2nthrt (problem 3.4.6)

Percentage Accurate: 53.6% → 98.2%

Time: 58.6s

Alternatives: 19

Speedup: 1.8×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -60000000000 \lor \neg \left(n \leq 470000\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -60000000000.0) (not (<= n 470000.0)))
   (/ (log1p (/ 1.0 x)) n)
   (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -60000000000.0) || !(n <= 470000.0)) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -60000000000.0) || !(n <= 470000.0)) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -60000000000.0) or not (n <= 470000.0):
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -60000000000.0) || !(n <= 470000.0))
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -60000000000.0], N[Not[LessEqual[n, 470000.0]], $MachinePrecision]], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -6e10 or 4.7e5 < n

   1. Initial program 29.1%

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

   3. Taylor expanded in n around inf 79.4%

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

      1. associate--l+ 79.4%

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

      2. log1p-define 79.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 79.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 79.4%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 79.4%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 79.3%

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

      1. sub-neg 79.3%

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

      2. log1p-define 79.3%

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

      3. exp-diff 79.3%

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

      4. log1p-define 79.3%

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

      5. rem-exp-log 57.3%

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

      6. +-commutative 57.3%

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

      7. rem-exp-log 79.5%

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

      8. metadata-eval 79.5%

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

   10. Simplified 79.5%

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

   11. Taylor expanded in n around 0 79.5%

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

       1. log1p-define 99.5%

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

   13. Simplified 99.5%

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

   if -6e10 < n < 4.7e5

   1. Initial program 79.9%

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

   3. Taylor expanded in n around 0 79.9%

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

      1. log1p-define 96.9%

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

      2. *-rgt-identity 96.9%

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

      3. associate-*l/ 96.9%

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

      4. associate-/l* 96.9%

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

      5. exp-to-pow 96.9%

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

   5. Simplified 96.9%

      \[\leadsto \color{blue}{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -60000000000 \lor \neg \left(n \leq 470000\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -10:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x \cdot \left(x \cdot -0.5 + 1\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -10.0)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-15)
       (/ (log1p (/ 1.0 x)) n)
       (- (exp (/ (* x (+ (* x -0.5) 1.0)) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -10.0) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = exp(((x * ((x * -0.5) + 1.0)) / 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) <= -10.0) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = Math.exp(((x * ((x * -0.5) + 1.0)) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -10.0:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-15:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = math.exp(((x * ((x * -0.5) + 1.0)) / 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) <= -10.0)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(exp(Float64(Float64(x * Float64(Float64(x * -0.5) + 1.0)) / 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], -10.0], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[(x * N[(N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -10

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate-/r* 100.0%

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

      2. mul-1-neg 100.0%

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

      3. log-rec 100.0%

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

      4. mul-1-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. mul-1-neg 100.0%

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

      7. remove-double-neg 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. associate-/l* 100.0%

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

      10. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   if -10 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 29.2%

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

   3. Taylor expanded in n around inf 79.4%

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

      1. associate--l+ 79.4%

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

      2. log1p-define 79.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 79.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 79.4%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 79.4%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 79.2%

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

      1. sub-neg 79.2%

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

      2. log1p-define 79.2%

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

      3. exp-diff 79.2%

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

      4. log1p-define 79.2%

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

      5. rem-exp-log 57.2%

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

      6. +-commutative 57.2%

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

      7. rem-exp-log 79.4%

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

      8. metadata-eval 79.4%

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

   10. Simplified 79.4%

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

   11. Taylor expanded in n around 0 79.4%

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

       1. log1p-define 99.4%

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

   13. Simplified 99.4%

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

   if 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 48.8%

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

   3. Taylor expanded in n around 0 48.8%

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

      1. log1p-define 91.8%

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

      2. *-rgt-identity 91.8%

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

      3. associate-*l/ 91.8%

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

      4. associate-/l* 91.8%

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

      5. exp-to-pow 91.8%

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

   5. Simplified 91.8%

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

   6. Taylor expanded in x around 0 91.8%

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

   7. Taylor expanded in n around 0 91.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x \cdot \left(x \cdot -0.5 + 1\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := 1 - t\_0\\ \mathbf{if}\;\frac{1}{n} \leq -10:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+149}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n + n \cdot \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n \cdot n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- 1.0 t_0)))
   (if (<= (/ 1.0 n) -10.0)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-15)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 5e+115)
         t_1
         (if (<= (/ 1.0 n) 2e+129)
           (/
            (/
             (- (* (* n x) 2.0) n)
             (* n (/ (* n x) (+ 0.5 (/ -0.3333333333333333 x)))))
            x)
           (if (<= (/ 1.0 n) 1e+149)
             t_1
             (/
              (/ (+ n (* n (/ (+ -0.5 (/ 0.3333333333333333 x)) x))) (* n n))
              x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = 1.0 - t_0;
	double tmp;
	if ((1.0 / n) <= -10.0) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-15) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+115) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+129) {
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x;
	} else if ((1.0 / n) <= 1e+149) {
		tmp = t_1;
	} else {
		tmp = ((n + (n * ((-0.5 + (0.3333333333333333 / x)) / x))) / (n * n)) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = 1.0 - t_0;
	double tmp;
	if ((1.0 / n) <= -10.0) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-15) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+115) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+129) {
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x;
	} else if ((1.0 / n) <= 1e+149) {
		tmp = t_1;
	} else {
		tmp = ((n + (n * ((-0.5 + (0.3333333333333333 / x)) / x))) / (n * n)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = 1.0 - t_0
	tmp = 0
	if (1.0 / n) <= -10.0:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-15:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+115:
		tmp = t_1
	elif (1.0 / n) <= 2e+129:
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x
	elif (1.0 / n) <= 1e+149:
		tmp = t_1
	else:
		tmp = ((n + (n * ((-0.5 + (0.3333333333333333 / x)) / x))) / (n * n)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(1.0 - t_0)
	tmp = 0.0
	if (Float64(1.0 / n) <= -10.0)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+115)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e+129)
		tmp = Float64(Float64(Float64(Float64(Float64(n * x) * 2.0) - n) / Float64(n * Float64(Float64(n * x) / Float64(0.5 + Float64(-0.3333333333333333 / x))))) / x);
	elseif (Float64(1.0 / n) <= 1e+149)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(n + Float64(n * Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x))) / Float64(n * n)) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -10.0], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+115], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+129], N[(N[(N[(N[(N[(n * x), $MachinePrecision] * 2.0), $MachinePrecision] - n), $MachinePrecision] / N[(n * N[(N[(n * x), $MachinePrecision] / N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+149], t$95$1, N[(N[(N[(n + N[(n * N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -10

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate-/r* 100.0%

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

      2. mul-1-neg 100.0%

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

      3. log-rec 100.0%

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

      4. mul-1-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. mul-1-neg 100.0%

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

      7. remove-double-neg 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. associate-/l* 100.0%

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

      10. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   if -10 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 29.2%

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

   3. Taylor expanded in n around inf 79.4%

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

      1. associate--l+ 79.4%

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

      2. log1p-define 79.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 79.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 79.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 79.4%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 79.4%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 79.2%

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

      1. sub-neg 79.2%

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

      2. log1p-define 79.2%

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

      3. exp-diff 79.2%

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

      4. log1p-define 79.2%

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

      5. rem-exp-log 57.2%

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

      6. +-commutative 57.2%

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

      7. rem-exp-log 79.4%

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

      8. metadata-eval 79.4%

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

   10. Simplified 79.4%

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

   11. Taylor expanded in n around 0 79.4%

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

       1. log1p-define 99.4%

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

   13. Simplified 99.4%

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

   if 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000008e115 or 2e129 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e149

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0 82.9%

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

      1. *-rgt-identity 82.9%

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

      2. associate-*l/ 82.9%

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

      3. associate-/l* 82.9%

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

      4. exp-to-pow 82.9%

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

   5. Simplified 82.9%

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

   if 5.00000000000000008e115 < (/.f64 #s(literal 1 binary64) n) < 2e129

   1. Initial program 3.1%

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

   3. Taylor expanded in n around inf 0.1%

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

      1. associate--l+ 0.1%

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

      2. log1p-define 0.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 0.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 0.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 0.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 0.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 0.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 0.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 0.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 0.0%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 4.3%

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

      1. sub-neg 4.3%

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

      2. log1p-define 4.3%

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

      3. exp-diff 4.3%

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

      4. log1p-define 4.3%

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

      5. rem-exp-log 4.3%

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

      6. +-commutative 4.3%

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

      7. rem-exp-log 4.3%

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

      8. metadata-eval 4.3%

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

   10. Simplified 4.3%

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

   11. Taylor expanded in n around 0 4.3%

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

       1. log1p-define 4.3%

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

   13. Simplified 4.3%

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

   14. Taylor expanded in x around inf 81.7%

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

   16. Simplified 81.7%

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

       1. frac-2neg 81.7%

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

       2. metadata-eval 81.7%

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

       3. clear-num 81.7%

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

       4. frac-add 100.0%

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

       5. associate-/l* 100.0%

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

       6. associate-/l* 100.0%

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

   18. Applied egg-rr 100.0%

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

       1. *-rgt-identity 100.0%

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

       2. unsub-neg 100.0%

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

       3. mul-1-neg 100.0%

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

       4. associate-*r/ 100.0%

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

       5. *-commutative 100.0%

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

       6. distribute-neg-frac2 100.0%

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

       7. *-commutative 100.0%

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

       8. +-commutative 100.0%

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

       9. distribute-neg-in 100.0%

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

       10. metadata-eval 100.0%

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

       11. distribute-neg-frac 100.0%

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

       12. metadata-eval 100.0%

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

       13. distribute-lft-neg-out 100.0%

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

       14. distribute-rgt-neg-in 100.0%

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

       15. associate-*r/ 100.0%

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

   20. Simplified 100.0%

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

   21. Taylor expanded in x around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. *-commutative 100.0%

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

   23. Simplified 100.0%

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

   if 1.00000000000000005e149 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 21.8%

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

   3. Taylor expanded in n around inf 0.1%

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

      1. associate--l+ 0.1%

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

      2. log1p-define 0.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 0.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 0.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 0.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 0.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 0.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 0.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 0.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 0.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 0.1%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 6.8%

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

      1. sub-neg 6.8%

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

      2. log1p-define 6.8%

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

      3. exp-diff 6.8%

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

      4. log1p-define 6.8%

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

      5. rem-exp-log 6.8%

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

      6. +-commutative 6.8%

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

      7. rem-exp-log 6.8%

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

      8. metadata-eval 6.8%

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

   10. Simplified 6.8%

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

   11. Taylor expanded in n around 0 6.8%

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

       1. log1p-define 6.8%

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

   13. Simplified 6.8%

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

   14. Taylor expanded in x around inf 12.7%

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

   16. Simplified 68.9%

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

       1. frac-2neg 68.9%

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

       2. metadata-eval 68.9%

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

       3. associate-/r* 68.9%

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

       4. frac-add 78.5%

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

       5. neg-mul-1 78.5%

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

   18. Applied egg-rr 78.5%

       \[\leadsto \frac{\color{blue}{\frac{\left(-n\right) + \left(-n\right) \cdot \frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{\left(-n\right) \cdot n}}}{x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+149}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n + n \cdot \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n \cdot n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+248}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -500000000000:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+248)
   (/ 0.0 n)
   (if (<= (/ 1.0 n) -500000000000.0)
     (/ 0.3333333333333333 (* n (pow x 3.0)))
     (if (<= (/ 1.0 n) 2e-14)
       (/ (log1p (/ 1.0 x)) n)
       (if (<= (/ 1.0 n) 5e+115)
         (- 1.0 (pow x (/ 1.0 n)))
         (/
          (/
           (- (* (* n x) 2.0) n)
           (* n (/ (* n x) (+ 0.5 (/ -0.3333333333333333 x)))))
          x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+248) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -500000000000.0) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if ((1.0 / n) <= 2e-14) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+115) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+248) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -500000000000.0) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if ((1.0 / n) <= 2e-14) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+115) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+248:
		tmp = 0.0 / n
	elif (1.0 / n) <= -500000000000.0:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif (1.0 / n) <= 2e-14:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+115:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+248)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= -500000000000.0)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (Float64(1.0 / n) <= 2e-14)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+115)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(n * x) * 2.0) - n) / Float64(n * Float64(Float64(n * x) / Float64(0.5 + Float64(-0.3333333333333333 / x))))) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+248], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -500000000000.0], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-14], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+115], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(n * x), $MachinePrecision] * 2.0), $MachinePrecision] - n), $MachinePrecision] / N[(n * N[(N[(n * x), $MachinePrecision] / N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000005e248

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 94.8%

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

      1. associate--l+ 32.7%

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

      2. log1p-define 32.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 32.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 94.8%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 94.8%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 94.8%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 94.8%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 94.8%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 94.8%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 94.8%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 66.1%

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

      1. sub-neg 66.1%

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

      2. log1p-define 66.1%

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

      3. exp-diff 66.1%

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

      4. log1p-define 66.1%

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

      5. rem-exp-log 4.1%

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

      6. +-commutative 4.1%

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

      7. rem-exp-log 66.1%

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

      8. metadata-eval 66.1%

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

   10. Simplified 66.1%

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

   11. Taylor expanded in x around inf 68.9%

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

   if -1.00000000000000005e248 < (/.f64 #s(literal 1 binary64) n) < -5e11

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 64.3%

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

      1. associate--l+ 23.1%

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

      2. log1p-define 23.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 23.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 64.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 64.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 64.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 64.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 64.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 100.0%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 44.8%

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

      1. sub-neg 44.8%

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

      2. log1p-define 44.8%

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

      3. exp-diff 44.8%

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

      4. log1p-define 44.8%

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

      5. rem-exp-log 3.7%

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

      6. +-commutative 3.7%

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

      7. rem-exp-log 44.8%

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

      8. metadata-eval 44.8%

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

   10. Simplified 44.8%

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

   11. Taylor expanded in x around inf 28.6%

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

   13. Simplified 53.1%

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

   14. Taylor expanded in x around 0 72.4%

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

   if -5e11 < (/.f64 #s(literal 1 binary64) n) < 2e-14

   1. Initial program 30.7%

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

   3. Taylor expanded in n around inf 78.7%

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

      1. associate--l+ 78.0%

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

      2. log1p-define 78.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 78.0%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 78.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 78.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 78.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 78.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 78.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 80.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 80.0%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 80.0%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 78.3%

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

      1. sub-neg 78.3%

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

      2. log1p-define 78.3%

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

      3. exp-diff 78.3%

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

      4. log1p-define 78.3%

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

      5. rem-exp-log 56.3%

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

      6. +-commutative 56.3%

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

      7. rem-exp-log 78.5%

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

      8. metadata-eval 78.5%

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

   10. Simplified 78.5%

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

   11. Taylor expanded in n around 0 78.5%

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

       1. log1p-define 97.3%

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

   13. Simplified 97.3%

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

   if 2e-14 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000008e115

   1. Initial program 83.2%

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

   3. Taylor expanded in x around 0 83.2%

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

      1. *-rgt-identity 83.2%

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

      2. associate-*l/ 83.2%

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

      3. associate-/l* 83.2%

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

      4. exp-to-pow 83.2%

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

   5. Simplified 83.2%

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

   if 5.00000000000000008e115 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 24.3%

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

   3. Taylor expanded in n around inf 0.2%

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

      1. associate--l+ 0.2%

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

      2. log1p-define 0.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 0.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 0.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 0.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 0.2%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 6.2%

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

      1. sub-neg 6.2%

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

      2. log1p-define 6.2%

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

      3. exp-diff 6.2%

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

      4. log1p-define 6.2%

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

      5. rem-exp-log 6.2%

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

      6. +-commutative 6.2%

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

      7. rem-exp-log 6.2%

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

      8. metadata-eval 6.2%

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

   10. Simplified 6.2%

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

   11. Taylor expanded in n around 0 6.2%

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

       1. log1p-define 6.2%

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

   13. Simplified 6.2%

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

   14. Taylor expanded in x around inf 25.5%

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

   16. Simplified 66.2%

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

       1. frac-2neg 66.2%

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

       2. metadata-eval 66.2%

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

       3. clear-num 66.2%

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

       4. frac-add 76.8%

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

       5. associate-/l* 76.8%

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

       6. associate-/l* 76.8%

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

   18. Applied egg-rr 76.8%

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

       1. *-rgt-identity 76.8%

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

       2. unsub-neg 76.8%

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

       3. mul-1-neg 76.8%

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

       4. associate-*r/ 76.8%

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

       5. *-commutative 76.8%

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

       6. distribute-neg-frac2 76.8%

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

       7. *-commutative 76.8%

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

       8. +-commutative 76.8%

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

       9. distribute-neg-in 76.8%

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

       10. metadata-eval 76.8%

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

       11. distribute-neg-frac 76.8%

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

       12. metadata-eval 76.8%

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

       13. distribute-lft-neg-out 76.8%

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

       14. distribute-rgt-neg-in 76.8%

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

       15. associate-*r/ 76.8%

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

   20. Simplified 76.8%

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

   21. Taylor expanded in x around inf 76.8%

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

       1. *-commutative 76.8%

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

       2. *-commutative 76.8%

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

   23. Simplified 76.8%

       \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot n\right) \cdot 2} - n}{n \cdot \frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}}}}{x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+248}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -500000000000:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500000000000:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -500000000000.0)
   (/ 0.3333333333333333 (* n (pow x 3.0)))
   (if (<= (/ 1.0 n) 2e-14)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 5e+115)
       (- 1.0 (pow x (/ 1.0 n)))
       (/
        (/
         (- (* (* n x) 2.0) n)
         (* n (/ (* n x) (+ 0.5 (/ -0.3333333333333333 x)))))
        x)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500000000000.0) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if ((1.0 / n) <= 2e-14) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+115) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500000000000.0) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if ((1.0 / n) <= 2e-14) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+115) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -500000000000.0:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif (1.0 / n) <= 2e-14:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+115:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500000000000.0)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (Float64(1.0 / n) <= 2e-14)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+115)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(n * x) * 2.0) - n) / Float64(n * Float64(Float64(n * x) / Float64(0.5 + Float64(-0.3333333333333333 / x))))) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -500000000000.0], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-14], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+115], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(n * x), $MachinePrecision] * 2.0), $MachinePrecision] - n), $MachinePrecision] / N[(n * N[(N[(n * x), $MachinePrecision] / N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e11

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 73.4%

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

      1. associate--l+ 26.0%

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

      2. log1p-define 26.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 26.0%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 73.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 73.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 73.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 73.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 73.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 98.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 98.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 98.5%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 51.2%

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

      1. sub-neg 51.2%

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

      2. log1p-define 51.2%

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

      3. exp-diff 51.2%

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

      4. log1p-define 51.2%

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

      5. rem-exp-log 3.8%

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

      6. +-commutative 3.8%

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

      7. rem-exp-log 51.2%

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

      8. metadata-eval 51.2%

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

   10. Simplified 51.2%

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

   11. Taylor expanded in x around inf 20.7%

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

   13. Simplified 45.7%

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

   14. Taylor expanded in x around 0 63.9%

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

   if -5e11 < (/.f64 #s(literal 1 binary64) n) < 2e-14

   1. Initial program 30.7%

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

   3. Taylor expanded in n around inf 78.7%

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

      1. associate--l+ 78.0%

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

      2. log1p-define 78.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 78.0%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 78.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 78.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 78.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 78.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 78.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 80.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 80.0%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 80.0%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 78.3%

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

      1. sub-neg 78.3%

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

      2. log1p-define 78.3%

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

      3. exp-diff 78.3%

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

      4. log1p-define 78.3%

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

      5. rem-exp-log 56.3%

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

      6. +-commutative 56.3%

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

      7. rem-exp-log 78.5%

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

      8. metadata-eval 78.5%

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

   10. Simplified 78.5%

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

   11. Taylor expanded in n around 0 78.5%

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

       1. log1p-define 97.3%

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

   13. Simplified 97.3%

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

   if 2e-14 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000008e115

   1. Initial program 83.2%

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

   3. Taylor expanded in x around 0 83.2%

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

      1. *-rgt-identity 83.2%

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

      2. associate-*l/ 83.2%

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

      3. associate-/l* 83.2%

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

      4. exp-to-pow 83.2%

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

   5. Simplified 83.2%

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

   if 5.00000000000000008e115 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 24.3%

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

   3. Taylor expanded in n around inf 0.2%

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

      1. associate--l+ 0.2%

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

      2. log1p-define 0.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 0.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 0.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 0.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 0.2%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 6.2%

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

      1. sub-neg 6.2%

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

      2. log1p-define 6.2%

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

      3. exp-diff 6.2%

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

      4. log1p-define 6.2%

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

      5. rem-exp-log 6.2%

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

      6. +-commutative 6.2%

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

      7. rem-exp-log 6.2%

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

      8. metadata-eval 6.2%

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

   10. Simplified 6.2%

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

   11. Taylor expanded in n around 0 6.2%

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

       1. log1p-define 6.2%

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

   13. Simplified 6.2%

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

   14. Taylor expanded in x around inf 25.5%

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

   16. Simplified 66.2%

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

       1. frac-2neg 66.2%

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

       2. metadata-eval 66.2%

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

       3. clear-num 66.2%

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

       4. frac-add 76.8%

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

       5. associate-/l* 76.8%

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

       6. associate-/l* 76.8%

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

   18. Applied egg-rr 76.8%

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

       1. *-rgt-identity 76.8%

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

       2. unsub-neg 76.8%

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

       3. mul-1-neg 76.8%

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

       4. associate-*r/ 76.8%

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

       5. *-commutative 76.8%

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

       6. distribute-neg-frac2 76.8%

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

       7. *-commutative 76.8%

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

       8. +-commutative 76.8%

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

       9. distribute-neg-in 76.8%

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

       10. metadata-eval 76.8%

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

       11. distribute-neg-frac 76.8%

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

       12. metadata-eval 76.8%

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

       13. distribute-lft-neg-out 76.8%

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

       14. distribute-rgt-neg-in 76.8%

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

       15. associate-*r/ 76.8%

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

   20. Simplified 76.8%

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

   21. Taylor expanded in x around inf 76.8%

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

       1. *-commutative 76.8%

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

       2. *-commutative 76.8%

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

   23. Simplified 76.8%

       \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot n\right) \cdot 2} - n}{n \cdot \frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}}}}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500000000000:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -50:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -50.0)
   (* 0.3333333333333333 (/ (pow x -3.0) n))
   (if (<= (/ 1.0 n) 2e-14)
     (/ (log1p (/ 1.0 x)) n)
     (if (<= (/ 1.0 n) 5e+115)
       (- 1.0 (pow x (/ 1.0 n)))
       (/
        (/
         (- (* (* n x) 2.0) n)
         (* n (/ (* n x) (+ 0.5 (/ -0.3333333333333333 x)))))
        x)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -50.0) {
		tmp = 0.3333333333333333 * (pow(x, -3.0) / n);
	} else if ((1.0 / n) <= 2e-14) {
		tmp = log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+115) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -50.0) {
		tmp = 0.3333333333333333 * (Math.pow(x, -3.0) / n);
	} else if ((1.0 / n) <= 2e-14) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else if ((1.0 / n) <= 5e+115) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -50.0:
		tmp = 0.3333333333333333 * (math.pow(x, -3.0) / n)
	elif (1.0 / n) <= 2e-14:
		tmp = math.log1p((1.0 / x)) / n
	elif (1.0 / n) <= 5e+115:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -50.0)
		tmp = Float64(0.3333333333333333 * Float64((x ^ -3.0) / n));
	elseif (Float64(1.0 / n) <= 2e-14)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+115)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(n * x) * 2.0) - n) / Float64(n * Float64(Float64(n * x) / Float64(0.5 + Float64(-0.3333333333333333 / x))))) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -50.0], N[(0.3333333333333333 * N[(N[Power[x, -3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-14], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+115], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(n * x), $MachinePrecision] * 2.0), $MachinePrecision] - n), $MachinePrecision] / N[(n * N[(N[(n * x), $MachinePrecision] / N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -50

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 72.7%

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

      1. associate--l+ 25.3%

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

      2. log1p-define 25.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 25.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 72.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 72.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 72.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 72.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 98.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 98.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 98.5%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 51.2%

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

      1. sub-neg 51.2%

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

      2. log1p-define 51.2%

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

      3. exp-diff 51.2%

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

      4. log1p-define 51.2%

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

      5. rem-exp-log 3.8%

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

      6. +-commutative 3.8%

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

      7. rem-exp-log 51.2%

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

      8. metadata-eval 51.2%

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

   10. Simplified 51.2%

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

   11. Taylor expanded in n around 0 51.2%

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

       1. log1p-define 4.2%

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

   13. Simplified 4.2%

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

   14. Taylor expanded in x around inf 44.5%

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

       1. associate--l+ 44.5%

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

       2. metadata-eval 44.5%

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

       3. associate-*r/ 44.5%

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

       4. associate-*r/ 44.5%

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

       5. metadata-eval 44.5%

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

       6. unpow2 44.5%

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

       7. associate-/r* 44.5%

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

       8. associate-*r/ 44.5%

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

       9. metadata-eval 44.5%

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

       10. div-sub 44.5%

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

       11. metadata-eval 44.5%

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

       12. associate-*r/ 44.5%

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

       13. sub-neg 44.5%

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

       14. associate-*r/ 44.5%

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

       15. metadata-eval 44.5%

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

       16. metadata-eval 44.5%

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

   16. Simplified 44.5%

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

   17. Taylor expanded in x around 0 63.6%

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

       1. associate-/r* 63.6%

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

   19. Simplified 63.6%

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

       1. *-un-lft-identity 63.6%

          \[\leadsto \color{blue}{1 \cdot \frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]

       2. div-inv 63.6%

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

       3. pow-flip 62.1%

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

       4. metadata-eval 62.1%

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

   21. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{1 \cdot \left(\frac{0.3333333333333333}{n} \cdot {x}^{-3}\right)} \]
   22. Step-by-step derivation

       1. *-lft-identity 62.1%

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

       2. associate-*l/ 62.1%

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

       3. associate-/l* 62.1%

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

   23. Simplified 62.1%

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

   if -50 < (/.f64 #s(literal 1 binary64) n) < 2e-14

   1. Initial program 29.7%

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

   3. Taylor expanded in n around inf 79.1%

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

      1. associate--l+ 79.1%

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

      2. log1p-define 79.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 79.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 79.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 79.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 79.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 79.1%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 79.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 79.7%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 79.7%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 78.7%

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

      1. sub-neg 78.7%

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

      2. log1p-define 78.7%

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

      3. exp-diff 78.7%

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

      4. log1p-define 78.7%

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

      5. rem-exp-log 57.0%

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

      6. +-commutative 57.0%

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

      7. rem-exp-log 78.9%

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

      8. metadata-eval 78.9%

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

   10. Simplified 78.9%

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

   11. Taylor expanded in n around 0 78.9%

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

       1. log1p-define 98.6%

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

   13. Simplified 98.6%

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

   if 2e-14 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000008e115

   1. Initial program 83.2%

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

   3. Taylor expanded in x around 0 83.2%

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

      1. *-rgt-identity 83.2%

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

      2. associate-*l/ 83.2%

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

      3. associate-/l* 83.2%

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

      4. exp-to-pow 83.2%

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

   5. Simplified 83.2%

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

   if 5.00000000000000008e115 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 24.3%

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

   3. Taylor expanded in n around inf 0.2%

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

      1. associate--l+ 0.2%

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

      2. log1p-define 0.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 0.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 0.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 0.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 0.2%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 0.2%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 6.2%

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

      1. sub-neg 6.2%

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

      2. log1p-define 6.2%

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

      3. exp-diff 6.2%

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

      4. log1p-define 6.2%

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

      5. rem-exp-log 6.2%

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

      6. +-commutative 6.2%

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

      7. rem-exp-log 6.2%

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

      8. metadata-eval 6.2%

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

   10. Simplified 6.2%

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

   11. Taylor expanded in n around 0 6.2%

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

       1. log1p-define 6.2%

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

   13. Simplified 6.2%

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

   14. Taylor expanded in x around inf 25.5%

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

   16. Simplified 66.2%

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

       1. frac-2neg 66.2%

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

       2. metadata-eval 66.2%

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

       3. clear-num 66.2%

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

       4. frac-add 76.8%

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

       5. associate-/l* 76.8%

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

       6. associate-/l* 76.8%

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

   18. Applied egg-rr 76.8%

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

       1. *-rgt-identity 76.8%

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

       2. unsub-neg 76.8%

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

       3. mul-1-neg 76.8%

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

       4. associate-*r/ 76.8%

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

       5. *-commutative 76.8%

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

       6. distribute-neg-frac2 76.8%

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

       7. *-commutative 76.8%

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

       8. +-commutative 76.8%

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

       9. distribute-neg-in 76.8%

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

       10. metadata-eval 76.8%

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

       11. distribute-neg-frac 76.8%

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

       12. metadata-eval 76.8%

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

       13. distribute-lft-neg-out 76.8%

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

       14. distribute-rgt-neg-in 76.8%

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

       15. associate-*r/ 76.8%

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

   20. Simplified 76.8%

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

   21. Taylor expanded in x around inf 76.8%

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

       1. *-commutative 76.8%

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

       2. *-commutative 76.8%

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

   23. Simplified 76.8%

       \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot n\right) \cdot 2} - n}{n \cdot \frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}}}}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -50:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 1.2 \cdot 10^{-152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-25}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 1.2e-152)
     t_0
     (if (<= x 5.1e-101)
       (/
        (/
         (- (* (* n x) 2.0) n)
         (* n (/ (* n x) (+ 0.5 (/ -0.3333333333333333 x)))))
        x)
       (if (<= x 1.6e-25)
         t_0
         (/ (/ (+ (/ (+ -0.5 (/ 0.3333333333333333 x)) x) 1.0) n) x))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 1.2e-152) {
		tmp = t_0;
	} else if (x <= 5.1e-101) {
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x;
	} else if (x <= 1.6e-25) {
		tmp = t_0;
	} else {
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 1.2d-152) then
        tmp = t_0
    else if (x <= 5.1d-101) then
        tmp = ((((n * x) * 2.0d0) - n) / (n * ((n * x) / (0.5d0 + ((-0.3333333333333333d0) / x))))) / x
    else if (x <= 1.6d-25) then
        tmp = t_0
    else
        tmp = (((((-0.5d0) + (0.3333333333333333d0 / x)) / x) + 1.0d0) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 1.2e-152) {
		tmp = t_0;
	} else if (x <= 5.1e-101) {
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x;
	} else if (x <= 1.6e-25) {
		tmp = t_0;
	} else {
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 1.2e-152:
		tmp = t_0
	elif x <= 5.1e-101:
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x
	elif x <= 1.6e-25:
		tmp = t_0
	else:
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 1.2e-152)
		tmp = t_0;
	elseif (x <= 5.1e-101)
		tmp = Float64(Float64(Float64(Float64(Float64(n * x) * 2.0) - n) / Float64(n * Float64(Float64(n * x) / Float64(0.5 + Float64(-0.3333333333333333 / x))))) / x);
	elseif (x <= 1.6e-25)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) + 1.0) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 1.2e-152)
		tmp = t_0;
	elseif (x <= 5.1e-101)
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x;
	elseif (x <= 1.6e-25)
		tmp = t_0;
	else
		tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 1.2e-152], t$95$0, If[LessEqual[x, 5.1e-101], N[(N[(N[(N[(N[(n * x), $MachinePrecision] * 2.0), $MachinePrecision] - n), $MachinePrecision] / N[(n * N[(N[(n * x), $MachinePrecision] / N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.6e-25], t$95$0, N[(N[(N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.2e-152 or 5.1000000000000002e-101 < x < 1.6000000000000001e-25

   1. Initial program 38.7%

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

   3. Taylor expanded in x around 0 38.7%

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

      1. *-rgt-identity 38.7%

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

      2. associate-*l/ 38.7%

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

      3. associate-/l* 38.7%

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

      4. exp-to-pow 38.7%

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

   5. Simplified 38.7%

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

   6. Taylor expanded in n around inf 56.1%

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

      1. mul-1-neg 56.1%

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

      2. distribute-frac-neg2 56.1%

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

   8. Simplified 56.1%

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

   if 1.2e-152 < x < 5.1000000000000002e-101

   1. Initial program 50.6%

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

   3. Taylor expanded in n around inf 50.2%

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

      1. associate--l+ 50.2%

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

      2. log1p-define 50.2%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 50.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 50.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 50.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 50.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 50.2%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 64.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 64.9%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 64.9%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 36.8%

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

      1. sub-neg 36.8%

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

      2. log1p-define 36.8%

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

      3. exp-diff 36.8%

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

      4. log1p-define 36.8%

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

      5. rem-exp-log 36.8%

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

      6. +-commutative 36.8%

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

      7. rem-exp-log 36.8%

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

      8. metadata-eval 36.8%

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

   10. Simplified 36.8%

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

   11. Taylor expanded in n around 0 36.8%

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

       1. log1p-define 36.8%

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

   13. Simplified 36.8%

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

   14. Taylor expanded in x around inf 29.4%

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

   16. Simplified 53.4%

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

       1. frac-2neg 53.4%

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

       2. metadata-eval 53.4%

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

       3. clear-num 53.4%

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

       4. frac-add 54.0%

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

       5. associate-/l* 54.0%

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

       6. associate-/l* 54.0%

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

   18. Applied egg-rr 54.0%

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

       1. *-rgt-identity 54.0%

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

       2. unsub-neg 54.0%

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

       3. mul-1-neg 54.0%

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

       4. associate-*r/ 54.0%

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

       5. *-commutative 54.0%

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

       6. distribute-neg-frac2 54.0%

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

       7. *-commutative 54.0%

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

       8. +-commutative 54.0%

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

       9. distribute-neg-in 54.0%

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

       10. metadata-eval 54.0%

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

       11. distribute-neg-frac 54.0%

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

       12. metadata-eval 54.0%

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

       13. distribute-lft-neg-out 54.0%

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

       14. distribute-rgt-neg-in 54.0%

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

       15. associate-*r/ 54.0%

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

   20. Simplified 54.0%

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

   21. Taylor expanded in x around inf 54.0%

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

       1. *-commutative 54.0%

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

       2. *-commutative 54.0%

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

   23. Simplified 54.0%

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

   if 1.6000000000000001e-25 < x

   1. Initial program 63.9%

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

   3. Taylor expanded in n around inf 68.9%

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

      1. associate--l+ 41.5%

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

      2. log1p-define 41.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 41.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 68.9%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 68.9%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 68.9%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 68.9%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 68.9%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 69.8%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 69.8%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 69.8%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 68.2%

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

      1. sub-neg 68.2%

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

      2. log1p-define 68.2%

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

      3. exp-diff 68.2%

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

      4. log1p-define 68.2%

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

      5. rem-exp-log 12.7%

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

      6. +-commutative 12.7%

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

      7. rem-exp-log 68.4%

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

      8. metadata-eval 68.4%

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

   10. Simplified 68.4%

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

   11. Taylor expanded in n around 0 68.4%

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

       1. log1p-define 66.6%

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

   13. Simplified 66.6%

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

   14. Taylor expanded in x around inf 61.9%

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

   16. Simplified 61.9%

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

   17. Taylor expanded in n around 0 61.9%

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

       1. associate--l+ 61.9%

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

       2. +-commutative 61.9%

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

       3. associate-*r/ 61.9%

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

       4. metadata-eval 61.9%

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

       5. unpow2 61.9%

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

       6. associate-/r* 61.9%

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

       7. associate-*r/ 61.9%

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

       8. metadata-eval 61.9%

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

       9. div-sub 61.9%

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

       10. sub-neg 61.9%

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

       11. metadata-eval 61.9%

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

       12. +-commutative 61.9%

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

       13. +-commutative 61.9%

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

   19. Simplified 61.9%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-152}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}\\ \mathbf{if}\;\frac{1}{n} \leq -50:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 - n}{n \cdot t\_0}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (* n x) (+ 0.5 (/ -0.3333333333333333 x)))))
   (if (<= (/ 1.0 n) -50.0)
     (* 0.3333333333333333 (/ (pow x -3.0) n))
     (if (<= (/ 1.0 n) 5e-29)
       (/ (log1p (/ 1.0 x)) n)
       (/ (/ (- t_0 n) (* n t_0)) x)))))

C

double code(double x, double n) {
	double t_0 = (n * x) / (0.5 + (-0.3333333333333333 / x));
	double tmp;
	if ((1.0 / n) <= -50.0) {
		tmp = 0.3333333333333333 * (pow(x, -3.0) / n);
	} else if ((1.0 / n) <= 5e-29) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = ((t_0 - n) / (n * t_0)) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (n * x) / (0.5 + (-0.3333333333333333 / x));
	double tmp;
	if ((1.0 / n) <= -50.0) {
		tmp = 0.3333333333333333 * (Math.pow(x, -3.0) / n);
	} else if ((1.0 / n) <= 5e-29) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = ((t_0 - n) / (n * t_0)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (n * x) / (0.5 + (-0.3333333333333333 / x))
	tmp = 0
	if (1.0 / n) <= -50.0:
		tmp = 0.3333333333333333 * (math.pow(x, -3.0) / n)
	elif (1.0 / n) <= 5e-29:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = ((t_0 - n) / (n * t_0)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(n * x) / Float64(0.5 + Float64(-0.3333333333333333 / x)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -50.0)
		tmp = Float64(0.3333333333333333 * Float64((x ^ -3.0) / n));
	elseif (Float64(1.0 / n) <= 5e-29)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(Float64(t_0 - n) / Float64(n * t_0)) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(n * x), $MachinePrecision] / N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -50.0], N[(0.3333333333333333 * N[(N[Power[x, -3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-29], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(t$95$0 - n), $MachinePrecision] / N[(n * t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -50

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 72.7%

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

      1. associate--l+ 25.3%

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

      2. log1p-define 25.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 25.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 72.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 72.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 72.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 72.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 98.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 98.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 98.5%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 51.2%

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

      1. sub-neg 51.2%

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

      2. log1p-define 51.2%

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

      3. exp-diff 51.2%

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

      4. log1p-define 51.2%

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

      5. rem-exp-log 3.8%

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

      6. +-commutative 3.8%

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

      7. rem-exp-log 51.2%

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

      8. metadata-eval 51.2%

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

   10. Simplified 51.2%

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

   11. Taylor expanded in n around 0 51.2%

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

       1. log1p-define 4.2%

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

   13. Simplified 4.2%

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

   14. Taylor expanded in x around inf 44.5%

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

       1. associate--l+ 44.5%

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

       2. metadata-eval 44.5%

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

       3. associate-*r/ 44.5%

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

       4. associate-*r/ 44.5%

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

       5. metadata-eval 44.5%

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

       6. unpow2 44.5%

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

       7. associate-/r* 44.5%

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

       8. associate-*r/ 44.5%

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

       9. metadata-eval 44.5%

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

       10. div-sub 44.5%

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

       11. metadata-eval 44.5%

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

       12. associate-*r/ 44.5%

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

       13. sub-neg 44.5%

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

       14. associate-*r/ 44.5%

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

       15. metadata-eval 44.5%

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

       16. metadata-eval 44.5%

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

   16. Simplified 44.5%

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

   17. Taylor expanded in x around 0 63.6%

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

       1. associate-/r* 63.6%

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

   19. Simplified 63.6%

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

       1. *-un-lft-identity 63.6%

          \[\leadsto \color{blue}{1 \cdot \frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]

       2. div-inv 63.6%

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

       3. pow-flip 62.1%

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

       4. metadata-eval 62.1%

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

   21. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{1 \cdot \left(\frac{0.3333333333333333}{n} \cdot {x}^{-3}\right)} \]
   22. Step-by-step derivation

       1. *-lft-identity 62.1%

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

       2. associate-*l/ 62.1%

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

       3. associate-/l* 62.1%

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

   23. Simplified 62.1%

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

   if -50 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999986e-29

   1. Initial program 30.7%

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

   3. Taylor expanded in n around inf 80.7%

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

      1. associate--l+ 80.7%

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

      2. log1p-define 80.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 80.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 80.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 80.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 80.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 80.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 81.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 81.4%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 81.4%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 80.5%

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

      1. sub-neg 80.5%

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

      2. log1p-define 80.5%

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

      3. exp-diff 80.5%

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

      4. log1p-define 80.5%

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

      5. rem-exp-log 57.8%

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

      6. +-commutative 57.8%

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

      7. rem-exp-log 80.7%

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

      8. metadata-eval 80.7%

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

   10. Simplified 80.7%

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

   11. Taylor expanded in n around 0 80.7%

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

       1. log1p-define 98.8%

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

   13. Simplified 98.8%

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

   if 4.99999999999999986e-29 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 43.5%

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

   3. Taylor expanded in n around inf 7.5%

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

      1. associate--l+ 7.6%

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

      2. log1p-define 7.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 7.6%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 7.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 7.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 7.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 7.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 7.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 7.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 7.3%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 7.3%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 12.2%

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

      1. sub-neg 12.2%

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

      2. log1p-define 12.2%

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

      3. exp-diff 12.2%

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

      4. log1p-define 12.2%

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

      5. rem-exp-log 12.0%

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

      6. +-commutative 12.0%

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

      7. rem-exp-log 12.2%

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

      8. metadata-eval 12.2%

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

   10. Simplified 12.2%

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

   11. Taylor expanded in n around 0 12.2%

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

       1. log1p-define 19.3%

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

   13. Simplified 19.3%

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

   14. Taylor expanded in x around inf 21.6%

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

   16. Simplified 42.1%

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

       1. frac-2neg 42.1%

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

       2. metadata-eval 42.1%

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

       3. clear-num 42.1%

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

       4. frac-add 47.3%

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

       5. associate-/l* 47.3%

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

       6. associate-/l* 47.3%

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

   18. Applied egg-rr 47.3%

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

       1. *-rgt-identity 47.3%

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

       2. unsub-neg 47.3%

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

       3. mul-1-neg 47.3%

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

       4. associate-*r/ 47.3%

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

       5. *-commutative 47.3%

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

       6. distribute-neg-frac2 47.3%

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

       7. *-commutative 47.3%

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

       8. +-commutative 47.3%

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

       9. distribute-neg-in 47.3%

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

       10. metadata-eval 47.3%

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

       11. distribute-neg-frac 47.3%

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

       12. metadata-eval 47.3%

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

       13. distribute-lft-neg-out 47.3%

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

       14. distribute-rgt-neg-in 47.3%

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

       15. associate-*r/ 47.3%

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

   20. Simplified 47.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -50:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-3}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}} - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}\\ t_1 := n \cdot t\_0\\ \mathbf{if}\;\frac{1}{n} \leq -500000000000:\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{t\_1}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 - n}{t\_1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (* n x) (+ 0.5 (/ -0.3333333333333333 x)))) (t_1 (* n t_0)))
   (if (<= (/ 1.0 n) -500000000000.0)
     (/ (/ (- (* (* n x) 2.0) n) t_1) x)
     (if (<= (/ 1.0 n) 5e-29)
       (/ (log1p (/ 1.0 x)) n)
       (/ (/ (- t_0 n) t_1) x)))))

C

double code(double x, double n) {
	double t_0 = (n * x) / (0.5 + (-0.3333333333333333 / x));
	double t_1 = n * t_0;
	double tmp;
	if ((1.0 / n) <= -500000000000.0) {
		tmp = ((((n * x) * 2.0) - n) / t_1) / x;
	} else if ((1.0 / n) <= 5e-29) {
		tmp = log1p((1.0 / x)) / n;
	} else {
		tmp = ((t_0 - n) / t_1) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (n * x) / (0.5 + (-0.3333333333333333 / x));
	double t_1 = n * t_0;
	double tmp;
	if ((1.0 / n) <= -500000000000.0) {
		tmp = ((((n * x) * 2.0) - n) / t_1) / x;
	} else if ((1.0 / n) <= 5e-29) {
		tmp = Math.log1p((1.0 / x)) / n;
	} else {
		tmp = ((t_0 - n) / t_1) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (n * x) / (0.5 + (-0.3333333333333333 / x))
	t_1 = n * t_0
	tmp = 0
	if (1.0 / n) <= -500000000000.0:
		tmp = ((((n * x) * 2.0) - n) / t_1) / x
	elif (1.0 / n) <= 5e-29:
		tmp = math.log1p((1.0 / x)) / n
	else:
		tmp = ((t_0 - n) / t_1) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(n * x) / Float64(0.5 + Float64(-0.3333333333333333 / x)))
	t_1 = Float64(n * t_0)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500000000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(n * x) * 2.0) - n) / t_1) / x);
	elseif (Float64(1.0 / n) <= 5e-29)
		tmp = Float64(log1p(Float64(1.0 / x)) / n);
	else
		tmp = Float64(Float64(Float64(t_0 - n) / t_1) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(n * x), $MachinePrecision] / N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(n * t$95$0), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -500000000000.0], N[(N[(N[(N[(N[(n * x), $MachinePrecision] * 2.0), $MachinePrecision] - n), $MachinePrecision] / t$95$1), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-29], N[(N[Log[1 + N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(t$95$0 - n), $MachinePrecision] / t$95$1), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e11

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 73.4%

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

      1. associate--l+ 26.0%

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

      2. log1p-define 26.0%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 26.0%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 73.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 73.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 73.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 73.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 73.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 98.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 98.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 98.5%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 51.2%

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

      1. sub-neg 51.2%

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

      2. log1p-define 51.2%

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

      3. exp-diff 51.2%

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

      4. log1p-define 51.2%

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

      5. rem-exp-log 3.8%

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

      6. +-commutative 3.8%

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

      7. rem-exp-log 51.2%

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

      8. metadata-eval 51.2%

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

   10. Simplified 51.2%

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

   11. Taylor expanded in n around 0 51.2%

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

       1. log1p-define 4.1%

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

   13. Simplified 4.1%

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

   14. Taylor expanded in x around inf 20.7%

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

   16. Simplified 45.7%

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

       1. frac-2neg 45.7%

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

       2. metadata-eval 45.7%

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

       3. clear-num 45.7%

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

       4. frac-add 48.4%

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

       5. associate-/l* 48.4%

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

       6. associate-/l* 48.4%

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

   18. Applied egg-rr 48.4%

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

       1. *-rgt-identity 48.4%

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

       2. unsub-neg 48.4%

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

       3. mul-1-neg 48.4%

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

       4. associate-*r/ 48.4%

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

       5. *-commutative 48.4%

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

       6. distribute-neg-frac2 48.4%

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

       7. *-commutative 48.4%

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

       8. +-commutative 48.4%

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

       9. distribute-neg-in 48.4%

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

       10. metadata-eval 48.4%

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

       11. distribute-neg-frac 48.4%

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

       12. metadata-eval 48.4%

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

       13. distribute-lft-neg-out 48.4%

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

       14. distribute-rgt-neg-in 48.4%

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

       15. associate-*r/ 48.4%

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

   20. Simplified 48.4%

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

   21. Taylor expanded in x around inf 48.4%

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

       1. *-commutative 48.4%

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

       2. *-commutative 48.4%

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

   23. Simplified 48.4%

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

   if -5e11 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999986e-29

   1. Initial program 31.7%

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

   3. Taylor expanded in n around inf 80.3%

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

      1. associate--l+ 79.6%

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

      2. log1p-define 79.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 79.6%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 80.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 80.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 80.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 80.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 80.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 81.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 81.7%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 81.7%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 80.1%

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

      1. sub-neg 80.1%

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

      2. log1p-define 80.1%

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

      3. exp-diff 80.1%

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

      4. log1p-define 80.1%

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

      5. rem-exp-log 57.0%

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

      6. +-commutative 57.0%

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

      7. rem-exp-log 80.2%

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

      8. metadata-eval 80.2%

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

   10. Simplified 80.2%

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

   11. Taylor expanded in n around 0 80.2%

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

       1. log1p-define 97.5%

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

   13. Simplified 97.5%

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

   if 4.99999999999999986e-29 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 43.5%

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

   3. Taylor expanded in n around inf 7.5%

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

      1. associate--l+ 7.6%

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

      2. log1p-define 7.6%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 7.6%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 7.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 7.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 7.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 7.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 7.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 7.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 7.3%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 7.3%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 12.2%

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

      1. sub-neg 12.2%

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

      2. log1p-define 12.2%

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

      3. exp-diff 12.2%

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

      4. log1p-define 12.2%

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

      5. rem-exp-log 12.0%

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

      6. +-commutative 12.0%

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

      7. rem-exp-log 12.2%

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

      8. metadata-eval 12.2%

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

   10. Simplified 12.2%

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

   11. Taylor expanded in n around 0 12.2%

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

       1. log1p-define 19.3%

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

   13. Simplified 19.3%

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

   14. Taylor expanded in x around inf 21.6%

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

   16. Simplified 42.1%

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

       1. frac-2neg 42.1%

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

       2. metadata-eval 42.1%

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

       3. clear-num 42.1%

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

       4. frac-add 47.3%

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

       5. associate-/l* 47.3%

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

       6. associate-/l* 47.3%

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

   18. Applied egg-rr 47.3%

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

       1. *-rgt-identity 47.3%

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

       2. unsub-neg 47.3%

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

       3. mul-1-neg 47.3%

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

       4. associate-*r/ 47.3%

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

       5. *-commutative 47.3%

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

       6. distribute-neg-frac2 47.3%

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

       7. *-commutative 47.3%

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

       8. +-commutative 47.3%

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

       9. distribute-neg-in 47.3%

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

       10. metadata-eval 47.3%

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

       11. distribute-neg-frac 47.3%

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

       12. metadata-eval 47.3%

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

       13. distribute-lft-neg-out 47.3%

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

       14. distribute-rgt-neg-in 47.3%

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

       15. associate-*r/ 47.3%

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

   20. Simplified 47.3%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500000000000:\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}} - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}\\ t_1 := n \cdot t\_0\\ \mathbf{if}\;\frac{1}{n} \leq -10:\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{t\_1}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{n \cdot \left(x \cdot \left(\frac{0.5}{x} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 - n}{t\_1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (* n x) (+ 0.5 (/ -0.3333333333333333 x)))) (t_1 (* n t_0)))
   (if (<= (/ 1.0 n) -10.0)
     (/ (/ (- (* (* n x) 2.0) n) t_1) x)
     (if (<= (/ 1.0 n) 5e-70)
       (/ 1.0 (* n (* x (+ (/ 0.5 x) 1.0))))
       (/ (/ (- t_0 n) t_1) x)))))

C

double code(double x, double n) {
	double t_0 = (n * x) / (0.5 + (-0.3333333333333333 / x));
	double t_1 = n * t_0;
	double tmp;
	if ((1.0 / n) <= -10.0) {
		tmp = ((((n * x) * 2.0) - n) / t_1) / x;
	} else if ((1.0 / n) <= 5e-70) {
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)));
	} else {
		tmp = ((t_0 - n) / t_1) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (n * x) / (0.5d0 + ((-0.3333333333333333d0) / x))
    t_1 = n * t_0
    if ((1.0d0 / n) <= (-10.0d0)) then
        tmp = ((((n * x) * 2.0d0) - n) / t_1) / x
    else if ((1.0d0 / n) <= 5d-70) then
        tmp = 1.0d0 / (n * (x * ((0.5d0 / x) + 1.0d0)))
    else
        tmp = ((t_0 - n) / t_1) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (n * x) / (0.5 + (-0.3333333333333333 / x));
	double t_1 = n * t_0;
	double tmp;
	if ((1.0 / n) <= -10.0) {
		tmp = ((((n * x) * 2.0) - n) / t_1) / x;
	} else if ((1.0 / n) <= 5e-70) {
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)));
	} else {
		tmp = ((t_0 - n) / t_1) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (n * x) / (0.5 + (-0.3333333333333333 / x))
	t_1 = n * t_0
	tmp = 0
	if (1.0 / n) <= -10.0:
		tmp = ((((n * x) * 2.0) - n) / t_1) / x
	elif (1.0 / n) <= 5e-70:
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)))
	else:
		tmp = ((t_0 - n) / t_1) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(n * x) / Float64(0.5 + Float64(-0.3333333333333333 / x)))
	t_1 = Float64(n * t_0)
	tmp = 0.0
	if (Float64(1.0 / n) <= -10.0)
		tmp = Float64(Float64(Float64(Float64(Float64(n * x) * 2.0) - n) / t_1) / x);
	elseif (Float64(1.0 / n) <= 5e-70)
		tmp = Float64(1.0 / Float64(n * Float64(x * Float64(Float64(0.5 / x) + 1.0))));
	else
		tmp = Float64(Float64(Float64(t_0 - n) / t_1) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (n * x) / (0.5 + (-0.3333333333333333 / x));
	t_1 = n * t_0;
	tmp = 0.0;
	if ((1.0 / n) <= -10.0)
		tmp = ((((n * x) * 2.0) - n) / t_1) / x;
	elseif ((1.0 / n) <= 5e-70)
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)));
	else
		tmp = ((t_0 - n) / t_1) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(n * x), $MachinePrecision] / N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(n * t$95$0), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -10.0], N[(N[(N[(N[(N[(n * x), $MachinePrecision] * 2.0), $MachinePrecision] - n), $MachinePrecision] / t$95$1), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-70], N[(1.0 / N[(n * N[(x * N[(N[(0.5 / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 - n), $MachinePrecision] / t$95$1), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -10

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 71.7%

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

      1. associate--l+ 24.9%

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

      2. log1p-define 24.9%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 24.9%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 71.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 71.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 71.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 71.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 98.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 98.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 98.5%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 50.4%

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

      1. sub-neg 50.4%

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

      2. log1p-define 50.4%

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

      3. exp-diff 50.4%

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

      4. log1p-define 50.4%

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

      5. rem-exp-log 3.8%

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

      6. +-commutative 3.8%

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

      7. rem-exp-log 50.4%

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

      8. metadata-eval 50.4%

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

   10. Simplified 50.4%

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

   11. Taylor expanded in n around 0 50.4%

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

       1. log1p-define 4.1%

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

   13. Simplified 4.1%

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

   14. Taylor expanded in x around inf 21.4%

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

   16. Simplified 45.3%

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

       1. frac-2neg 45.3%

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

       2. metadata-eval 45.3%

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

       3. clear-num 45.3%

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

       4. frac-add 47.9%

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

       5. associate-/l* 47.9%

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

       6. associate-/l* 47.9%

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

   18. Applied egg-rr 47.9%

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

       1. *-rgt-identity 47.9%

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

       2. unsub-neg 47.9%

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

       3. mul-1-neg 47.9%

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

       4. associate-*r/ 47.9%

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

       5. *-commutative 47.9%

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

       6. distribute-neg-frac2 47.9%

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

       7. *-commutative 47.9%

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

       8. +-commutative 47.9%

          \[\leadsto \frac{\frac{\frac{x \cdot n}{-\color{blue}{\left(-0.5 + \frac{0.3333333333333333}{x}\right)}} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       9. distribute-neg-in 47.9%

          \[\leadsto \frac{\frac{\frac{x \cdot n}{\color{blue}{\left(--0.5\right) + \left(-\frac{0.3333333333333333}{x}\right)}} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       10. metadata-eval 47.9%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{\color{blue}{0.5} + \left(-\frac{0.3333333333333333}{x}\right)} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       11. distribute-neg-frac 47.9%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \color{blue}{\frac{-0.3333333333333333}{x}}} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       12. metadata-eval 47.9%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \frac{\color{blue}{-0.3333333333333333}}{x}} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       13. distribute-lft-neg-out 47.9%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}} - n}{\color{blue}{-n \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}}{x} \]

       14. distribute-rgt-neg-in 47.9%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}} - n}{\color{blue}{n \cdot \left(-x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}}{x} \]

       15. associate-*r/ 47.9%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}} - n}{n \cdot \left(-\color{blue}{\frac{x \cdot n}{\frac{0.3333333333333333}{x} + -0.5}}\right)}}{x} \]

   20. Simplified 47.9%

       \[\leadsto \frac{\color{blue}{\frac{\frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}} - n}{n \cdot \frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}}}}}{x} \]

   21. Taylor expanded in x around inf 47.9%

       \[\leadsto \frac{\frac{\color{blue}{2 \cdot \left(n \cdot x\right)} - n}{n \cdot \frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}}}}{x} \]
   22. Step-by-step derivation

       1. *-commutative 47.9%

          \[\leadsto \frac{\frac{\color{blue}{\left(n \cdot x\right) \cdot 2} - n}{n \cdot \frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}}}}{x} \]

       2. *-commutative 47.9%

          \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot n\right)} \cdot 2 - n}{n \cdot \frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}}}}{x} \]

   23. Simplified 47.9%

       \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot n\right) \cdot 2} - n}{n \cdot \frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}}}}{x} \]

   if -10 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e-70

   1. Initial program 31.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 82.4%

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. associate--l+ 82.4%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}}{n} \]

      2. log1p-define 82.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 82.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 82.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 82.3%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 82.3%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 82.1%

      \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} - 1\right)}\right)}{n} \]
   9. Step-by-step derivation

      1. sub-neg 82.1%

         \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} + \left(-1\right)\right)}\right)}{n} \]

      2. log1p-define 82.1%

         \[\leadsto \frac{\log \left(1 + \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x} + \left(-1\right)\right)\right)}{n} \]

      3. exp-diff 82.1%

         \[\leadsto \frac{\log \left(1 + \left(\color{blue}{\frac{e^{\mathsf{log1p}\left(x\right)}}{e^{\log x}}} + \left(-1\right)\right)\right)}{n} \]

      4. log1p-define 82.1%

         \[\leadsto \frac{\log \left(1 + \left(\frac{e^{\color{blue}{\log \left(1 + x\right)}}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      5. rem-exp-log 57.8%

         \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{1 + x}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      6. +-commutative 57.8%

         \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{x + 1}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      7. rem-exp-log 82.3%

         \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{\color{blue}{x}} + \left(-1\right)\right)\right)}{n} \]

      8. metadata-eval 82.3%

         \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{x} + \color{blue}{-1}\right)\right)}{n} \]

   10. Simplified 82.3%

       \[\leadsto \frac{\log \left(1 + \color{blue}{\left(\frac{x + 1}{x} + -1\right)}\right)}{n} \]

   11. Taylor expanded in n around 0 82.3%

       \[\leadsto \color{blue}{\frac{\log \left(1 + \frac{1}{x}\right)}{n}} \]
   12. Step-by-step derivation

       1. log1p-define 99.5%

          \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}}{n} \]

   13. Simplified 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}} \]
   14. Step-by-step derivation

       1. clear-num 99.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}} \]

       2. inv-pow 99.1%

          \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}\right)}^{-1}} \]

   15. Applied egg-rr 99.1%

       \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}\right)}^{-1}} \]
   16. Step-by-step derivation

       1. unpow-1 99.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}} \]

   17. Simplified 99.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}} \]
   18. Step-by-step derivation

       1. clear-num 99.1%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}}}} \]

       2. associate-/r/ 99.0%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{log1p}\left(\frac{1}{x}\right)} \cdot n}} \]

   19. Applied egg-rr 99.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{log1p}\left(\frac{1}{x}\right)} \cdot n}} \]

   20. Taylor expanded in x around inf 54.2%

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(1 + 0.5 \cdot \frac{1}{x}\right)\right)} \cdot n} \]
   21. Step-by-step derivation

       1. associate-*r/ 54.2%

          \[\leadsto \frac{1}{\left(x \cdot \left(1 + \color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) \cdot n} \]

       2. metadata-eval 54.2%

          \[\leadsto \frac{1}{\left(x \cdot \left(1 + \frac{\color{blue}{0.5}}{x}\right)\right) \cdot n} \]

   22. Simplified 54.2%

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(1 + \frac{0.5}{x}\right)\right)} \cdot n} \]

   if 4.9999999999999998e-70 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 38.1%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 15.3%

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. associate--l+ 15.3%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}}{n} \]

      2. log1p-define 15.3%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 15.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 15.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 15.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 15.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 15.3%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 15.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 15.1%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 15.1%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 15.1%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 19.3%

      \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} - 1\right)}\right)}{n} \]
   9. Step-by-step derivation

      1. sub-neg 19.3%

         \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} + \left(-1\right)\right)}\right)}{n} \]

      2. log1p-define 19.3%

         \[\leadsto \frac{\log \left(1 + \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x} + \left(-1\right)\right)\right)}{n} \]

      3. exp-diff 19.2%

         \[\leadsto \frac{\log \left(1 + \left(\color{blue}{\frac{e^{\mathsf{log1p}\left(x\right)}}{e^{\log x}}} + \left(-1\right)\right)\right)}{n} \]

      4. log1p-define 19.2%

         \[\leadsto \frac{\log \left(1 + \left(\frac{e^{\color{blue}{\log \left(1 + x\right)}}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      5. rem-exp-log 19.1%

         \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{1 + x}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      6. +-commutative 19.1%

         \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{x + 1}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      7. rem-exp-log 19.3%

         \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{\color{blue}{x}} + \left(-1\right)\right)\right)}{n} \]

      8. metadata-eval 19.3%

         \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{x} + \color{blue}{-1}\right)\right)}{n} \]

   10. Simplified 19.3%

       \[\leadsto \frac{\log \left(1 + \color{blue}{\left(\frac{x + 1}{x} + -1\right)}\right)}{n} \]

   11. Taylor expanded in n around 0 19.3%

       \[\leadsto \color{blue}{\frac{\log \left(1 + \frac{1}{x}\right)}{n}} \]
   12. Step-by-step derivation

       1. log1p-define 30.4%

          \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}}{n} \]

   13. Simplified 30.4%

       \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}} \]

   14. Taylor expanded in x around inf 24.9%

       \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]
   15. Step-by-step derivation

   16. Simplified 42.5%

       \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot n}}{x}} \]
   17. Step-by-step derivation

       1. frac-2neg 42.5%

          \[\leadsto \frac{\color{blue}{\frac{-1}{-n}} + \frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot n}}{x} \]

       2. metadata-eval 42.5%

          \[\leadsto \frac{\frac{\color{blue}{-1}}{-n} + \frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot n}}{x} \]

       3. clear-num 42.5%

          \[\leadsto \frac{\frac{-1}{-n} + \color{blue}{\frac{1}{\frac{x \cdot n}{\frac{0.3333333333333333}{x} + -0.5}}}}{x} \]

       4. frac-add 47.0%

          \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \frac{x \cdot n}{\frac{0.3333333333333333}{x} + -0.5} + \left(-n\right) \cdot 1}{\left(-n\right) \cdot \frac{x \cdot n}{\frac{0.3333333333333333}{x} + -0.5}}}}{x} \]

       5. associate-/l* 47.0%

          \[\leadsto \frac{\frac{-1 \cdot \color{blue}{\left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)} + \left(-n\right) \cdot 1}{\left(-n\right) \cdot \frac{x \cdot n}{\frac{0.3333333333333333}{x} + -0.5}}}{x} \]

       6. associate-/l* 47.0%

          \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right) + \left(-n\right) \cdot 1}{\left(-n\right) \cdot \color{blue}{\left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}}{x} \]

   18. Applied egg-rr 47.0%

       \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right) + \left(-n\right) \cdot 1}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}}{x} \]
   19. Step-by-step derivation

       1. *-rgt-identity 47.0%

          \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right) + \color{blue}{\left(-n\right)}}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       2. unsub-neg 47.0%

          \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right) - n}}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       3. mul-1-neg 47.0%

          \[\leadsto \frac{\frac{\color{blue}{\left(-x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       4. associate-*r/ 47.0%

          \[\leadsto \frac{\frac{\left(-\color{blue}{\frac{x \cdot n}{\frac{0.3333333333333333}{x} + -0.5}}\right) - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       5. *-commutative 47.0%

          \[\leadsto \frac{\frac{\left(-\frac{\color{blue}{n \cdot x}}{\frac{0.3333333333333333}{x} + -0.5}\right) - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       6. distribute-neg-frac2 47.0%

          \[\leadsto \frac{\frac{\color{blue}{\frac{n \cdot x}{-\left(\frac{0.3333333333333333}{x} + -0.5\right)}} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       7. *-commutative 47.0%

          \[\leadsto \frac{\frac{\frac{\color{blue}{x \cdot n}}{-\left(\frac{0.3333333333333333}{x} + -0.5\right)} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       8. +-commutative 47.0%

          \[\leadsto \frac{\frac{\frac{x \cdot n}{-\color{blue}{\left(-0.5 + \frac{0.3333333333333333}{x}\right)}} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       9. distribute-neg-in 47.0%

          \[\leadsto \frac{\frac{\frac{x \cdot n}{\color{blue}{\left(--0.5\right) + \left(-\frac{0.3333333333333333}{x}\right)}} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       10. metadata-eval 47.0%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{\color{blue}{0.5} + \left(-\frac{0.3333333333333333}{x}\right)} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       11. distribute-neg-frac 47.0%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \color{blue}{\frac{-0.3333333333333333}{x}}} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       12. metadata-eval 47.0%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \frac{\color{blue}{-0.3333333333333333}}{x}} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       13. distribute-lft-neg-out 47.0%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}} - n}{\color{blue}{-n \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}}{x} \]

       14. distribute-rgt-neg-in 47.0%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}} - n}{\color{blue}{n \cdot \left(-x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}}{x} \]

       15. associate-*r/ 47.0%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}} - n}{n \cdot \left(-\color{blue}{\frac{x \cdot n}{\frac{0.3333333333333333}{x} + -0.5}}\right)}}{x} \]

   20. Simplified 47.0%

       \[\leadsto \frac{\color{blue}{\frac{\frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}} - n}{n \cdot \frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}}}}}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10:\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{n \cdot \left(x \cdot \left(\frac{0.5}{x} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}} - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10 \lor \neg \left(\frac{1}{n} \leq 5 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(x \cdot \left(\frac{0.5}{x} + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= (/ 1.0 n) -10.0) (not (<= (/ 1.0 n) 5e-70)))
   (/
    (/
     (- (* (* n x) 2.0) n)
     (* n (/ (* n x) (+ 0.5 (/ -0.3333333333333333 x)))))
    x)
   (/ 1.0 (* n (* x (+ (/ 0.5 x) 1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (((1.0 / n) <= -10.0) || !((1.0 / n) <= 5e-70)) {
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x;
	} else {
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (((1.0d0 / n) <= (-10.0d0)) .or. (.not. ((1.0d0 / n) <= 5d-70))) then
        tmp = ((((n * x) * 2.0d0) - n) / (n * ((n * x) / (0.5d0 + ((-0.3333333333333333d0) / x))))) / x
    else
        tmp = 1.0d0 / (n * (x * ((0.5d0 / x) + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (((1.0 / n) <= -10.0) || !((1.0 / n) <= 5e-70)) {
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x;
	} else {
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if ((1.0 / n) <= -10.0) or not ((1.0 / n) <= 5e-70):
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x
	else:
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((Float64(1.0 / n) <= -10.0) || !(Float64(1.0 / n) <= 5e-70))
		tmp = Float64(Float64(Float64(Float64(Float64(n * x) * 2.0) - n) / Float64(n * Float64(Float64(n * x) / Float64(0.5 + Float64(-0.3333333333333333 / x))))) / x);
	else
		tmp = Float64(1.0 / Float64(n * Float64(x * Float64(Float64(0.5 / x) + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (((1.0 / n) <= -10.0) || ~(((1.0 / n) <= 5e-70)))
		tmp = ((((n * x) * 2.0) - n) / (n * ((n * x) / (0.5 + (-0.3333333333333333 / x))))) / x;
	else
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[N[(1.0 / n), $MachinePrecision], -10.0], N[Not[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-70]], $MachinePrecision]], N[(N[(N[(N[(N[(n * x), $MachinePrecision] * 2.0), $MachinePrecision] - n), $MachinePrecision] / N[(n * N[(N[(n * x), $MachinePrecision] / N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(n * N[(x * N[(N[(0.5 / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -10 or 4.9999999999999998e-70 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 71.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 45.5%

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. associate--l+ 20.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}}{n} \]

      2. log1p-define 20.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 20.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 45.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 45.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 45.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 45.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 45.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 59.8%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 59.8%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 59.8%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 36.0%

      \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} - 1\right)}\right)}{n} \]
   9. Step-by-step derivation

      1. sub-neg 36.0%

         \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} + \left(-1\right)\right)}\right)}{n} \]

      2. log1p-define 36.0%

         \[\leadsto \frac{\log \left(1 + \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x} + \left(-1\right)\right)\right)}{n} \]

      3. exp-diff 36.0%

         \[\leadsto \frac{\log \left(1 + \left(\color{blue}{\frac{e^{\mathsf{log1p}\left(x\right)}}{e^{\log x}}} + \left(-1\right)\right)\right)}{n} \]

      4. log1p-define 36.0%

         \[\leadsto \frac{\log \left(1 + \left(\frac{e^{\color{blue}{\log \left(1 + x\right)}}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      5. rem-exp-log 10.9%

         \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{1 + x}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      6. +-commutative 10.9%

         \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{x + 1}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      7. rem-exp-log 36.0%

         \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{\color{blue}{x}} + \left(-1\right)\right)\right)}{n} \]

      8. metadata-eval 36.0%

         \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{x} + \color{blue}{-1}\right)\right)}{n} \]

   10. Simplified 36.0%

       \[\leadsto \frac{\log \left(1 + \color{blue}{\left(\frac{x + 1}{x} + -1\right)}\right)}{n} \]

   11. Taylor expanded in n around 0 36.0%

       \[\leadsto \color{blue}{\frac{\log \left(1 + \frac{1}{x}\right)}{n}} \]
   12. Step-by-step derivation

       1. log1p-define 16.3%

          \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}}{n} \]

   13. Simplified 16.3%

       \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}} \]

   14. Taylor expanded in x around inf 23.0%

       \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]
   15. Step-by-step derivation

   16. Simplified 44.0%

       \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot n}}{x}} \]
   17. Step-by-step derivation

       1. frac-2neg 44.0%

          \[\leadsto \frac{\color{blue}{\frac{-1}{-n}} + \frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot n}}{x} \]

       2. metadata-eval 44.0%

          \[\leadsto \frac{\frac{\color{blue}{-1}}{-n} + \frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot n}}{x} \]

       3. clear-num 44.0%

          \[\leadsto \frac{\frac{-1}{-n} + \color{blue}{\frac{1}{\frac{x \cdot n}{\frac{0.3333333333333333}{x} + -0.5}}}}{x} \]

       4. frac-add 47.5%

          \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \frac{x \cdot n}{\frac{0.3333333333333333}{x} + -0.5} + \left(-n\right) \cdot 1}{\left(-n\right) \cdot \frac{x \cdot n}{\frac{0.3333333333333333}{x} + -0.5}}}}{x} \]

       5. associate-/l* 47.5%

          \[\leadsto \frac{\frac{-1 \cdot \color{blue}{\left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)} + \left(-n\right) \cdot 1}{\left(-n\right) \cdot \frac{x \cdot n}{\frac{0.3333333333333333}{x} + -0.5}}}{x} \]

       6. associate-/l* 47.5%

          \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right) + \left(-n\right) \cdot 1}{\left(-n\right) \cdot \color{blue}{\left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}}{x} \]

   18. Applied egg-rr 47.5%

       \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right) + \left(-n\right) \cdot 1}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}}{x} \]
   19. Step-by-step derivation

       1. *-rgt-identity 47.5%

          \[\leadsto \frac{\frac{-1 \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right) + \color{blue}{\left(-n\right)}}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       2. unsub-neg 47.5%

          \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right) - n}}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       3. mul-1-neg 47.5%

          \[\leadsto \frac{\frac{\color{blue}{\left(-x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       4. associate-*r/ 47.5%

          \[\leadsto \frac{\frac{\left(-\color{blue}{\frac{x \cdot n}{\frac{0.3333333333333333}{x} + -0.5}}\right) - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       5. *-commutative 47.5%

          \[\leadsto \frac{\frac{\left(-\frac{\color{blue}{n \cdot x}}{\frac{0.3333333333333333}{x} + -0.5}\right) - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       6. distribute-neg-frac2 47.5%

          \[\leadsto \frac{\frac{\color{blue}{\frac{n \cdot x}{-\left(\frac{0.3333333333333333}{x} + -0.5\right)}} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       7. *-commutative 47.5%

          \[\leadsto \frac{\frac{\frac{\color{blue}{x \cdot n}}{-\left(\frac{0.3333333333333333}{x} + -0.5\right)} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       8. +-commutative 47.5%

          \[\leadsto \frac{\frac{\frac{x \cdot n}{-\color{blue}{\left(-0.5 + \frac{0.3333333333333333}{x}\right)}} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       9. distribute-neg-in 47.5%

          \[\leadsto \frac{\frac{\frac{x \cdot n}{\color{blue}{\left(--0.5\right) + \left(-\frac{0.3333333333333333}{x}\right)}} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       10. metadata-eval 47.5%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{\color{blue}{0.5} + \left(-\frac{0.3333333333333333}{x}\right)} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       11. distribute-neg-frac 47.5%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \color{blue}{\frac{-0.3333333333333333}{x}}} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       12. metadata-eval 47.5%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \frac{\color{blue}{-0.3333333333333333}}{x}} - n}{\left(-n\right) \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}{x} \]

       13. distribute-lft-neg-out 47.5%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}} - n}{\color{blue}{-n \cdot \left(x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}}{x} \]

       14. distribute-rgt-neg-in 47.5%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}} - n}{\color{blue}{n \cdot \left(-x \cdot \frac{n}{\frac{0.3333333333333333}{x} + -0.5}\right)}}}{x} \]

       15. associate-*r/ 47.5%

           \[\leadsto \frac{\frac{\frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}} - n}{n \cdot \left(-\color{blue}{\frac{x \cdot n}{\frac{0.3333333333333333}{x} + -0.5}}\right)}}{x} \]

   20. Simplified 47.5%

       \[\leadsto \frac{\color{blue}{\frac{\frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}} - n}{n \cdot \frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}}}}}{x} \]

   21. Taylor expanded in x around inf 47.4%

       \[\leadsto \frac{\frac{\color{blue}{2 \cdot \left(n \cdot x\right)} - n}{n \cdot \frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}}}}{x} \]
   22. Step-by-step derivation

       1. *-commutative 47.4%

          \[\leadsto \frac{\frac{\color{blue}{\left(n \cdot x\right) \cdot 2} - n}{n \cdot \frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}}}}{x} \]

       2. *-commutative 47.4%

          \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot n\right)} \cdot 2 - n}{n \cdot \frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}}}}{x} \]

   23. Simplified 47.4%

       \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot n\right) \cdot 2} - n}{n \cdot \frac{x \cdot n}{0.5 + \frac{-0.3333333333333333}{x}}}}{x} \]

   if -10 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e-70

   1. Initial program 31.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 82.4%

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. associate--l+ 82.4%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}}{n} \]

      2. log1p-define 82.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 82.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 82.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 82.3%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 82.3%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 82.1%

      \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} - 1\right)}\right)}{n} \]
   9. Step-by-step derivation

      1. sub-neg 82.1%

         \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} + \left(-1\right)\right)}\right)}{n} \]

      2. log1p-define 82.1%

         \[\leadsto \frac{\log \left(1 + \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x} + \left(-1\right)\right)\right)}{n} \]

      3. exp-diff 82.1%

         \[\leadsto \frac{\log \left(1 + \left(\color{blue}{\frac{e^{\mathsf{log1p}\left(x\right)}}{e^{\log x}}} + \left(-1\right)\right)\right)}{n} \]

      4. log1p-define 82.1%

         \[\leadsto \frac{\log \left(1 + \left(\frac{e^{\color{blue}{\log \left(1 + x\right)}}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      5. rem-exp-log 57.8%

         \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{1 + x}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      6. +-commutative 57.8%

         \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{x + 1}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      7. rem-exp-log 82.3%

         \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{\color{blue}{x}} + \left(-1\right)\right)\right)}{n} \]

      8. metadata-eval 82.3%

         \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{x} + \color{blue}{-1}\right)\right)}{n} \]

   10. Simplified 82.3%

       \[\leadsto \frac{\log \left(1 + \color{blue}{\left(\frac{x + 1}{x} + -1\right)}\right)}{n} \]

   11. Taylor expanded in n around 0 82.3%

       \[\leadsto \color{blue}{\frac{\log \left(1 + \frac{1}{x}\right)}{n}} \]
   12. Step-by-step derivation

       1. log1p-define 99.5%

          \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}}{n} \]

   13. Simplified 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}} \]
   14. Step-by-step derivation

       1. clear-num 99.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}} \]

       2. inv-pow 99.1%

          \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}\right)}^{-1}} \]

   15. Applied egg-rr 99.1%

       \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}\right)}^{-1}} \]
   16. Step-by-step derivation

       1. unpow-1 99.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}} \]

   17. Simplified 99.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}} \]
   18. Step-by-step derivation

       1. clear-num 99.1%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}}}} \]

       2. associate-/r/ 99.0%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{log1p}\left(\frac{1}{x}\right)} \cdot n}} \]

   19. Applied egg-rr 99.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{log1p}\left(\frac{1}{x}\right)} \cdot n}} \]

   20. Taylor expanded in x around inf 54.2%

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(1 + 0.5 \cdot \frac{1}{x}\right)\right)} \cdot n} \]
   21. Step-by-step derivation

       1. associate-*r/ 54.2%

          \[\leadsto \frac{1}{\left(x \cdot \left(1 + \color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) \cdot n} \]

       2. metadata-eval 54.2%

          \[\leadsto \frac{1}{\left(x \cdot \left(1 + \frac{\color{blue}{0.5}}{x}\right)\right) \cdot n} \]

   22. Simplified 54.2%

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(1 + \frac{0.5}{x}\right)\right)} \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10 \lor \neg \left(\frac{1}{n} \leq 5 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{\frac{\left(n \cdot x\right) \cdot 2 - n}{n \cdot \frac{n \cdot x}{0.5 + \frac{-0.3333333333333333}{x}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(x \cdot \left(\frac{0.5}{x} + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10 \lor \neg \left(\frac{1}{n} \leq 5 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{\frac{n + n \cdot \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(x \cdot \left(\frac{0.5}{x} + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= (/ 1.0 n) -10.0) (not (<= (/ 1.0 n) 5e-70)))
   (/ (/ (+ n (* n (/ (+ -0.5 (/ 0.3333333333333333 x)) x))) (* n n)) x)
   (/ 1.0 (* n (* x (+ (/ 0.5 x) 1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (((1.0 / n) <= -10.0) || !((1.0 / n) <= 5e-70)) {
		tmp = ((n + (n * ((-0.5 + (0.3333333333333333 / x)) / x))) / (n * n)) / x;
	} else {
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (((1.0d0 / n) <= (-10.0d0)) .or. (.not. ((1.0d0 / n) <= 5d-70))) then
        tmp = ((n + (n * (((-0.5d0) + (0.3333333333333333d0 / x)) / x))) / (n * n)) / x
    else
        tmp = 1.0d0 / (n * (x * ((0.5d0 / x) + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (((1.0 / n) <= -10.0) || !((1.0 / n) <= 5e-70)) {
		tmp = ((n + (n * ((-0.5 + (0.3333333333333333 / x)) / x))) / (n * n)) / x;
	} else {
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if ((1.0 / n) <= -10.0) or not ((1.0 / n) <= 5e-70):
		tmp = ((n + (n * ((-0.5 + (0.3333333333333333 / x)) / x))) / (n * n)) / x
	else:
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((Float64(1.0 / n) <= -10.0) || !(Float64(1.0 / n) <= 5e-70))
		tmp = Float64(Float64(Float64(n + Float64(n * Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x))) / Float64(n * n)) / x);
	else
		tmp = Float64(1.0 / Float64(n * Float64(x * Float64(Float64(0.5 / x) + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (((1.0 / n) <= -10.0) || ~(((1.0 / n) <= 5e-70)))
		tmp = ((n + (n * ((-0.5 + (0.3333333333333333 / x)) / x))) / (n * n)) / x;
	else
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[N[(1.0 / n), $MachinePrecision], -10.0], N[Not[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-70]], $MachinePrecision]], N[(N[(N[(n + N[(n * N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(n * N[(x * N[(N[(0.5 / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -10 or 4.9999999999999998e-70 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 71.3%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 45.5%

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. associate--l+ 20.5%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}}{n} \]

      2. log1p-define 20.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 20.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 45.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 45.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 45.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 45.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 45.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 59.8%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 59.8%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 59.8%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 36.0%

      \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} - 1\right)}\right)}{n} \]
   9. Step-by-step derivation

      1. sub-neg 36.0%

         \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} + \left(-1\right)\right)}\right)}{n} \]

      2. log1p-define 36.0%

         \[\leadsto \frac{\log \left(1 + \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x} + \left(-1\right)\right)\right)}{n} \]

      3. exp-diff 36.0%

         \[\leadsto \frac{\log \left(1 + \left(\color{blue}{\frac{e^{\mathsf{log1p}\left(x\right)}}{e^{\log x}}} + \left(-1\right)\right)\right)}{n} \]

      4. log1p-define 36.0%

         \[\leadsto \frac{\log \left(1 + \left(\frac{e^{\color{blue}{\log \left(1 + x\right)}}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      5. rem-exp-log 10.9%

         \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{1 + x}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      6. +-commutative 10.9%

         \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{x + 1}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      7. rem-exp-log 36.0%

         \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{\color{blue}{x}} + \left(-1\right)\right)\right)}{n} \]

      8. metadata-eval 36.0%

         \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{x} + \color{blue}{-1}\right)\right)}{n} \]

   10. Simplified 36.0%

       \[\leadsto \frac{\log \left(1 + \color{blue}{\left(\frac{x + 1}{x} + -1\right)}\right)}{n} \]

   11. Taylor expanded in n around 0 36.0%

       \[\leadsto \color{blue}{\frac{\log \left(1 + \frac{1}{x}\right)}{n}} \]
   12. Step-by-step derivation

       1. log1p-define 16.3%

          \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}}{n} \]

   13. Simplified 16.3%

       \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}} \]

   14. Taylor expanded in x around inf 23.0%

       \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]
   15. Step-by-step derivation

   16. Simplified 44.0%

       \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot n}}{x}} \]
   17. Step-by-step derivation

       1. frac-2neg 44.0%

          \[\leadsto \frac{\color{blue}{\frac{-1}{-n}} + \frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot n}}{x} \]

       2. metadata-eval 44.0%

          \[\leadsto \frac{\frac{\color{blue}{-1}}{-n} + \frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot n}}{x} \]

       3. associate-/r* 44.0%

          \[\leadsto \frac{\frac{-1}{-n} + \color{blue}{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{n}}}{x} \]

       4. frac-add 46.7%

          \[\leadsto \frac{\color{blue}{\frac{-1 \cdot n + \left(-n\right) \cdot \frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{\left(-n\right) \cdot n}}}{x} \]

       5. neg-mul-1 46.7%

          \[\leadsto \frac{\frac{\color{blue}{\left(-n\right)} + \left(-n\right) \cdot \frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{\left(-n\right) \cdot n}}{x} \]

   18. Applied egg-rr 46.7%

       \[\leadsto \frac{\color{blue}{\frac{\left(-n\right) + \left(-n\right) \cdot \frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{\left(-n\right) \cdot n}}}{x} \]

   if -10 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999998e-70

   1. Initial program 31.8%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 82.4%

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. associate--l+ 82.4%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}}{n} \]

      2. log1p-define 82.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 82.4%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 82.4%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 82.4%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 82.3%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 82.3%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 82.1%

      \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} - 1\right)}\right)}{n} \]
   9. Step-by-step derivation

      1. sub-neg 82.1%

         \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} + \left(-1\right)\right)}\right)}{n} \]

      2. log1p-define 82.1%

         \[\leadsto \frac{\log \left(1 + \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x} + \left(-1\right)\right)\right)}{n} \]

      3. exp-diff 82.1%

         \[\leadsto \frac{\log \left(1 + \left(\color{blue}{\frac{e^{\mathsf{log1p}\left(x\right)}}{e^{\log x}}} + \left(-1\right)\right)\right)}{n} \]

      4. log1p-define 82.1%

         \[\leadsto \frac{\log \left(1 + \left(\frac{e^{\color{blue}{\log \left(1 + x\right)}}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      5. rem-exp-log 57.8%

         \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{1 + x}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      6. +-commutative 57.8%

         \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{x + 1}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      7. rem-exp-log 82.3%

         \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{\color{blue}{x}} + \left(-1\right)\right)\right)}{n} \]

      8. metadata-eval 82.3%

         \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{x} + \color{blue}{-1}\right)\right)}{n} \]

   10. Simplified 82.3%

       \[\leadsto \frac{\log \left(1 + \color{blue}{\left(\frac{x + 1}{x} + -1\right)}\right)}{n} \]

   11. Taylor expanded in n around 0 82.3%

       \[\leadsto \color{blue}{\frac{\log \left(1 + \frac{1}{x}\right)}{n}} \]
   12. Step-by-step derivation

       1. log1p-define 99.5%

          \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}}{n} \]

   13. Simplified 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}} \]
   14. Step-by-step derivation

       1. clear-num 99.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}} \]

       2. inv-pow 99.1%

          \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}\right)}^{-1}} \]

   15. Applied egg-rr 99.1%

       \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}\right)}^{-1}} \]
   16. Step-by-step derivation

       1. unpow-1 99.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}} \]

   17. Simplified 99.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}} \]
   18. Step-by-step derivation

       1. clear-num 99.1%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}}}} \]

       2. associate-/r/ 99.0%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{log1p}\left(\frac{1}{x}\right)} \cdot n}} \]

   19. Applied egg-rr 99.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{log1p}\left(\frac{1}{x}\right)} \cdot n}} \]

   20. Taylor expanded in x around inf 54.2%

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(1 + 0.5 \cdot \frac{1}{x}\right)\right)} \cdot n} \]
   21. Step-by-step derivation

       1. associate-*r/ 54.2%

          \[\leadsto \frac{1}{\left(x \cdot \left(1 + \color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) \cdot n} \]

       2. metadata-eval 54.2%

          \[\leadsto \frac{1}{\left(x \cdot \left(1 + \frac{\color{blue}{0.5}}{x}\right)\right) \cdot n} \]

   22. Simplified 54.2%

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(1 + \frac{0.5}{x}\right)\right)} \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10 \lor \neg \left(\frac{1}{n} \leq 5 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{\frac{n + n \cdot \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n \cdot n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(x \cdot \left(\frac{0.5}{x} + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.6% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.56 \cdot 10^{-192}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(x \cdot \left(\frac{0.5}{x} + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n 1.56e-192) (/ (/ 1.0 n) x) (/ 1.0 (* n (* x (+ (/ 0.5 x) 1.0))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= 1.56e-192) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1.56d-192) then
        tmp = (1.0d0 / n) / x
    else
        tmp = 1.0d0 / (n * (x * ((0.5d0 / x) + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= 1.56e-192) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= 1.56e-192:
		tmp = (1.0 / n) / x
	else:
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= 1.56e-192)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(1.0 / Float64(n * Float64(x * Float64(Float64(0.5 / x) + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= 1.56e-192)
		tmp = (1.0 / n) / x;
	else
		tmp = 1.0 / (n * (x * ((0.5 / x) + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, 1.56e-192], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(n * N[(x * N[(N[(0.5 / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.5599999999999999e-192

   1. Initial program 60.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 68.1%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. associate-/r* 68.5%

         \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n}}{x}} \]

      2. mul-1-neg 68.5%

         \[\leadsto \frac{\frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n}}{x} \]

      3. log-rec 68.5%

         \[\leadsto \frac{\frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n}}{x} \]

      4. mul-1-neg 68.5%

         \[\leadsto \frac{\frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n}}{x} \]

      5. distribute-neg-frac 68.5%

         \[\leadsto \frac{\frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n}}{x} \]

      6. mul-1-neg 68.5%

         \[\leadsto \frac{\frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n}}{x} \]

      7. remove-double-neg 68.5%

         \[\leadsto \frac{\frac{e^{\frac{\color{blue}{\log x}}{n}}}{n}}{x} \]

      8. *-rgt-identity 68.5%

         \[\leadsto \frac{\frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n}}{x} \]

      9. associate-/l* 68.5%

         \[\leadsto \frac{\frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n}}{x} \]

      10. exp-to-pow 68.5%

          \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x} \]

   5. Simplified 68.5%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]

   6. Taylor expanded in n around inf 43.9%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   7. Step-by-step derivation

      1. associate-/r* 44.2%

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

   8. Simplified 44.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

   if 1.5599999999999999e-192 < n

   1. Initial program 35.2%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf 55.6%

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. associate--l+ 55.7%

         \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}}{n} \]

      2. log1p-define 55.7%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

      3. +-commutative 55.7%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

      4. associate--r+ 55.6%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

      5. distribute-lft-out-- 55.6%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

      6. div-sub 55.6%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

      7. log1p-define 55.6%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

   5. Simplified 55.6%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 55.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

      2. log1p-undefine 55.6%

         \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   7. Applied egg-rr 55.6%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   8. Taylor expanded in n around inf 57.0%

      \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} - 1\right)}\right)}{n} \]
   9. Step-by-step derivation

      1. sub-neg 57.0%

         \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} + \left(-1\right)\right)}\right)}{n} \]

      2. log1p-define 57.0%

         \[\leadsto \frac{\log \left(1 + \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x} + \left(-1\right)\right)\right)}{n} \]

      3. exp-diff 57.0%

         \[\leadsto \frac{\log \left(1 + \left(\color{blue}{\frac{e^{\mathsf{log1p}\left(x\right)}}{e^{\log x}}} + \left(-1\right)\right)\right)}{n} \]

      4. log1p-define 57.0%

         \[\leadsto \frac{\log \left(1 + \left(\frac{e^{\color{blue}{\log \left(1 + x\right)}}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      5. rem-exp-log 45.0%

         \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{1 + x}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      6. +-commutative 45.0%

         \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{x + 1}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

      7. rem-exp-log 57.1%

         \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{\color{blue}{x}} + \left(-1\right)\right)\right)}{n} \]

      8. metadata-eval 57.1%

         \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{x} + \color{blue}{-1}\right)\right)}{n} \]

   10. Simplified 57.1%

       \[\leadsto \frac{\log \left(1 + \color{blue}{\left(\frac{x + 1}{x} + -1\right)}\right)}{n} \]

   11. Taylor expanded in n around 0 57.1%

       \[\leadsto \color{blue}{\frac{\log \left(1 + \frac{1}{x}\right)}{n}} \]
   12. Step-by-step derivation

       1. log1p-define 71.3%

          \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}}{n} \]

   13. Simplified 71.3%

       \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}} \]
   14. Step-by-step derivation

       1. clear-num 71.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}} \]

       2. inv-pow 71.3%

          \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}\right)}^{-1}} \]

   15. Applied egg-rr 71.3%

       \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}\right)}^{-1}} \]
   16. Step-by-step derivation

       1. unpow-1 71.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}} \]

   17. Simplified 71.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(\frac{1}{x}\right)}}} \]
   18. Step-by-step derivation

       1. clear-num 71.3%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}}}} \]

       2. associate-/r/ 71.3%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{log1p}\left(\frac{1}{x}\right)} \cdot n}} \]

   19. Applied egg-rr 71.3%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{log1p}\left(\frac{1}{x}\right)} \cdot n}} \]

   20. Taylor expanded in x around inf 36.7%

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(1 + 0.5 \cdot \frac{1}{x}\right)\right)} \cdot n} \]
   21. Step-by-step derivation

       1. associate-*r/ 36.7%

          \[\leadsto \frac{1}{\left(x \cdot \left(1 + \color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) \cdot n} \]

       2. metadata-eval 36.7%

          \[\leadsto \frac{1}{\left(x \cdot \left(1 + \frac{\color{blue}{0.5}}{x}\right)\right) \cdot n} \]

   22. Simplified 36.7%

       \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(1 + \frac{0.5}{x}\right)\right)} \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.56 \cdot 10^{-192}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot \left(x \cdot \left(\frac{0.5}{x} + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.2% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n} + \frac{-0.5 + \frac{0.3333333333333333}{x}}{n \cdot x}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (+ (/ 1.0 n) (/ (+ -0.5 (/ 0.3333333333333333 x)) (* n x))) x))

C

double code(double x, double n) {
	return ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (n * x))) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 / n) + (((-0.5d0) + (0.3333333333333333d0 / x)) / (n * x))) / x
end function

Java

public static double code(double x, double n) {
	return ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (n * x))) / x;
}

Python

def code(x, n):
	return ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (n * x))) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 / n) + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / Float64(n * x))) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (n * x))) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 51.1%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 64.4%

   \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
4. Step-by-step derivation

   1. associate--l+ 52.1%

      \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}}{n} \]

   2. log1p-define 52.1%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

   3. +-commutative 52.1%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

   4. associate--r+ 64.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

   5. distribute-lft-out-- 64.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

   6. div-sub 64.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

   7. log1p-define 64.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

5. Simplified 64.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
6. Step-by-step derivation

   1. log1p-expm1-u 71.4%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   2. log1p-undefine 71.4%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

7. Applied egg-rr 71.4%

   \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

8. Taylor expanded in n around inf 59.6%

   \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} - 1\right)}\right)}{n} \]
9. Step-by-step derivation

   1. sub-neg 59.6%

      \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} + \left(-1\right)\right)}\right)}{n} \]

   2. log1p-define 59.6%

      \[\leadsto \frac{\log \left(1 + \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x} + \left(-1\right)\right)\right)}{n} \]

   3. exp-diff 59.6%

      \[\leadsto \frac{\log \left(1 + \left(\color{blue}{\frac{e^{\mathsf{log1p}\left(x\right)}}{e^{\log x}}} + \left(-1\right)\right)\right)}{n} \]

   4. log1p-define 59.6%

      \[\leadsto \frac{\log \left(1 + \left(\frac{e^{\color{blue}{\log \left(1 + x\right)}}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

   5. rem-exp-log 34.9%

      \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{1 + x}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

   6. +-commutative 34.9%

      \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{x + 1}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

   7. rem-exp-log 59.7%

      \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{\color{blue}{x}} + \left(-1\right)\right)\right)}{n} \]

   8. metadata-eval 59.7%

      \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{x} + \color{blue}{-1}\right)\right)}{n} \]

10. Simplified 59.7%

    \[\leadsto \frac{\log \left(1 + \color{blue}{\left(\frac{x + 1}{x} + -1\right)}\right)}{n} \]

11. Taylor expanded in n around 0 59.7%

    \[\leadsto \color{blue}{\frac{\log \left(1 + \frac{1}{x}\right)}{n}} \]
12. Step-by-step derivation

    1. log1p-define 58.9%

       \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}}{n} \]

13. Simplified 58.9%

    \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}} \]

14. Taylor expanded in x around inf 36.2%

    \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]
15. Step-by-step derivation

16. Simplified 46.4%

    \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot n}}{x}} \]

17. Final simplification 46.4%

    \[\leadsto \frac{\frac{1}{n} + \frac{-0.5 + \frac{0.3333333333333333}{x}}{n \cdot x}}{x} \]
18. Add Preprocessing

Alternative 15: 46.2% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{n}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (/ (+ (/ (+ -0.5 (/ 0.3333333333333333 x)) x) 1.0) n) x))

C

double code(double x, double n) {
	return ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (((((-0.5d0) + (0.3333333333333333d0 / x)) / x) + 1.0d0) / n) / x
end function

Java

public static double code(double x, double n) {
	return ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x;
}

Python

def code(x, n):
	return ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) + 1.0) / n) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / n) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 51.1%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 64.4%

   \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
4. Step-by-step derivation

   1. associate--l+ 52.1%

      \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}}{n} \]

   2. log1p-define 52.1%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

   3. +-commutative 52.1%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

   4. associate--r+ 64.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

   5. distribute-lft-out-- 64.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

   6. div-sub 64.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

   7. log1p-define 64.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

5. Simplified 64.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
6. Step-by-step derivation

   1. log1p-expm1-u 71.4%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   2. log1p-undefine 71.4%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

7. Applied egg-rr 71.4%

   \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

8. Taylor expanded in n around inf 59.6%

   \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} - 1\right)}\right)}{n} \]
9. Step-by-step derivation

   1. sub-neg 59.6%

      \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} + \left(-1\right)\right)}\right)}{n} \]

   2. log1p-define 59.6%

      \[\leadsto \frac{\log \left(1 + \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x} + \left(-1\right)\right)\right)}{n} \]

   3. exp-diff 59.6%

      \[\leadsto \frac{\log \left(1 + \left(\color{blue}{\frac{e^{\mathsf{log1p}\left(x\right)}}{e^{\log x}}} + \left(-1\right)\right)\right)}{n} \]

   4. log1p-define 59.6%

      \[\leadsto \frac{\log \left(1 + \left(\frac{e^{\color{blue}{\log \left(1 + x\right)}}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

   5. rem-exp-log 34.9%

      \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{1 + x}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

   6. +-commutative 34.9%

      \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{x + 1}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

   7. rem-exp-log 59.7%

      \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{\color{blue}{x}} + \left(-1\right)\right)\right)}{n} \]

   8. metadata-eval 59.7%

      \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{x} + \color{blue}{-1}\right)\right)}{n} \]

10. Simplified 59.7%

    \[\leadsto \frac{\log \left(1 + \color{blue}{\left(\frac{x + 1}{x} + -1\right)}\right)}{n} \]

11. Taylor expanded in n around 0 59.7%

    \[\leadsto \color{blue}{\frac{\log \left(1 + \frac{1}{x}\right)}{n}} \]
12. Step-by-step derivation

    1. log1p-define 58.9%

       \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}}{n} \]

13. Simplified 58.9%

    \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}} \]

14. Taylor expanded in x around inf 36.2%

    \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]
15. Step-by-step derivation

16. Simplified 46.4%

    \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot n}}{x}} \]

17. Taylor expanded in n around 0 46.4%

    \[\leadsto \frac{\color{blue}{\frac{\left(1 + 0.3333333333333333 \cdot \frac{1}{{x}^{2}}\right) - 0.5 \cdot \frac{1}{x}}{n}}}{x} \]
18. Step-by-step derivation

    1. associate--l+ 46.4%

       \[\leadsto \frac{\frac{\color{blue}{1 + \left(0.3333333333333333 \cdot \frac{1}{{x}^{2}} - 0.5 \cdot \frac{1}{x}\right)}}{n}}{x} \]

    2. +-commutative 46.4%

       \[\leadsto \frac{\frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{2}} - 0.5 \cdot \frac{1}{x}\right) + 1}}{n}}{x} \]

    3. associate-*r/ 46.4%

       \[\leadsto \frac{\frac{\left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{2}}} - 0.5 \cdot \frac{1}{x}\right) + 1}{n}}{x} \]

    4. metadata-eval 46.4%

       \[\leadsto \frac{\frac{\left(\frac{\color{blue}{0.3333333333333333}}{{x}^{2}} - 0.5 \cdot \frac{1}{x}\right) + 1}{n}}{x} \]

    5. unpow2 46.4%

       \[\leadsto \frac{\frac{\left(\frac{0.3333333333333333}{\color{blue}{x \cdot x}} - 0.5 \cdot \frac{1}{x}\right) + 1}{n}}{x} \]

    6. associate-/r* 46.4%

       \[\leadsto \frac{\frac{\left(\color{blue}{\frac{\frac{0.3333333333333333}{x}}{x}} - 0.5 \cdot \frac{1}{x}\right) + 1}{n}}{x} \]

    7. associate-*r/ 46.4%

       \[\leadsto \frac{\frac{\left(\frac{\frac{0.3333333333333333}{x}}{x} - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) + 1}{n}}{x} \]

    8. metadata-eval 46.4%

       \[\leadsto \frac{\frac{\left(\frac{\frac{0.3333333333333333}{x}}{x} - \frac{\color{blue}{0.5}}{x}\right) + 1}{n}}{x} \]

    9. div-sub 46.4%

       \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{0.3333333333333333}{x} - 0.5}{x}} + 1}{n}}{x} \]

    10. sub-neg 46.4%

        \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{0.3333333333333333}{x} + \left(-0.5\right)}}{x} + 1}{n}}{x} \]

    11. metadata-eval 46.4%

        \[\leadsto \frac{\frac{\frac{\frac{0.3333333333333333}{x} + \color{blue}{-0.5}}{x} + 1}{n}}{x} \]

    12. +-commutative 46.4%

        \[\leadsto \frac{\frac{\color{blue}{1 + \frac{\frac{0.3333333333333333}{x} + -0.5}{x}}}{n}}{x} \]

    13. +-commutative 46.4%

        \[\leadsto \frac{\frac{1 + \frac{\color{blue}{-0.5 + \frac{0.3333333333333333}{x}}}{x}}{n}}{x} \]

19. Simplified 46.4%

    \[\leadsto \frac{\color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{n}}}{x} \]

20. Final simplification 46.4%

    \[\leadsto \frac{\frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{n}}{x} \]
21. Add Preprocessing

Alternative 16: 45.8% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{n \cdot x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (+ (/ (+ -0.5 (/ 0.3333333333333333 x)) x) 1.0) (* n x)))

C

double code(double x, double n) {
	return (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (n * x);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((((-0.5d0) + (0.3333333333333333d0 / x)) / x) + 1.0d0) / (n * x)
end function

Java

public static double code(double x, double n) {
	return (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (n * x);
}

Python

def code(x, n):
	return (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (n * x)

Julia

function code(x, n)
	return Float64(Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) + 1.0) / Float64(n * x))
end

MATLAB

function tmp = code(x, n)
	tmp = (((-0.5 + (0.3333333333333333 / x)) / x) + 1.0) / (n * x);
end

Wolfram

code[x_, n_] := N[(N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 64.4%

   \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
4. Step-by-step derivation

   1. associate--l+ 52.1%

      \[\leadsto \frac{\color{blue}{\log \left(1 + x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}}{n} \]

   2. log1p-define 52.1%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)\right)}{n} \]

   3. +-commutative 52.1%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - \color{blue}{\left(0.5 \cdot \frac{{\log x}^{2}}{n} + \log x\right)}\right)}{n} \]

   4. associate--r+ 64.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \color{blue}{\left(\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x\right)}}{n} \]

   5. distribute-lft-out-- 64.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(\color{blue}{0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{n} - \frac{{\log x}^{2}}{n}\right)} - \log x\right)}{n} \]

   6. div-sub 64.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \color{blue}{\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{n}} - \log x\right)}{n} \]

   7. log1p-define 64.4%

      \[\leadsto \frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}}^{2} - {\log x}^{2}}{n} - \log x\right)}{n} \]

5. Simplified 64.4%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}} \]
6. Step-by-step derivation

   1. log1p-expm1-u 71.4%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

   2. log1p-undefine 71.4%

      \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

7. Applied egg-rr 71.4%

   \[\leadsto \frac{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)\right)\right)}}{n} \]

8. Taylor expanded in n around inf 59.6%

   \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} - 1\right)}\right)}{n} \]
9. Step-by-step derivation

   1. sub-neg 59.6%

      \[\leadsto \frac{\log \left(1 + \color{blue}{\left(e^{\log \left(1 + x\right) - \log x} + \left(-1\right)\right)}\right)}{n} \]

   2. log1p-define 59.6%

      \[\leadsto \frac{\log \left(1 + \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x} + \left(-1\right)\right)\right)}{n} \]

   3. exp-diff 59.6%

      \[\leadsto \frac{\log \left(1 + \left(\color{blue}{\frac{e^{\mathsf{log1p}\left(x\right)}}{e^{\log x}}} + \left(-1\right)\right)\right)}{n} \]

   4. log1p-define 59.6%

      \[\leadsto \frac{\log \left(1 + \left(\frac{e^{\color{blue}{\log \left(1 + x\right)}}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

   5. rem-exp-log 34.9%

      \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{1 + x}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

   6. +-commutative 34.9%

      \[\leadsto \frac{\log \left(1 + \left(\frac{\color{blue}{x + 1}}{e^{\log x}} + \left(-1\right)\right)\right)}{n} \]

   7. rem-exp-log 59.7%

      \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{\color{blue}{x}} + \left(-1\right)\right)\right)}{n} \]

   8. metadata-eval 59.7%

      \[\leadsto \frac{\log \left(1 + \left(\frac{x + 1}{x} + \color{blue}{-1}\right)\right)}{n} \]

10. Simplified 59.7%

    \[\leadsto \frac{\log \left(1 + \color{blue}{\left(\frac{x + 1}{x} + -1\right)}\right)}{n} \]

11. Taylor expanded in n around 0 59.7%

    \[\leadsto \color{blue}{\frac{\log \left(1 + \frac{1}{x}\right)}{n}} \]
12. Step-by-step derivation

    1. log1p-define 58.9%

       \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)}}{n} \]

13. Simplified 58.9%

    \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(\frac{1}{x}\right)}{n}} \]

14. Taylor expanded in x around inf 36.2%

    \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]
15. Step-by-step derivation

16. Simplified 46.4%

    \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x} + -0.5}{x \cdot n}}{x}} \]

17. Taylor expanded in n around 0 46.2%

    \[\leadsto \color{blue}{\frac{\left(1 + 0.3333333333333333 \cdot \frac{1}{{x}^{2}}\right) - 0.5 \cdot \frac{1}{x}}{n \cdot x}} \]
18. Step-by-step derivation

    1. associate--l+ 46.2%

       \[\leadsto \frac{\color{blue}{1 + \left(0.3333333333333333 \cdot \frac{1}{{x}^{2}} - 0.5 \cdot \frac{1}{x}\right)}}{n \cdot x} \]

    2. sub-neg 46.2%

       \[\leadsto \frac{1 + \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{2}} + \left(-0.5 \cdot \frac{1}{x}\right)\right)}}{n \cdot x} \]

    3. sub-neg 46.2%

       \[\leadsto \frac{1 + \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{2}} - 0.5 \cdot \frac{1}{x}\right)}}{n \cdot x} \]

    4. associate-*r/ 46.2%

       \[\leadsto \frac{1 + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{2}}} - 0.5 \cdot \frac{1}{x}\right)}{n \cdot x} \]

    5. metadata-eval 46.2%

       \[\leadsto \frac{1 + \left(\frac{\color{blue}{0.3333333333333333}}{{x}^{2}} - 0.5 \cdot \frac{1}{x}\right)}{n \cdot x} \]

    6. unpow2 46.2%

       \[\leadsto \frac{1 + \left(\frac{0.3333333333333333}{\color{blue}{x \cdot x}} - 0.5 \cdot \frac{1}{x}\right)}{n \cdot x} \]

    7. associate-/r* 46.2%

       \[\leadsto \frac{1 + \left(\color{blue}{\frac{\frac{0.3333333333333333}{x}}{x}} - 0.5 \cdot \frac{1}{x}\right)}{n \cdot x} \]

    8. associate-*r/ 46.2%

       \[\leadsto \frac{1 + \left(\frac{\frac{0.3333333333333333}{x}}{x} - \color{blue}{\frac{0.5 \cdot 1}{x}}\right)}{n \cdot x} \]

    9. metadata-eval 46.2%

       \[\leadsto \frac{1 + \left(\frac{\frac{0.3333333333333333}{x}}{x} - \frac{\color{blue}{0.5}}{x}\right)}{n \cdot x} \]

    10. div-sub 46.2%

        \[\leadsto \frac{1 + \color{blue}{\frac{\frac{0.3333333333333333}{x} - 0.5}{x}}}{n \cdot x} \]

    11. metadata-eval 46.2%

        \[\leadsto \frac{1 + \frac{\frac{\color{blue}{0.3333333333333333 \cdot 1}}{x} - 0.5}{x}}{n \cdot x} \]

    12. associate-*r/ 46.2%

        \[\leadsto \frac{1 + \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{x}} - 0.5}{x}}{n \cdot x} \]

    13. sub-neg 46.2%

        \[\leadsto \frac{1 + \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{x} + \left(-0.5\right)}}{x}}{n \cdot x} \]

    14. associate-*r/ 46.2%

        \[\leadsto \frac{1 + \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{x}} + \left(-0.5\right)}{x}}{n \cdot x} \]

    15. metadata-eval 46.2%

        \[\leadsto \frac{1 + \frac{\frac{\color{blue}{0.3333333333333333}}{x} + \left(-0.5\right)}{x}}{n \cdot x} \]

    16. metadata-eval 46.2%

        \[\leadsto \frac{1 + \frac{\frac{0.3333333333333333}{x} + \color{blue}{-0.5}}{x}}{n \cdot x} \]

    17. +-commutative 46.2%

        \[\leadsto \frac{1 + \frac{\color{blue}{-0.5 + \frac{0.3333333333333333}{x}}}{x}}{n \cdot x} \]

19. Simplified 46.2%

    \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}} \]

20. Final simplification 46.2%

    \[\leadsto \frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x} + 1}{n \cdot x} \]
21. Add Preprocessing

Alternative 17: 40.4% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n}}{x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 n) x))

C

double code(double x, double n) {
	return (1.0 / n) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / n) / x
end function

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

function code(x, n)
	return Float64(Float64(1.0 / n) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / n) / x;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 51.1%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 55.5%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
4. Step-by-step derivation

   1. associate-/r* 55.8%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n}}{x}} \]

   2. mul-1-neg 55.8%

      \[\leadsto \frac{\frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n}}{x} \]

   3. log-rec 55.8%

      \[\leadsto \frac{\frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n}}{x} \]

   4. mul-1-neg 55.8%

      \[\leadsto \frac{\frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n}}{x} \]

   5. distribute-neg-frac 55.8%

      \[\leadsto \frac{\frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n}}{x} \]

   6. mul-1-neg 55.8%

      \[\leadsto \frac{\frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n}}{x} \]

   7. remove-double-neg 55.8%

      \[\leadsto \frac{\frac{e^{\frac{\color{blue}{\log x}}{n}}}{n}}{x} \]

   8. *-rgt-identity 55.8%

      \[\leadsto \frac{\frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n}}{x} \]

   9. associate-/l* 55.8%

      \[\leadsto \frac{\frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n}}{x} \]

   10. exp-to-pow 55.8%

       \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x} \]

5. Simplified 55.8%

   \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]

6. Taylor expanded in n around inf 39.4%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
7. Step-by-step derivation

   1. associate-/r* 39.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

8. Simplified 39.6%

   \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]
9. Add Preprocessing

Alternative 18: 40.0% 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 51.1%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 55.5%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
4. Step-by-step derivation

   1. mul-1-neg 55.5%

      \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

   2. log-rec 55.5%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 55.5%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   4. distribute-neg-frac 55.5%

      \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

   5. mul-1-neg 55.5%

      \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

   6. remove-double-neg 55.5%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

   7. *-commutative 55.5%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

5. Simplified 55.5%

   \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

6. Taylor expanded in n around inf 39.4%

   \[\leadsto \frac{\color{blue}{1}}{x \cdot n} \]

7. Final simplification 39.4%

   \[\leadsto \frac{1}{n \cdot x} \]
8. Add Preprocessing

Alternative 19: 4.5% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ x n))

C

double code(double x, double n) {
	return x / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = x / n
end function

Java

public static double code(double x, double n) {
	return x / n;
}

Python

def code(x, n):
	return x / n

Julia

function code(x, n)
	return Float64(x / n)
end

MATLAB

function tmp = code(x, n)
	tmp = x / n;
end

Wolfram

code[x_, n_] := N[(x / n), $MachinePrecision]

Derivation

1. Initial program 51.1%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 30.3%

   \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

4. Taylor expanded in x around inf 4.7%

   \[\leadsto \color{blue}{\frac{x}{n}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))