;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-56)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-142)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 10.0)
		tmp = t_1;
	else
		tmp = 1.0 - t_0;
	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[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-56], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-142], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], t$95$1, N[(1.0 - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -4.99999999999999997e-56 or 5.0000000000000002e-142 < (/.f64 1 n) < 10

   1. Initial program 64.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 88.3%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. mul-1-neg 88.3%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 88.3%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 88.3%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 88.3%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 88.3%

         \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

      6. remove-double-neg 88.3%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 88.3%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   4. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 58.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{\frac{\log x}{n}}}{x \cdot n}\right)\right)} \]

      2. expm1-udef 35.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{\frac{\log x}{n}}}{x \cdot n}\right)} - 1} \]

      3. associate-/r* 35.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{e^{\frac{\log x}{n}}}{x}}{n}}\right)} - 1 \]

      4. div-inv 35.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{x}}{n}\right)} - 1 \]

      5. pow-to-exp 35.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x}}{n}\right)} - 1 \]

      6. pow1 35.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{{x}^{1}}}}{n}\right)} - 1 \]

      7. pow-div 35.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{x}^{\left(\frac{1}{n} - 1\right)}}}{n}\right)} - 1 \]

   6. Applied egg-rr 35.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 59.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n}\right)\right)} \]

      2. expm1-log1p 88.9%

         \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n}} \]

      3. sub-neg 88.9%

         \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n} + \left(-1\right)\right)}}}{n} \]

      4. metadata-eval 88.9%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n} + \color{blue}{-1}\right)}}{n} \]

   8. Simplified 88.9%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}} \]
   9. Step-by-step derivation

      1. unpow-prod-up 88.7%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot {x}^{-1}}}{n} \]

      2. inv-pow 88.7%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)} \cdot \color{blue}{\frac{1}{x}}}{n} \]

      3. *-un-lft-identity 88.7%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x}}{\color{blue}{1 \cdot n}} \]

      4. times-frac 88.7%

         \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{1} \cdot \frac{\frac{1}{x}}{n}} \]

   10. Applied egg-rr 88.7%

       \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{1} \cdot \frac{\frac{1}{x}}{n}} \]
   11. Step-by-step derivation

       1. /-rgt-identity 88.7%

          \[\leadsto \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \cdot \frac{\frac{1}{x}}{n} \]

       2. associate-/l/ 88.3%

          \[\leadsto {x}^{\left(\frac{1}{n}\right)} \cdot \color{blue}{\frac{1}{n \cdot x}} \]

       3. associate-/r* 88.7%

          \[\leadsto {x}^{\left(\frac{1}{n}\right)} \cdot \color{blue}{\frac{\frac{1}{n}}{x}} \]

       4. associate-*r/ 88.7%

          \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{n}}{x}} \]

       5. associate-*r/ 88.7%

          \[\leadsto \frac{\color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)} \cdot 1}{n}}}{x} \]

       6. *-rgt-identity 88.7%

          \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x} \]

   12. Simplified 88.7%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]

   if -4.99999999999999997e-56 < (/.f64 1 n) < 5.0000000000000002e-142

   1. Initial program 44.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 91.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 91.6%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 91.6%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 91.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 91.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. log1p-udef 91.6%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      2. diff-log 91.8%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   6. Applied egg-rr 91.8%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   if 10 < (/.f64 1 n)

   1. Initial program 54.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 48.6%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 10: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-56)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-142)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 10.0)
         (/ (pow x (+ (/ 1.0 n) -1.0)) n)
         (- 1.0 t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-56)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-142) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 10.0d0) then
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    else
        tmp = 1.0d0 - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-56) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-142) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 10.0) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-56:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-142:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 10.0:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	else:
		tmp = 1.0 - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-56)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-142)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 10.0)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	else
		tmp = Float64(1.0 - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-56)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-142)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 10.0)
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	else
		tmp = 1.0 - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-56], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-142], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 10.0], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(1.0 - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -4.99999999999999997e-56

   1. Initial program 85.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 95.7%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. mul-1-neg 95.7%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 95.7%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 95.7%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 95.7%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 95.7%

         \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

      6. remove-double-neg 95.7%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 95.7%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   4. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 54.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{\frac{\log x}{n}}}{x \cdot n}\right)\right)} \]

      2. expm1-udef 44.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{\frac{\log x}{n}}}{x \cdot n}\right)} - 1} \]

      3. associate-/r* 44.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{e^{\frac{\log x}{n}}}{x}}{n}}\right)} - 1 \]

      4. div-inv 44.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{x}}{n}\right)} - 1 \]

      5. pow-to-exp 44.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x}}{n}\right)} - 1 \]

      6. pow1 44.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{{x}^{1}}}}{n}\right)} - 1 \]

      7. pow-div 44.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{x}^{\left(\frac{1}{n} - 1\right)}}}{n}\right)} - 1 \]

   6. Applied egg-rr 44.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 54.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n}\right)\right)} \]

      2. expm1-log1p 95.1%

         \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n}} \]

      3. sub-neg 95.1%

         \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n} + \left(-1\right)\right)}}}{n} \]

      4. metadata-eval 95.1%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n} + \color{blue}{-1}\right)}}{n} \]

   8. Simplified 95.1%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}} \]
   9. Step-by-step derivation

      1. unpow-prod-up 95.7%

         \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot {x}^{-1}}}{n} \]

      2. inv-pow 95.7%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)} \cdot \color{blue}{\frac{1}{x}}}{n} \]

      3. *-un-lft-identity 95.7%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x}}{\color{blue}{1 \cdot n}} \]

      4. times-frac 95.8%

         \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{1} \cdot \frac{\frac{1}{x}}{n}} \]

   10. Applied egg-rr 95.8%

       \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{1} \cdot \frac{\frac{1}{x}}{n}} \]
   11. Step-by-step derivation

       1. /-rgt-identity 95.8%

          \[\leadsto \color{blue}{{x}^{\left(\frac{1}{n}\right)}} \cdot \frac{\frac{1}{x}}{n} \]

       2. associate-/l/ 95.7%

          \[\leadsto {x}^{\left(\frac{1}{n}\right)} \cdot \color{blue}{\frac{1}{n \cdot x}} \]

       3. associate-/r* 95.7%

          \[\leadsto {x}^{\left(\frac{1}{n}\right)} \cdot \color{blue}{\frac{\frac{1}{n}}{x}} \]

       4. associate-*r/ 95.7%

          \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{n}}{x}} \]

       5. associate-*r/ 95.8%

          \[\leadsto \frac{\color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)} \cdot 1}{n}}}{x} \]

       6. *-rgt-identity 95.8%

          \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x} \]

   12. Simplified 95.8%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]

   if -4.99999999999999997e-56 < (/.f64 1 n) < 5.0000000000000002e-142

   1. Initial program 44.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 91.6%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 91.6%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 91.6%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 91.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 91.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   5. Step-by-step derivation

      1. log1p-udef 91.6%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      2. diff-log 91.8%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   6. Applied egg-rr 91.8%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]

   if 5.0000000000000002e-142 < (/.f64 1 n) < 10

   1. Initial program 8.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around inf 68.3%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   3. Step-by-step derivation

      1. mul-1-neg 68.3%

         \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

      2. log-rec 68.3%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

      3. mul-1-neg 68.3%

         \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

      4. distribute-neg-frac 68.3%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 68.3%

         \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

      6. remove-double-neg 68.3%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 68.3%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   4. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 68.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{\frac{\log x}{n}}}{x \cdot n}\right)\right)} \]

      2. expm1-udef 11.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{\frac{\log x}{n}}}{x \cdot n}\right)} - 1} \]

      3. associate-/r* 11.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{e^{\frac{\log x}{n}}}{x}}{n}}\right)} - 1 \]

      4. div-inv 11.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{x}}{n}\right)} - 1 \]

      5. pow-to-exp 11.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x}}{n}\right)} - 1 \]

      6. pow1 11.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{{x}^{1}}}}{n}\right)} - 1 \]

      7. pow-div 14.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{x}^{\left(\frac{1}{n} - 1\right)}}}{n}\right)} - 1 \]

   6. Applied egg-rr 14.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 72.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n}\right)\right)} \]

      2. expm1-log1p 72.3%

         \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n}} \]

      3. sub-neg 72.3%

         \[\leadsto \frac{{x}^{\color{blue}{\left(\frac{1}{n} + \left(-1\right)\right)}}}{n} \]

      4. metadata-eval 72.3%

         \[\leadsto \frac{{x}^{\left(\frac{1}{n} + \color{blue}{-1}\right)}}{n} \]

   8. Simplified 72.3%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}} \]

   if 10 < (/.f64 1 n)

   1. Initial program 54.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 48.6%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 11: 59.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-171}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.21:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+212}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.5e-171)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 0.21)
     (/ (- x (log x)) n)
     (if (<= x 1.65e+212)
       (/ (/ 1.0 x) n)
       (/ (/ 0.3333333333333333 (pow x 3.0)) n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.5e-171) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.21) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.65e+212) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (0.3333333333333333 / pow(x, 3.0)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.5d-171) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.21d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.65d+212) then
        tmp = (1.0d0 / x) / n
    else
        tmp = (0.3333333333333333d0 / (x ** 3.0d0)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.5e-171) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.21) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.65e+212) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (0.3333333333333333 / Math.pow(x, 3.0)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.5e-171:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.21:
		tmp = (x - math.log(x)) / n
	elif x <= 1.65e+212:
		tmp = (1.0 / x) / n
	else:
		tmp = (0.3333333333333333 / math.pow(x, 3.0)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.5e-171)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.21)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.65e+212)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(0.3333333333333333 / (x ^ 3.0)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.5e-171)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.21)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.65e+212)
		tmp = (1.0 / x) / n;
	else
		tmp = (0.3333333333333333 / (x ^ 3.0)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.5e-171], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.21], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.65e+212], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.5000000000000004e-171

   1. Initial program 63.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 63.3%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   if 4.5000000000000004e-171 < x < 0.209999999999999992

   1. Initial program 30.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 57.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 57.3%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 57.3%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 57.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 56.9%

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 56.9%

         \[\leadsto \frac{x + \color{blue}{\left(-\log x\right)}}{n} \]

      2. unsub-neg 56.9%

         \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   7. Simplified 56.9%

      \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   if 0.209999999999999992 < x < 1.65e212

   1. Initial program 58.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 57.8%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 57.8%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 57.8%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 57.8%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 67.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if 1.65e212 < x

   1. Initial program 89.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 89.5%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 89.5%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 89.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 89.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 89.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 61.5%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. associate--l+ 61.5%

         \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. associate-*r/ 61.5%

         \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. metadata-eval 61.5%

         \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      4. associate-*r/ 61.5%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)}{n} \]

      5. metadata-eval 61.5%

         \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}\right)}{n} \]

   7. Simplified 61.5%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}}{n} \]

   8. Taylor expanded in x around 0 89.5%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{{x}^{3}}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-171}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.21:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+212}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}}}{n}\\ \end{array} \]

Alternative 12: 56.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-171}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.21:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.5e-171)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 0.21) (/ (- x (log x)) n) (/ (/ 1.0 x) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.5e-171) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.21) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.5d-171) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.21d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.5e-171) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.21) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.5e-171:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.21:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.5e-171)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.21)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.5e-171)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.21)
		tmp = (x - log(x)) / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.5e-171], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.21], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.5000000000000004e-171

   1. Initial program 63.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 63.3%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   if 4.5000000000000004e-171 < x < 0.209999999999999992

   1. Initial program 30.4%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 57.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 57.3%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 57.3%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 57.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 56.9%

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 56.9%

         \[\leadsto \frac{x + \color{blue}{\left(-\log x\right)}}{n} \]

      2. unsub-neg 56.9%

         \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   7. Simplified 56.9%

      \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   if 0.209999999999999992 < x

   1. Initial program 67.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 66.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 66.7%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 66.7%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 66.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 66.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 65.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-171}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.21:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 13: 57.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.21:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.21) (/ (- x (log x)) n) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.21) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.21d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.21) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.21:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.21)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.21)
		tmp = (x - log(x)) / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.21], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.209999999999999992

   1. Initial program 44.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 51.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 51.2%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 51.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 51.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 51.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around 0 51.0%

      \[\leadsto \frac{\color{blue}{x + -1 \cdot \log x}}{n} \]
   6. Step-by-step derivation

      1. neg-mul-1 51.0%

         \[\leadsto \frac{x + \color{blue}{\left(-\log x\right)}}{n} \]

      2. unsub-neg 51.0%

         \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   7. Simplified 51.0%

      \[\leadsto \frac{\color{blue}{x - \log x}}{n} \]

   if 0.209999999999999992 < x

   1. Initial program 67.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 66.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 66.7%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 66.7%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 66.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 66.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 65.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.21:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 14: 57.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.21:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.21) (/ (- (log x)) n) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.21) {
		tmp = -log(x) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.21d0) then
        tmp = -log(x) / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.21) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.21:
		tmp = -math.log(x) / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.21)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.21)
		tmp = -log(x) / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.21], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.209999999999999992

   1. Initial program 44.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in x around 0 42.5%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Taylor expanded in n around inf 50.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   4. Step-by-step derivation

      1. neg-mul-1 50.8%

         \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

      2. distribute-neg-frac 50.8%

         \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   5. Simplified 50.8%

      \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   if 0.209999999999999992 < x

   1. Initial program 67.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   2. Taylor expanded in n around inf 66.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   3. Step-by-step derivation

      1. +-rgt-identity 66.7%

         \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

      2. +-rgt-identity 66.7%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

      3. log1p-def 66.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Simplified 66.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

   5. Taylor expanded in x around inf 65.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.21:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 15: 39.7% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{n \cdot x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* n x)))

C

double code(double x, double n) {
	return 1.0 / (n * x);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 1.0d0 / (n * x)
end function

Java

public static double code(double x, double n) {
	return 1.0 / (n * x);
}

Python

def code(x, n):
	return 1.0 / (n * x)

Julia

function code(x, n)
	return Float64(1.0 / Float64(n * x))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (n * x);
end

Wolfram

code[x_, n_] := N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

2. Taylor expanded in x around inf 62.7%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
3. Step-by-step derivation

   1. mul-1-neg 62.7%

      \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

   2. log-rec 62.7%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 62.7%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   4. distribute-neg-frac 62.7%

      \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

   5. mul-1-neg 62.7%

      \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

   6. remove-double-neg 62.7%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

   7. *-commutative 62.7%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

4. Simplified 62.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

5. Taylor expanded in n around inf 43.5%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
6. Step-by-step derivation

   1. *-commutative 43.5%

      \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

7. Simplified 43.5%

   \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

8. Final simplification 43.5%

   \[\leadsto \frac{1}{n \cdot x} \]

Alternative 16: 40.2% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 x) n))

C

double code(double x, double n) {
	return (1.0 / x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / x) / n
end function

Java

public static double code(double x, double n) {
	return (1.0 / x) / n;
}

Python

def code(x, n):
	return (1.0 / x) / n

Julia

function code(x, n)
	return Float64(Float64(1.0 / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

2. Taylor expanded in n around inf 58.7%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
3. Step-by-step derivation

   1. +-rgt-identity 58.7%

      \[\leadsto \frac{\color{blue}{\left(\log \left(1 + x\right) + 0\right)} - \log x}{n} \]

   2. +-rgt-identity 58.7%

      \[\leadsto \frac{\color{blue}{\log \left(1 + x\right)} - \log x}{n} \]

   3. log1p-def 58.7%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

4. Simplified 58.7%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

5. Taylor expanded in x around inf 44.0%

   \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

6. Final simplification 44.0%

   \[\leadsto \frac{\frac{1}{x}}{n} \]

Reproduce

herbie shell --seed 2023318 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))