2nthrt (problem 3.4.6)

Percentage Accurate: 54.0% → 85.4%

Time: 42.5s

Alternatives: 21

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: 54.0% 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: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (exp (/ (log x) n)) (* n x))))
   (if (<= (/ 1.0 n) -5e-61)
     t_0
     (if (<= (/ 1.0 n) 4e-65)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 1e-10)
         t_0
         (log (exp (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = t_0;
	} else {
		tmp = log(exp((exp((log1p(x) / n)) - pow(x, (1.0 / n)))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.exp((Math.log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = t_0;
	} else {
		tmp = Math.log(Math.exp((Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n)))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.exp((math.log(x) / n)) / (n * x)
	tmp = 0
	if (1.0 / n) <= -5e-61:
		tmp = t_0
	elif (1.0 / n) <= 4e-65:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 1e-10:
		tmp = t_0
	else:
		tmp = math.log(math.exp((math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n)))))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log(x) / n)) / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-61)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-65)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = t_0;
	else
		tmp = log(exp(Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)))));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-61], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-65], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], t$95$0, N[Log[N[Exp[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-61 or 3.99999999999999969e-65 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 77.2%

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

   3. Taylor expanded in x around inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. log-rec 91.5%

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

      3. mul-1-neg 91.5%

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

      4. distribute-neg-frac 91.5%

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

      5. mul-1-neg 91.5%

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

      6. remove-double-neg 91.5%

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

      7. *-commutative 91.5%

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

   5. Simplified 91.5%

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

   if -4.9999999999999999e-61 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999969e-65

   1. Initial program 23.1%

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

   3. Taylor expanded in n around inf 83.6%

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

   5. Simplified 83.6%

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

      1. add-log-exp 83.6%

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

      2. associate-+r- 83.6%

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

      3. exp-diff 83.6%

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

      4. add-exp-log 67.0%

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

   7. Applied egg-rr 67.0%

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

      1. associate-*r/ 67.0%

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

      2. *-commutative 67.0%

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

      3. associate-/l* 67.0%

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

   9. Simplified 67.0%

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

   10. Taylor expanded in n around inf 83.9%

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

       1. +-commutative 83.9%

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

   12. Simplified 83.9%

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

       1. clear-num 83.9%

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

       2. log-div 84.0%

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

       3. metadata-eval 84.0%

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

   14. Applied egg-rr 84.0%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 84.0%

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

   16. Simplified 84.0%

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

   if 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 57.5%

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

      1. add-log-exp 57.6%

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

      2. pow-to-exp 57.6%

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

      3. un-div-inv 57.6%

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

      4. +-commutative 57.6%

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

      5. log1p-define 99.7%

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (exp (/ (log x) n)) (* n x))))
   (if (<= (/ 1.0 n) -5e-61)
     t_0
     (if (<= (/ 1.0 n) 4e-65)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 1e-10)
         t_0
         (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = t_0;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.exp((Math.log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = t_0;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.exp((math.log(x) / n)) / (n * x)
	tmp = 0
	if (1.0 / n) <= -5e-61:
		tmp = t_0
	elif (1.0 / n) <= 4e-65:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 1e-10:
		tmp = t_0
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log(x) / n)) / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-61)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-65)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = t_0;
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-61], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-65], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], t$95$0, 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 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-61 or 3.99999999999999969e-65 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 77.2%

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

   3. Taylor expanded in x around inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. log-rec 91.5%

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

      3. mul-1-neg 91.5%

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

      4. distribute-neg-frac 91.5%

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

      5. mul-1-neg 91.5%

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

      6. remove-double-neg 91.5%

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

      7. *-commutative 91.5%

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

   5. Simplified 91.5%

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

   if -4.9999999999999999e-61 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999969e-65

   1. Initial program 23.1%

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

   3. Taylor expanded in n around inf 83.6%

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

   5. Simplified 83.6%

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

      1. add-log-exp 83.6%

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

      2. associate-+r- 83.6%

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

      3. exp-diff 83.6%

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

      4. add-exp-log 67.0%

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

   7. Applied egg-rr 67.0%

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

      1. associate-*r/ 67.0%

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

      2. *-commutative 67.0%

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

      3. associate-/l* 67.0%

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

   9. Simplified 67.0%

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

   10. Taylor expanded in n around inf 83.9%

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

       1. +-commutative 83.9%

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

   12. Simplified 83.9%

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

       1. clear-num 83.9%

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

       2. log-div 84.0%

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

       3. metadata-eval 84.0%

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

   14. Applied egg-rr 84.0%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 84.0%

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

   16. Simplified 84.0%

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

   if 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 57.5%

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

   3. Taylor expanded in n around 0 57.5%

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

      1. log1p-define 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (exp (/ (log x) n)) (* n x))))
   (if (<= (/ 1.0 n) -5e-61)
     t_0
     (if (<= (/ 1.0 n) 4e-65)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 1e-10)
         t_0
         (-
          (+
           1.0
           (*
            x
            (+
             (/ 1.0 n)
             (* x (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))))
          (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - pow(x, (1.0 / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((log(x) / n)) / (n * x)
    if ((1.0d0 / n) <= (-5d-61)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 4d-65) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 1d-10) then
        tmp = t_0
    else
        tmp = (1.0d0 + (x * ((1.0d0 / n) + (x * ((0.5d0 * (1.0d0 / (n ** 2.0d0))) + (0.5d0 * ((-1.0d0) / n))))))) - (x ** (1.0d0 / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.exp((Math.log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.exp((math.log(x) / n)) / (n * x)
	tmp = 0
	if (1.0 / n) <= -5e-61:
		tmp = t_0
	elif (1.0 / n) <= 4e-65:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 1e-10:
		tmp = t_0
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log(x) / n)) / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-61)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-65)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))))) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = exp((log(x) / n)) / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-61)
		tmp = t_0;
	elseif ((1.0 / n) <= 4e-65)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 1e-10)
		tmp = t_0;
	else
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / (n ^ 2.0))) + (0.5 * (-1.0 / n))))))) - (x ^ (1.0 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-61], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-65], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], t$95$0, N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(x * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-61 or 3.99999999999999969e-65 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 77.2%

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

   3. Taylor expanded in x around inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. log-rec 91.5%

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

      3. mul-1-neg 91.5%

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

      4. distribute-neg-frac 91.5%

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

      5. mul-1-neg 91.5%

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

      6. remove-double-neg 91.5%

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

      7. *-commutative 91.5%

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

   5. Simplified 91.5%

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

   if -4.9999999999999999e-61 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999969e-65

   1. Initial program 23.1%

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

   3. Taylor expanded in n around inf 83.6%

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

   5. Simplified 83.6%

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

      1. add-log-exp 83.6%

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

      2. associate-+r- 83.6%

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

      3. exp-diff 83.6%

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

      4. add-exp-log 67.0%

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

   7. Applied egg-rr 67.0%

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

      1. associate-*r/ 67.0%

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

      2. *-commutative 67.0%

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

      3. associate-/l* 67.0%

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

   9. Simplified 67.0%

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

   10. Taylor expanded in n around inf 83.9%

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

       1. +-commutative 83.9%

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

   12. Simplified 83.9%

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

       1. clear-num 83.9%

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

       2. log-div 84.0%

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

       3. metadata-eval 84.0%

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

   14. Applied egg-rr 84.0%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 84.0%

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

   16. Simplified 84.0%

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

   if 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 57.5%

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

   3. Taylor expanded in x around 0 79.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (exp (/ (log x) n)) (* n x))))
   (if (<= (/ 1.0 n) -5e-61)
     t_0
     (if (<= (/ 1.0 n) 4e-65)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 1e-10)
         t_0
         (if (<= (/ 1.0 n) 5e+176)
           (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
           (+ (exp (/ (log1p x) n)) -1.0)))))))

C

double code(double x, double n) {
	double t_0 = exp((log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+176) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = exp((log1p(x) / n)) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.exp((Math.log(x) / n)) / (n * x);
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+176) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) + -1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.exp((math.log(x) / n)) / (n * x)
	tmp = 0
	if (1.0 / n) <= -5e-61:
		tmp = t_0
	elif (1.0 / n) <= 4e-65:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 1e-10:
		tmp = t_0
	elif (1.0 / n) <= 5e+176:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.exp((math.log1p(x) / n)) + -1.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(exp(Float64(log(x) / n)) / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-61)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-65)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e+176)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-61], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-65], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+176], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-61 or 3.99999999999999969e-65 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 77.2%

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

   3. Taylor expanded in x around inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. log-rec 91.5%

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

      3. mul-1-neg 91.5%

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

      4. distribute-neg-frac 91.5%

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

      5. mul-1-neg 91.5%

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

      6. remove-double-neg 91.5%

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

      7. *-commutative 91.5%

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

   5. Simplified 91.5%

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

   if -4.9999999999999999e-61 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999969e-65

   1. Initial program 23.1%

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

   3. Taylor expanded in n around inf 83.6%

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

   5. Simplified 83.6%

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

      1. add-log-exp 83.6%

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

      2. associate-+r- 83.6%

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

      3. exp-diff 83.6%

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

      4. add-exp-log 67.0%

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

   7. Applied egg-rr 67.0%

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

      1. associate-*r/ 67.0%

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

      2. *-commutative 67.0%

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

      3. associate-/l* 67.0%

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

   9. Simplified 67.0%

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

   10. Taylor expanded in n around inf 83.9%

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

       1. +-commutative 83.9%

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

   12. Simplified 83.9%

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

       1. clear-num 83.9%

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

       2. log-div 84.0%

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

       3. metadata-eval 84.0%

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

   14. Applied egg-rr 84.0%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 84.0%

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

   16. Simplified 84.0%

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

   if 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n) < 5e176

   1. Initial program 82.6%

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

   3. Taylor expanded in x around 0 82.7%

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

   if 5e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.1%

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

   3. Taylor expanded in n around 0 17.1%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in n around inf 84.7%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{1 + \frac{\log x}{n}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-61)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 4e-65)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 1e-10)
         (/ (+ 1.0 (/ (log x) n)) (* n x))
         (if (<= (/ 1.0 n) 5e+176)
           (- (+ 1.0 (/ x n)) t_0)
           (+ (exp (/ (log1p x) n)) -1.0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = (1.0 + (log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e+176) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = exp((log1p(x) / n)) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = (1.0 + (Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e+176) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) + -1.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-61:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 4e-65:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 1e-10:
		tmp = (1.0 + (math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 5e+176:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.exp((math.log1p(x) / n)) + -1.0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-61)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 4e-65)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = Float64(Float64(1.0 + Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e+176)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-61], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-65], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], N[(N[(1.0 + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+176], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-61

   1. Initial program 91.3%

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

   3. Taylor expanded in x around inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. log-rec 96.5%

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

      3. mul-1-neg 96.5%

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

      4. distribute-neg-frac 96.5%

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

      5. mul-1-neg 96.5%

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

      6. remove-double-neg 96.5%

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

      7. *-commutative 96.5%

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

   5. Simplified 96.5%

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

   6. Taylor expanded in x around 0 96.5%

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

      1. associate-/r* 96.4%

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

      2. *-rgt-identity 96.4%

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

      3. associate-*l/ 96.4%

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

      4. associate-*r/ 96.4%

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

      5. exp-to-pow 96.4%

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

   8. Simplified 96.4%

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

   if -4.9999999999999999e-61 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999969e-65

   1. Initial program 23.1%

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

   3. Taylor expanded in n around inf 83.6%

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

   5. Simplified 83.6%

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

      1. add-log-exp 83.6%

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

      2. associate-+r- 83.6%

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

      3. exp-diff 83.6%

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

      4. add-exp-log 67.0%

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

   7. Applied egg-rr 67.0%

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

      1. associate-*r/ 67.0%

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

      2. *-commutative 67.0%

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

      3. associate-/l* 67.0%

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

   9. Simplified 67.0%

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

   10. Taylor expanded in n around inf 83.9%

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

       1. +-commutative 83.9%

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

   12. Simplified 83.9%

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

       1. clear-num 83.9%

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

       2. log-div 84.0%

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

       3. metadata-eval 84.0%

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

   14. Applied egg-rr 84.0%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 84.0%

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

   16. Simplified 84.0%

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

   if 3.99999999999999969e-65 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 16.3%

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

   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. Taylor expanded in x around inf 69.9%

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

      1. mul-1-neg 69.9%

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

      2. log-rec 69.9%

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

      3. neg-mul-1 69.9%

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

      4. associate-*r/ 69.9%

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

      5. mul-1-neg 69.9%

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

      6. remove-double-neg 69.9%

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

      7. *-commutative 69.9%

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

   8. Simplified 69.9%

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

   if 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n) < 5e176

   1. Initial program 82.6%

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

   3. Taylor expanded in x around 0 82.7%

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

   if 5e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.1%

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

   3. Taylor expanded in n around 0 17.1%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in n around inf 84.7%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{1 + \frac{\log x}{n}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -400000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-90}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -2e+169)
     (/ (/ (/ n x) n) n)
     (if (<= (/ 1.0 n) -400000.0)
       t_0
       (if (<= (/ 1.0 n) 5e-90)
         (/ (- x (log x)) n)
         (if (<= (/ 1.0 n) 1e-10)
           (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)
           (if (<= (/ 1.0 n) 5e+176)
             t_0
             (/
              (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) x))
              x))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+169) {
		tmp = ((n / x) / n) / n;
	} else if ((1.0 / n) <= -400000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-90) {
		tmp = (x - log(x)) / n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if ((1.0 / n) <= 5e+176) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-2d+169)) then
        tmp = ((n / x) / n) / n
    else if ((1.0d0 / n) <= (-400000.0d0)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-90) then
        tmp = (x - log(x)) / n
    else if ((1.0d0 / n) <= 1d-10) then
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    else if ((1.0d0 / n) <= 5d+176) then
        tmp = t_0
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+169) {
		tmp = ((n / x) / n) / n;
	} else if ((1.0 / n) <= -400000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-90) {
		tmp = (x - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if ((1.0 / n) <= 5e+176) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e+169:
		tmp = ((n / x) / n) / n
	elif (1.0 / n) <= -400000.0:
		tmp = t_0
	elif (1.0 / n) <= 5e-90:
		tmp = (x - math.log(x)) / n
	elif (1.0 / n) <= 1e-10:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	elif (1.0 / n) <= 5e+176:
		tmp = t_0
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+169)
		tmp = Float64(Float64(Float64(n / x) / n) / n);
	elseif (Float64(1.0 / n) <= -400000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-90)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	elseif (Float64(1.0 / n) <= 5e+176)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -2e+169)
		tmp = ((n / x) / n) / n;
	elseif ((1.0 / n) <= -400000.0)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-90)
		tmp = (x - log(x)) / n;
	elseif ((1.0 / n) <= 1e-10)
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	elseif ((1.0 / n) <= 5e+176)
		tmp = t_0;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+169], N[(N[(N[(n / x), $MachinePrecision] / n), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -400000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-90], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+176], t$95$0, N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999987e169

   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 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in n around 0 97.4%

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

   7. Taylor expanded in x around inf 50.9%

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

      1. +-commutative 50.9%

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

      2. mul-1-neg 50.9%

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

      3. log-rec 50.9%

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

      4. remove-double-neg 50.9%

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

   9. Simplified 50.9%

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

   10. Taylor expanded in n around inf 67.3%

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

   if -1.99999999999999987e169 < (/.f64 #s(literal 1 binary64) n) < -4e5 or 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n) < 5e176

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0 68.4%

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

   if -4e5 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000019e-90

   1. Initial program 22.6%

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

   3. Taylor expanded in x around 0 13.4%

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

   4. Taylor expanded in n around inf 59.8%

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

   if 5.00000000000000019e-90 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 16.7%

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

   3. Taylor expanded in n around inf 51.7%

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

   5. Simplified 51.7%

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

      1. add-log-exp 51.7%

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

      2. associate-+r- 51.7%

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

      3. exp-diff 51.7%

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

      4. add-exp-log 39.7%

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

   7. Applied egg-rr 39.7%

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

      1. associate-*r/ 39.7%

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

      2. *-commutative 39.7%

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

      3. associate-/l* 39.7%

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

   9. Simplified 39.7%

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

   10. Taylor expanded in n around inf 51.9%

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

       1. +-commutative 51.9%

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

   12. Simplified 51.9%

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

   13. Taylor expanded in x around inf 62.5%

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

       1. associate--l+ 62.5%

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

       2. unpow2 62.5%

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

       3. associate-/r* 62.5%

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

       4. metadata-eval 62.5%

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

       5. associate-*r/ 62.5%

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

       6. associate-*r/ 62.5%

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

       7. metadata-eval 62.5%

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

       8. div-sub 62.5%

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

       9. sub-neg 62.5%

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

       10. metadata-eval 62.5%

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

       11. +-commutative 62.5%

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

       12. associate-*r/ 62.5%

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

       13. metadata-eval 62.5%

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

   15. Simplified 62.5%

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

   if 5e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.1%

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

   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. add-log-exp 0.1%

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

      2. associate-+r- 0.1%

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

      3. exp-diff 0.1%

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

      4. add-exp-log 0.1%

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

   7. Applied egg-rr 0.1%

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

      1. associate-*r/ 0.1%

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

      2. *-commutative 0.1%

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

      3. associate-/l* 0.1%

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

   9. Simplified 0.1%

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

   10. Taylor expanded in n around inf 7.1%

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

       1. +-commutative 7.1%

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

   12. Simplified 7.1%

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

   13. Taylor expanded in x around inf 6.1%

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

   15. Simplified 78.8%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -400000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-90}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{1 + \frac{\log x}{n}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-61)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 4e-65)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 1e-10)
         (/ (+ 1.0 (/ (log x) n)) (* n x))
         (if (<= (/ 1.0 n) 5e+176)
           (- (+ 1.0 (/ x n)) t_0)
           (/
            (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) x))
            x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = (1.0 + (log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e+176) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-61)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 4d-65) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 1d-10) then
        tmp = (1.0d0 + (log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 5d+176) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = (1.0 + (Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e+176) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-61:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 4e-65:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 1e-10:
		tmp = (1.0 + (math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 5e+176:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-61)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 4e-65)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = Float64(Float64(1.0 + Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e+176)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-61)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 4e-65)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 1e-10)
		tmp = (1.0 + (log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 5e+176)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-61], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-65], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], N[(N[(1.0 + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+176], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-61

   1. Initial program 91.3%

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

   3. Taylor expanded in x around inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. log-rec 96.5%

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

      3. mul-1-neg 96.5%

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

      4. distribute-neg-frac 96.5%

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

      5. mul-1-neg 96.5%

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

      6. remove-double-neg 96.5%

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

      7. *-commutative 96.5%

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

   5. Simplified 96.5%

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

   6. Taylor expanded in x around 0 96.5%

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

      1. associate-/r* 96.4%

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

      2. *-rgt-identity 96.4%

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

      3. associate-*l/ 96.4%

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

      4. associate-*r/ 96.4%

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

      5. exp-to-pow 96.4%

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

   8. Simplified 96.4%

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

   if -4.9999999999999999e-61 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999969e-65

   1. Initial program 23.1%

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

   3. Taylor expanded in n around inf 83.6%

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

   5. Simplified 83.6%

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

      1. add-log-exp 83.6%

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

      2. associate-+r- 83.6%

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

      3. exp-diff 83.6%

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

      4. add-exp-log 67.0%

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

   7. Applied egg-rr 67.0%

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

      1. associate-*r/ 67.0%

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

      2. *-commutative 67.0%

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

      3. associate-/l* 67.0%

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

   9. Simplified 67.0%

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

   10. Taylor expanded in n around inf 83.9%

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

       1. +-commutative 83.9%

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

   12. Simplified 83.9%

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

       1. clear-num 83.9%

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

       2. log-div 84.0%

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

       3. metadata-eval 84.0%

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

   14. Applied egg-rr 84.0%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 84.0%

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

   16. Simplified 84.0%

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

   if 3.99999999999999969e-65 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 16.3%

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

   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. Taylor expanded in x around inf 69.9%

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

      1. mul-1-neg 69.9%

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

      2. log-rec 69.9%

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

      3. neg-mul-1 69.9%

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

      4. associate-*r/ 69.9%

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

      5. mul-1-neg 69.9%

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

      6. remove-double-neg 69.9%

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

      7. *-commutative 69.9%

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

   8. Simplified 69.9%

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

   if 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n) < 5e176

   1. Initial program 82.6%

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

   3. Taylor expanded in x around 0 82.7%

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

   if 5e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.1%

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

   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. add-log-exp 0.1%

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

      2. associate-+r- 0.1%

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

      3. exp-diff 0.1%

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

      4. add-exp-log 0.1%

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

   7. Applied egg-rr 0.1%

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

      1. associate-*r/ 0.1%

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

      2. *-commutative 0.1%

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

      3. associate-/l* 0.1%

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

   9. Simplified 0.1%

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

   10. Taylor expanded in n around inf 7.1%

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

       1. +-commutative 7.1%

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

   12. Simplified 7.1%

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

   13. Taylor expanded in x around inf 6.1%

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

   15. Simplified 78.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{1 + \frac{\log x}{n}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{1 + \frac{\log x}{n}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-61)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 4e-65)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 1e-10)
         (/ (+ 1.0 (/ (log x) n)) (* n x))
         (if (<= (/ 1.0 n) 5e+176)
           (- 1.0 t_0)
           (/
            (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) x))
            x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = (1.0 + (log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e+176) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-61)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 4d-65) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 1d-10) then
        tmp = (1.0d0 + (log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 5d+176) then
        tmp = 1.0d0 - t_0
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = (1.0 + (Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e+176) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-61:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 4e-65:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 1e-10:
		tmp = (1.0 + (math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 5e+176:
		tmp = 1.0 - t_0
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-61)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 4e-65)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = Float64(Float64(1.0 + Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e+176)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-61)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 4e-65)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 1e-10)
		tmp = (1.0 + (log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 5e+176)
		tmp = 1.0 - t_0;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-61], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-65], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], N[(N[(1.0 + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+176], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-61

   1. Initial program 91.3%

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

   3. Taylor expanded in x around inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. log-rec 96.5%

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

      3. mul-1-neg 96.5%

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

      4. distribute-neg-frac 96.5%

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

      5. mul-1-neg 96.5%

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

      6. remove-double-neg 96.5%

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

      7. *-commutative 96.5%

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

   5. Simplified 96.5%

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

   6. Taylor expanded in x around 0 96.5%

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

      1. associate-/r* 96.4%

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

      2. *-rgt-identity 96.4%

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

      3. associate-*l/ 96.4%

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

      4. associate-*r/ 96.4%

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

      5. exp-to-pow 96.4%

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

   8. Simplified 96.4%

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

   if -4.9999999999999999e-61 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999969e-65

   1. Initial program 23.1%

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

   3. Taylor expanded in n around inf 83.6%

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

   5. Simplified 83.6%

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

      1. add-log-exp 83.6%

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

      2. associate-+r- 83.6%

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

      3. exp-diff 83.6%

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

      4. add-exp-log 67.0%

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

   7. Applied egg-rr 67.0%

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

      1. associate-*r/ 67.0%

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

      2. *-commutative 67.0%

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

      3. associate-/l* 67.0%

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

   9. Simplified 67.0%

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

   10. Taylor expanded in n around inf 83.9%

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

       1. +-commutative 83.9%

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

   12. Simplified 83.9%

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

       1. clear-num 83.9%

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

       2. log-div 84.0%

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

       3. metadata-eval 84.0%

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

   14. Applied egg-rr 84.0%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 84.0%

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

   16. Simplified 84.0%

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

   if 3.99999999999999969e-65 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 16.3%

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

   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. Taylor expanded in x around inf 69.9%

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

      1. mul-1-neg 69.9%

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

      2. log-rec 69.9%

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

      3. neg-mul-1 69.9%

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

      4. associate-*r/ 69.9%

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

      5. mul-1-neg 69.9%

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

      6. remove-double-neg 69.9%

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

      7. *-commutative 69.9%

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

   8. Simplified 69.9%

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

   if 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n) < 5e176

   1. Initial program 82.6%

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

   3. Taylor expanded in x around 0 82.6%

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

   if 5e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.1%

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

   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. add-log-exp 0.1%

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

      2. associate-+r- 0.1%

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

      3. exp-diff 0.1%

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

      4. add-exp-log 0.1%

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

   7. Applied egg-rr 0.1%

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

      1. associate-*r/ 0.1%

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

      2. *-commutative 0.1%

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

      3. associate-/l* 0.1%

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

   9. Simplified 0.1%

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

   10. Taylor expanded in n around inf 7.1%

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

       1. +-commutative 7.1%

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

   12. Simplified 7.1%

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

   13. Taylor expanded in x around inf 6.1%

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

   15. Simplified 78.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{1 + \frac{\log x}{n}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -5e-61)
     t_1
     (if (<= (/ 1.0 n) 4e-65)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 1e-10)
         t_1
         (if (<= (/ 1.0 n) 5e+176)
           (- 1.0 t_0)
           (/
            (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) x))
            x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+176) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / n) / x
    if ((1.0d0 / n) <= (-5d-61)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 4d-65) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 1d-10) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d+176) then
        tmp = 1.0d0 - t_0
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -5e-61) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e-65) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+176) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -5e-61:
		tmp = t_1
	elif (1.0 / n) <= 4e-65:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 1e-10:
		tmp = t_1
	elif (1.0 / n) <= 5e+176:
		tmp = 1.0 - t_0
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-61)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 4e-65)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+176)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / n) / x;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-61)
		tmp = t_1;
	elseif ((1.0 / n) <= 4e-65)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 1e-10)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e+176)
		tmp = 1.0 - t_0;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-61], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-65], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+176], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.9999999999999999e-61 or 3.99999999999999969e-65 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 77.2%

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

   3. Taylor expanded in x around inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. log-rec 91.5%

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

      3. mul-1-neg 91.5%

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

      4. distribute-neg-frac 91.5%

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

      5. mul-1-neg 91.5%

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

      6. remove-double-neg 91.5%

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

      7. *-commutative 91.5%

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

   5. Simplified 91.5%

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

   6. Taylor expanded in x around 0 91.5%

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

      1. associate-/r* 91.4%

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

      2. *-rgt-identity 91.4%

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

      3. associate-*l/ 91.4%

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

      4. associate-*r/ 91.4%

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

      5. exp-to-pow 91.4%

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

   8. Simplified 91.4%

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

   if -4.9999999999999999e-61 < (/.f64 #s(literal 1 binary64) n) < 3.99999999999999969e-65

   1. Initial program 23.1%

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

   3. Taylor expanded in n around inf 83.6%

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

   5. Simplified 83.6%

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

      1. add-log-exp 83.6%

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

      2. associate-+r- 83.6%

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

      3. exp-diff 83.6%

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

      4. add-exp-log 67.0%

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

   7. Applied egg-rr 67.0%

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

      1. associate-*r/ 67.0%

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

      2. *-commutative 67.0%

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

      3. associate-/l* 67.0%

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

   9. Simplified 67.0%

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

   10. Taylor expanded in n around inf 83.9%

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

       1. +-commutative 83.9%

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

   12. Simplified 83.9%

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

       1. clear-num 83.9%

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

       2. log-div 84.0%

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

       3. metadata-eval 84.0%

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

   14. Applied egg-rr 84.0%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 84.0%

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

   16. Simplified 84.0%

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

   if 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n) < 5e176

   1. Initial program 82.6%

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

   3. Taylor expanded in x around 0 82.6%

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

   if 5e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.1%

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

   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. add-log-exp 0.1%

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

      2. associate-+r- 0.1%

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

      3. exp-diff 0.1%

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

      4. add-exp-log 0.1%

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

   7. Applied egg-rr 0.1%

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

      1. associate-*r/ 0.1%

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

      2. *-commutative 0.1%

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

      3. associate-/l* 0.1%

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

   9. Simplified 0.1%

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

   10. Taylor expanded in n around inf 7.1%

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

       1. +-commutative 7.1%

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

   12. Simplified 7.1%

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

   13. Taylor expanded in x around inf 6.1%

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

   15. Simplified 78.8%

       \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -2e+169)
     (/ (/ (/ n x) n) n)
     (if (<= (/ 1.0 n) -5e+121)
       t_0
       (if (<= (/ 1.0 n) 1e-10)
         (/ (log (/ x (+ 1.0 x))) (- n))
         (if (<= (/ 1.0 n) 5e+176)
           t_0
           (/
            (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) x))
            x)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+169) {
		tmp = ((n / x) / n) / n;
	} else if ((1.0 / n) <= -5e+121) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 5e+176) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-2d+169)) then
        tmp = ((n / x) / n) / n
    else if ((1.0d0 / n) <= (-5d+121)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d-10) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 5d+176) then
        tmp = t_0
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+169) {
		tmp = ((n / x) / n) / n;
	} else if ((1.0 / n) <= -5e+121) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 5e+176) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e+169:
		tmp = ((n / x) / n) / n
	elif (1.0 / n) <= -5e+121:
		tmp = t_0
	elif (1.0 / n) <= 1e-10:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 5e+176:
		tmp = t_0
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+169)
		tmp = Float64(Float64(Float64(n / x) / n) / n);
	elseif (Float64(1.0 / n) <= -5e+121)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 5e+176)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -2e+169)
		tmp = ((n / x) / n) / n;
	elseif ((1.0 / n) <= -5e+121)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e-10)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 5e+176)
		tmp = t_0;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+169], N[(N[(N[(n / x), $MachinePrecision] / n), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+121], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+176], t$95$0, N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999987e169

   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 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in n around 0 97.4%

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

   7. Taylor expanded in x around inf 50.9%

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

      1. +-commutative 50.9%

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

      2. mul-1-neg 50.9%

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

      3. log-rec 50.9%

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

      4. remove-double-neg 50.9%

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

   9. Simplified 50.9%

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

   10. Taylor expanded in n around inf 67.3%

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

   if -1.99999999999999987e169 < (/.f64 #s(literal 1 binary64) n) < -5.00000000000000007e121 or 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n) < 5e176

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0 79.3%

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

   if -5.00000000000000007e121 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 32.4%

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

   3. Taylor expanded in n around inf 73.0%

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

   5. Simplified 72.9%

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

      1. add-log-exp 79.1%

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

      2. associate-+r- 79.1%

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

      3. exp-diff 79.1%

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

      4. add-exp-log 58.8%

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

   7. Applied egg-rr 58.8%

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

      1. associate-*r/ 58.8%

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

      2. *-commutative 58.8%

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

      3. associate-/l* 58.8%

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

   9. Simplified 58.8%

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

   10. Taylor expanded in n around inf 72.9%

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

       1. +-commutative 72.9%

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

   12. Simplified 72.9%

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

       1. clear-num 72.9%

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

       2. log-div 73.0%

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

       3. metadata-eval 73.0%

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

   14. Applied egg-rr 73.0%

       \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   15. Step-by-step derivation

       1. neg-sub0 73.0%

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

   16. Simplified 73.0%

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

   if 5e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.1%

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

   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. add-log-exp 0.1%

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

      2. associate-+r- 0.1%

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

      3. exp-diff 0.1%

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

      4. add-exp-log 0.1%

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

   7. Applied egg-rr 0.1%

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

      1. associate-*r/ 0.1%

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

      2. *-commutative 0.1%

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

      3. associate-/l* 0.1%

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

   9. Simplified 0.1%

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

   10. Taylor expanded in n around inf 7.1%

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

       1. +-commutative 7.1%

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

   12. Simplified 7.1%

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

   13. Taylor expanded in x around inf 6.1%

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

   15. Simplified 78.8%

       \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+121}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -2e+169)
     (/ (/ (/ n x) n) n)
     (if (<= (/ 1.0 n) -5e+121)
       t_0
       (if (<= (/ 1.0 n) 1e-10)
         (/ (log (/ (+ 1.0 x) x)) n)
         (if (<= (/ 1.0 n) 5e+176)
           t_0
           (/
            (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) x))
            x)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+169) {
		tmp = ((n / x) / n) / n;
	} else if ((1.0 / n) <= -5e+121) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+176) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-2d+169)) then
        tmp = ((n / x) / n) / n
    else if ((1.0d0 / n) <= (-5d+121)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d-10) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 5d+176) then
        tmp = t_0
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+169) {
		tmp = ((n / x) / n) / n;
	} else if ((1.0 / n) <= -5e+121) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+176) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e+169:
		tmp = ((n / x) / n) / n
	elif (1.0 / n) <= -5e+121:
		tmp = t_0
	elif (1.0 / n) <= 1e-10:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e+176:
		tmp = t_0
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+169)
		tmp = Float64(Float64(Float64(n / x) / n) / n);
	elseif (Float64(1.0 / n) <= -5e+121)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+176)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -2e+169)
		tmp = ((n / x) / n) / n;
	elseif ((1.0 / n) <= -5e+121)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e-10)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 5e+176)
		tmp = t_0;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+169], N[(N[(N[(n / x), $MachinePrecision] / n), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+121], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+176], t$95$0, N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999987e169

   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 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in n around 0 97.4%

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

   7. Taylor expanded in x around inf 50.9%

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

      1. +-commutative 50.9%

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

      2. mul-1-neg 50.9%

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

      3. log-rec 50.9%

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

      4. remove-double-neg 50.9%

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

   9. Simplified 50.9%

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

   10. Taylor expanded in n around inf 67.3%

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

   if -1.99999999999999987e169 < (/.f64 #s(literal 1 binary64) n) < -5.00000000000000007e121 or 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n) < 5e176

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0 79.3%

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

   if -5.00000000000000007e121 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 32.4%

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

   3. Taylor expanded in n around inf 73.0%

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

   5. Simplified 72.9%

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

      1. add-log-exp 79.1%

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

      2. associate-+r- 79.1%

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

      3. exp-diff 79.1%

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

      4. add-exp-log 58.8%

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

   7. Applied egg-rr 58.8%

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

      1. associate-*r/ 58.8%

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

      2. *-commutative 58.8%

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

      3. associate-/l* 58.8%

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

   9. Simplified 58.8%

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

   10. Taylor expanded in n around inf 72.9%

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

       1. +-commutative 72.9%

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

   12. Simplified 72.9%

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

   if 5e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.1%

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

   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. add-log-exp 0.1%

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

      2. associate-+r- 0.1%

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

      3. exp-diff 0.1%

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

      4. add-exp-log 0.1%

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

   7. Applied egg-rr 0.1%

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

      1. associate-*r/ 0.1%

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

      2. *-commutative 0.1%

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

      3. associate-/l* 0.1%

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

   9. Simplified 0.1%

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

   10. Taylor expanded in n around inf 7.1%

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

       1. +-commutative 7.1%

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

   12. Simplified 7.1%

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

   13. Taylor expanded in x around inf 6.1%

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

   15. Simplified 78.8%

       \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+121}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -2e+169)
     (/ (/ (/ n x) n) n)
     (if (<= (/ 1.0 n) -5e+121)
       t_0
       (if (<= (/ 1.0 n) 1e-10)
         (/ (log (+ 1.0 (/ 1.0 x))) n)
         (if (<= (/ 1.0 n) 5e+176)
           t_0
           (/
            (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) x))
            x)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+169) {
		tmp = ((n / x) / n) / n;
	} else if ((1.0 / n) <= -5e+121) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 5e+176) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-2d+169)) then
        tmp = ((n / x) / n) / n
    else if ((1.0d0 / n) <= (-5d+121)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d-10) then
        tmp = log((1.0d0 + (1.0d0 / x))) / n
    else if ((1.0d0 / n) <= 5d+176) then
        tmp = t_0
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+169) {
		tmp = ((n / x) / n) / n;
	} else if ((1.0 / n) <= -5e+121) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-10) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 5e+176) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e+169:
		tmp = ((n / x) / n) / n
	elif (1.0 / n) <= -5e+121:
		tmp = t_0
	elif (1.0 / n) <= 1e-10:
		tmp = math.log((1.0 + (1.0 / x))) / n
	elif (1.0 / n) <= 5e+176:
		tmp = t_0
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+169)
		tmp = Float64(Float64(Float64(n / x) / n) / n);
	elseif (Float64(1.0 / n) <= -5e+121)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	elseif (Float64(1.0 / n) <= 5e+176)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -2e+169)
		tmp = ((n / x) / n) / n;
	elseif ((1.0 / n) <= -5e+121)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e-10)
		tmp = log((1.0 + (1.0 / x))) / n;
	elseif ((1.0 / n) <= 5e+176)
		tmp = t_0;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+169], N[(N[(N[(n / x), $MachinePrecision] / n), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+121], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+176], t$95$0, N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999987e169

   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 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in n around 0 97.4%

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

   7. Taylor expanded in x around inf 50.9%

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

      1. +-commutative 50.9%

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

      2. mul-1-neg 50.9%

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

      3. log-rec 50.9%

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

      4. remove-double-neg 50.9%

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

   9. Simplified 50.9%

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

   10. Taylor expanded in n around inf 67.3%

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

   if -1.99999999999999987e169 < (/.f64 #s(literal 1 binary64) n) < -5.00000000000000007e121 or 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n) < 5e176

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0 79.3%

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

   if -5.00000000000000007e121 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 32.4%

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

   3. Taylor expanded in n around inf 73.0%

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

   5. Simplified 72.9%

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

      1. add-log-exp 79.1%

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

      2. associate-+r- 79.1%

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

      3. exp-diff 79.1%

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

      4. add-exp-log 58.8%

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

   7. Applied egg-rr 58.8%

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

      1. associate-*r/ 58.8%

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

      2. *-commutative 58.8%

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

      3. associate-/l* 58.8%

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

   9. Simplified 58.8%

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

   10. Taylor expanded in n around inf 72.9%

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

       1. +-commutative 72.9%

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

   12. Simplified 72.9%

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

   13. Taylor expanded in x around inf 72.9%

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

   if 5e176 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 17.1%

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

   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. add-log-exp 0.1%

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

      2. associate-+r- 0.1%

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

      3. exp-diff 0.1%

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

      4. add-exp-log 0.1%

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

   7. Applied egg-rr 0.1%

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

      1. associate-*r/ 0.1%

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

      2. *-commutative 0.1%

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

      3. associate-/l* 0.1%

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

   9. Simplified 0.1%

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

   10. Taylor expanded in n around inf 7.1%

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

       1. +-commutative 7.1%

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

   12. Simplified 7.1%

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

   13. Taylor expanded in x around inf 6.1%

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

   15. Simplified 78.8%

       \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+121}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+176}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-141}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.95e-141)
   (/ (log x) (- n))
   (if (<= x 4.8e-116)
     (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)
     (if (<= x 1.0) (/ (- x (log x)) n) (/ (/ (/ n x) n) n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.95e-141) {
		tmp = log(x) / -n;
	} else if (x <= 4.8e-116) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((n / x) / n) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.95d-141) then
        tmp = log(x) / -n
    else if (x <= 4.8d-116) then
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    else if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((n / x) / n) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.95e-141) {
		tmp = Math.log(x) / -n;
	} else if (x <= 4.8e-116) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((n / x) / n) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.95e-141:
		tmp = math.log(x) / -n
	elif x <= 4.8e-116:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	elif x <= 1.0:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((n / x) / n) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.95e-141)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 4.8e-116)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	elseif (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(n / x) / n) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.95e-141)
		tmp = log(x) / -n;
	elseif (x <= 4.8e-116)
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	elseif (x <= 1.0)
		tmp = (x - log(x)) / n;
	else
		tmp = ((n / x) / n) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.95e-141], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 4.8e-116], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(n / x), $MachinePrecision] / n), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.9499999999999999e-141

   1. Initial program 48.2%

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

   3. Taylor expanded in x around 0 48.2%

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

   4. Taylor expanded in n around inf 51.8%

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

      1. mul-1-neg 51.8%

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

   6. Simplified 51.8%

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

   if 1.9499999999999999e-141 < x < 4.79999999999999986e-116

   1. Initial program 70.2%

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

   3. Taylor expanded in n around inf 26.3%

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

   5. Simplified 26.3%

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

      1. add-log-exp 61.6%

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

      2. associate-+r- 61.6%

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

      3. exp-diff 61.6%

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

      4. add-exp-log 61.6%

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

   7. Applied egg-rr 61.6%

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

      1. associate-*r/ 61.6%

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

      2. *-commutative 61.6%

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

      3. associate-/l* 61.6%

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

   9. Simplified 61.6%

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

   10. Taylor expanded in n around inf 13.1%

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

       1. +-commutative 13.1%

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

   12. Simplified 13.1%

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

   13. Taylor expanded in x around inf 77.6%

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

       1. associate--l+ 77.6%

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

       2. unpow2 77.6%

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

       3. associate-/r* 77.6%

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

       4. metadata-eval 77.6%

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

       5. associate-*r/ 77.6%

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

       6. associate-*r/ 77.6%

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

       7. metadata-eval 77.6%

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

       8. div-sub 77.6%

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

       9. sub-neg 77.6%

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

       10. metadata-eval 77.6%

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

       11. +-commutative 77.6%

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

       12. associate-*r/ 77.6%

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

       13. metadata-eval 77.6%

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

   15. Simplified 77.6%

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

   if 4.79999999999999986e-116 < x < 1

   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 x around 0 32.0%

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

   4. Taylor expanded in n around inf 52.7%

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

   if 1 < x

   1. Initial program 61.8%

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

   3. Taylor expanded in n around inf 63.1%

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

   5. Simplified 63.0%

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

   6. Taylor expanded in n around 0 63.1%

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

   7. Taylor expanded in x around inf 60.7%

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

      1. +-commutative 60.7%

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

      2. mul-1-neg 60.7%

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

      3. log-rec 60.7%

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

      4. remove-double-neg 60.7%

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

   9. Simplified 60.7%

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

   10. Taylor expanded in n around inf 71.8%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-141}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 2.45 \cdot 10^{-141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 2.45e-141)
     t_0
     (if (<= x 4.4e-116)
       (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)
       (if (<= x 0.55) t_0 (/ (/ (/ n x) n) n))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 2.45e-141) {
		tmp = t_0;
	} else if (x <= 4.4e-116) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if (x <= 0.55) {
		tmp = t_0;
	} else {
		tmp = ((n / x) / n) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 2.45d-141) then
        tmp = t_0
    else if (x <= 4.4d-116) then
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    else if (x <= 0.55d0) then
        tmp = t_0
    else
        tmp = ((n / x) / n) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 2.45e-141) {
		tmp = t_0;
	} else if (x <= 4.4e-116) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if (x <= 0.55) {
		tmp = t_0;
	} else {
		tmp = ((n / x) / n) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 2.45e-141:
		tmp = t_0
	elif x <= 4.4e-116:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	elif x <= 0.55:
		tmp = t_0
	else:
		tmp = ((n / x) / n) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 2.45e-141)
		tmp = t_0;
	elseif (x <= 4.4e-116)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	elseif (x <= 0.55)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(n / x) / n) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 2.45e-141)
		tmp = t_0;
	elseif (x <= 4.4e-116)
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	elseif (x <= 0.55)
		tmp = t_0;
	else
		tmp = ((n / x) / n) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 2.45e-141], t$95$0, If[LessEqual[x, 4.4e-116], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 0.55], t$95$0, N[(N[(N[(n / x), $MachinePrecision] / n), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.45000000000000003e-141 or 4.4000000000000002e-116 < x < 0.55000000000000004

   1. Initial program 41.8%

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

   3. Taylor expanded in x around 0 41.8%

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

   4. Taylor expanded in n around inf 51.2%

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

      1. mul-1-neg 51.2%

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

   6. Simplified 51.2%

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

   if 2.45000000000000003e-141 < x < 4.4000000000000002e-116

   1. Initial program 70.2%

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

   3. Taylor expanded in n around inf 26.3%

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

   5. Simplified 26.3%

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

      1. add-log-exp 61.6%

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

      2. associate-+r- 61.6%

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

      3. exp-diff 61.6%

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

      4. add-exp-log 61.6%

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

   7. Applied egg-rr 61.6%

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

      1. associate-*r/ 61.6%

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

      2. *-commutative 61.6%

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

      3. associate-/l* 61.6%

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

   9. Simplified 61.6%

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

   10. Taylor expanded in n around inf 13.1%

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

       1. +-commutative 13.1%

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

   12. Simplified 13.1%

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

   13. Taylor expanded in x around inf 77.6%

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

       1. associate--l+ 77.6%

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

       2. unpow2 77.6%

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

       3. associate-/r* 77.6%

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

       4. metadata-eval 77.6%

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

       5. associate-*r/ 77.6%

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

       6. associate-*r/ 77.6%

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

       7. metadata-eval 77.6%

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

       8. div-sub 77.6%

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

       9. sub-neg 77.6%

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

       10. metadata-eval 77.6%

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

       11. +-commutative 77.6%

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

       12. associate-*r/ 77.6%

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

       13. metadata-eval 77.6%

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

   15. Simplified 77.6%

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

   if 0.55000000000000004 < x

   1. Initial program 61.8%

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

   3. Taylor expanded in n around inf 63.1%

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

   5. Simplified 63.0%

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

   6. Taylor expanded in n around 0 63.1%

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

   7. Taylor expanded in x around inf 60.7%

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

      1. +-commutative 60.7%

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

      2. mul-1-neg 60.7%

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

      3. log-rec 60.7%

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

      4. remove-double-neg 60.7%

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

   9. Simplified 60.7%

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

   10. Taylor expanded in n around inf 71.8%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-141}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -20000000000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+191)
   (/ (/ (/ n x) n) n)
   (if (<= (/ 1.0 n) -20000000000000.0)
     0.0
     (/ (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* n x)) (/ -0.5 n)) x)) x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+191) {
		tmp = ((n / x) / n) / n;
	} else if ((1.0 / n) <= -20000000000000.0) {
		tmp = 0.0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d+191)) then
        tmp = ((n / x) / n) / n
    else if ((1.0d0 / n) <= (-20000000000000.0d0)) then
        tmp = 0.0d0
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+191) {
		tmp = ((n / x) / n) / n;
	} else if ((1.0 / n) <= -20000000000000.0) {
		tmp = 0.0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+191:
		tmp = ((n / x) / n) / n
	elif (1.0 / n) <= -20000000000000.0:
		tmp = 0.0
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+191)
		tmp = Float64(Float64(Float64(n / x) / n) / n);
	elseif (Float64(1.0 / n) <= -20000000000000.0)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e+191)
		tmp = ((n / x) / n) / n;
	elseif ((1.0 / n) <= -20000000000000.0)
		tmp = 0.0;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+191], N[(N[(N[(n / x), $MachinePrecision] / n), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000000000000.0], 0.0, N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000007e191

   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 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in n around 0 95.9%

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

   7. Taylor expanded in x around inf 59.1%

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

      1. +-commutative 59.1%

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

      2. mul-1-neg 59.1%

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

      3. log-rec 59.1%

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

      4. remove-double-neg 59.1%

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

   9. Simplified 59.1%

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

   10. Taylor expanded in n around inf 76.4%

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

   if -1.00000000000000007e191 < (/.f64 #s(literal 1 binary64) n) < -2e13

   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 0 48.5%

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

   4. Taylor expanded in n around inf 53.9%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 53.9%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 53.9%

      \[\leadsto \color{blue}{0} \]

   if -2e13 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.9%

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

   3. Taylor expanded in n around inf 55.8%

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

   5. Simplified 55.8%

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

      1. add-log-exp 57.3%

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

      2. associate-+r- 57.3%

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

      3. exp-diff 57.3%

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

      4. add-exp-log 46.4%

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

   7. Applied egg-rr 46.4%

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

      1. associate-*r/ 46.4%

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

      2. *-commutative 46.4%

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

      3. associate-/l* 46.4%

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

   9. Simplified 46.4%

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

   10. Taylor expanded in n around inf 57.1%

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

       1. +-commutative 57.1%

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

   12. Simplified 57.1%

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

   13. Taylor expanded in x around inf 36.3%

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

   15. Simplified 43.7%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -20000000000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.3% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -20000000000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+191)
   (/ (/ (/ n x) n) n)
   (if (<= (/ 1.0 n) -20000000000000.0)
     0.0
     (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+191) {
		tmp = ((n / x) / n) / n;
	} else if ((1.0 / n) <= -20000000000000.0) {
		tmp = 0.0;
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d+191)) then
        tmp = ((n / x) / n) / n
    else if ((1.0d0 / n) <= (-20000000000000.0d0)) then
        tmp = 0.0d0
    else
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+191) {
		tmp = ((n / x) / n) / n;
	} else if ((1.0 / n) <= -20000000000000.0) {
		tmp = 0.0;
	} else {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+191:
		tmp = ((n / x) / n) / n
	elif (1.0 / n) <= -20000000000000.0:
		tmp = 0.0
	else:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+191)
		tmp = Float64(Float64(Float64(n / x) / n) / n);
	elseif (Float64(1.0 / n) <= -20000000000000.0)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e+191)
		tmp = ((n / x) / n) / n;
	elseif ((1.0 / n) <= -20000000000000.0)
		tmp = 0.0;
	else
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+191], N[(N[(N[(n / x), $MachinePrecision] / n), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000000000000.0], 0.0, N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000007e191

   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 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in n around 0 95.9%

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

   7. Taylor expanded in x around inf 59.1%

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

      1. +-commutative 59.1%

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

      2. mul-1-neg 59.1%

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

      3. log-rec 59.1%

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

      4. remove-double-neg 59.1%

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

   9. Simplified 59.1%

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

   10. Taylor expanded in n around inf 76.4%

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

   if -1.00000000000000007e191 < (/.f64 #s(literal 1 binary64) n) < -2e13

   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 0 48.5%

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

   4. Taylor expanded in n around inf 53.9%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 53.9%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 53.9%

      \[\leadsto \color{blue}{0} \]

   if -2e13 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.9%

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

   3. Taylor expanded in n around inf 55.8%

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

   5. Simplified 55.8%

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

      1. add-log-exp 57.3%

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

      2. associate-+r- 57.3%

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

      3. exp-diff 57.3%

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

      4. add-exp-log 46.4%

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

   7. Applied egg-rr 46.4%

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

      1. associate-*r/ 46.4%

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

      2. *-commutative 46.4%

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

      3. associate-/l* 46.4%

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

   9. Simplified 46.4%

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

   10. Taylor expanded in n around inf 57.1%

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

       1. +-commutative 57.1%

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

   12. Simplified 57.1%

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

   13. Taylor expanded in x around inf 43.6%

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

       1. associate--l+ 43.6%

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

       2. unpow2 43.6%

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

       3. associate-/r* 43.6%

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

       4. metadata-eval 43.6%

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

       5. associate-*r/ 43.6%

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

       6. associate-*r/ 43.6%

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

       7. metadata-eval 43.6%

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

       8. div-sub 43.6%

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

       9. sub-neg 43.6%

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

       10. metadata-eval 43.6%

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

       11. +-commutative 43.6%

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

       12. associate-*r/ 43.6%

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

       13. metadata-eval 43.6%

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

   15. Simplified 43.6%

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

Alternative 17: 49.6% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{\frac{n}{x}}{n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -20000000000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+191)
   (/ (/ (/ n x) n) n)
   (if (<= (/ 1.0 n) -20000000000000.0) 0.0 (/ (/ 1.0 x) n))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+191) {
		tmp = ((n / x) / n) / n;
	} else if ((1.0 / n) <= -20000000000000.0) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d+191)) then
        tmp = ((n / x) / n) / n
    else if ((1.0d0 / n) <= (-20000000000000.0d0)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+191) {
		tmp = ((n / x) / n) / n;
	} else if ((1.0 / n) <= -20000000000000.0) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+191:
		tmp = ((n / x) / n) / n
	elif (1.0 / n) <= -20000000000000.0:
		tmp = 0.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+191)
		tmp = Float64(Float64(Float64(n / x) / n) / n);
	elseif (Float64(1.0 / n) <= -20000000000000.0)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e+191)
		tmp = ((n / x) / n) / n;
	elseif ((1.0 / n) <= -20000000000000.0)
		tmp = 0.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+191], N[(N[(N[(n / x), $MachinePrecision] / n), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000000000000.0], 0.0, N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.00000000000000007e191

   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 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in n around 0 95.9%

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

   7. Taylor expanded in x around inf 59.1%

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

      1. +-commutative 59.1%

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

      2. mul-1-neg 59.1%

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

      3. log-rec 59.1%

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

      4. remove-double-neg 59.1%

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

   9. Simplified 59.1%

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

   10. Taylor expanded in n around inf 76.4%

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

   if -1.00000000000000007e191 < (/.f64 #s(literal 1 binary64) n) < -2e13

   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 0 48.5%

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

   4. Taylor expanded in n around inf 53.9%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 53.9%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 53.9%

      \[\leadsto \color{blue}{0} \]

   if -2e13 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.9%

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

   3. Taylor expanded in n around inf 55.8%

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

   5. Simplified 55.8%

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

      1. add-log-exp 57.3%

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

      2. associate-+r- 57.3%

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

      3. exp-diff 57.3%

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

      4. add-exp-log 46.4%

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

   7. Applied egg-rr 46.4%

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

      1. associate-*r/ 46.4%

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

      2. *-commutative 46.4%

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

      3. associate-/l* 46.4%

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

   9. Simplified 46.4%

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

   10. Taylor expanded in n around inf 57.1%

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

       1. +-commutative 57.1%

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

   12. Simplified 57.1%

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

   13. Taylor expanded in x around inf 40.5%

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

Alternative 18: 46.6% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-8} \lor \neg \left(n \leq -2.45 \cdot 10^{-232}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -1.4e-8) (not (<= n -2.45e-232))) (/ (/ 1.0 x) n) 0.0))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -1.4e-8) || !(n <= -2.45e-232)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.4d-8)) .or. (.not. (n <= (-2.45d-232)))) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -1.4e-8) || !(n <= -2.45e-232)) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -1.4e-8) or not (n <= -2.45e-232):
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -1.4e-8) || !(n <= -2.45e-232))
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -1.4e-8) || ~((n <= -2.45e-232)))
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -1.4e-8], N[Not[LessEqual[n, -2.45e-232]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.4e-8 or -2.4500000000000002e-232 < n

   1. Initial program 37.0%

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

   3. Taylor expanded in n around inf 59.1%

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

   5. Simplified 59.1%

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

      1. add-log-exp 60.5%

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

      2. associate-+r- 60.5%

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

      3. exp-diff 60.5%

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

      4. add-exp-log 48.4%

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

   7. Applied egg-rr 48.4%

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

      1. associate-*r/ 48.4%

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

      2. *-commutative 48.4%

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

      3. associate-/l* 48.4%

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

   9. Simplified 48.4%

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

   10. Taylor expanded in n around inf 55.7%

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

       1. +-commutative 55.7%

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

   12. Simplified 55.7%

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

   13. Taylor expanded in x around inf 42.2%

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

   if -1.4e-8 < n < -2.4500000000000002e-232

   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 0 46.4%

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

   4. Taylor expanded in n around inf 56.1%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 56.1%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 56.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-8} \lor \neg \left(n \leq -2.45 \cdot 10^{-232}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.1% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-8} \lor \neg \left(n \leq -6.8 \cdot 10^{-233}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -1.4e-8) (not (<= n -6.8e-233))) (/ 1.0 (* n x)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -1.4e-8) || !(n <= -6.8e-233)) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.4d-8)) .or. (.not. (n <= (-6.8d-233)))) then
        tmp = 1.0d0 / (n * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -1.4e-8) || !(n <= -6.8e-233)) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -1.4e-8) or not (n <= -6.8e-233):
		tmp = 1.0 / (n * x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -1.4e-8) || !(n <= -6.8e-233))
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -1.4e-8) || ~((n <= -6.8e-233)))
		tmp = 1.0 / (n * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -1.4e-8], N[Not[LessEqual[n, -6.8e-233]], $MachinePrecision]], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.4e-8 or -6.8000000000000004e-233 < n

   1. Initial program 37.0%

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

   3. Taylor expanded in x around inf 40.1%

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

      1. mul-1-neg 40.1%

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

      2. log-rec 40.1%

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

      3. mul-1-neg 40.1%

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

      4. distribute-neg-frac 40.1%

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

      5. mul-1-neg 40.1%

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

      6. remove-double-neg 40.1%

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

      7. *-commutative 40.1%

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

   5. Simplified 40.1%

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

   6. Taylor expanded in n around inf 42.1%

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

   if -1.4e-8 < n < -6.8000000000000004e-233

   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 0 46.4%

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

   4. Taylor expanded in n around inf 56.1%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 56.1%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 56.1%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-8} \lor \neg \left(n \leq -6.8 \cdot 10^{-233}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 46.3% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{-232}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -1.4e-8) (/ 1.0 (* n x)) (if (<= n -1.3e-232) 0.0 (/ (/ 1.0 n) x))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -1.4e-8) {
		tmp = 1.0 / (n * x);
	} else if (n <= -1.3e-232) {
		tmp = 0.0;
	} else {
		tmp = (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) :: tmp
    if (n <= (-1.4d-8)) then
        tmp = 1.0d0 / (n * x)
    else if (n <= (-1.3d-232)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -1.4e-8) {
		tmp = 1.0 / (n * x);
	} else if (n <= -1.3e-232) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -1.4e-8:
		tmp = 1.0 / (n * x)
	elif n <= -1.3e-232:
		tmp = 0.0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -1.4e-8)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (n <= -1.3e-232)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -1.4e-8)
		tmp = 1.0 / (n * x);
	elseif (n <= -1.3e-232)
		tmp = 0.0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -1.4e-8], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, -1.3e-232], 0.0, N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.4e-8

   1. Initial program 22.8%

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

   3. Taylor expanded in x around inf 48.5%

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

      1. mul-1-neg 48.5%

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

      2. log-rec 48.5%

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

      3. mul-1-neg 48.5%

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

      4. distribute-neg-frac 48.5%

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

      5. mul-1-neg 48.5%

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

      6. remove-double-neg 48.5%

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

      7. *-commutative 48.5%

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

   5. Simplified 48.5%

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

   6. Taylor expanded in n around inf 43.9%

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

   if -1.4e-8 < n < -1.29999999999999998e-232

   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 0 46.4%

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

   4. Taylor expanded in n around inf 56.1%

      \[\leadsto 1 - \color{blue}{1} \]
   5. Step-by-step derivation

      1. metadata-eval 56.1%

         \[\leadsto \color{blue}{0} \]

   6. Applied egg-rr 56.1%

      \[\leadsto \color{blue}{0} \]

   if -1.29999999999999998e-232 < n

   1. Initial program 44.3%

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

   3. Taylor expanded in x around inf 35.8%

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

      1. mul-1-neg 35.8%

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

      2. log-rec 35.8%

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

      3. mul-1-neg 35.8%

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

      4. distribute-neg-frac 35.8%

         \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

      5. mul-1-neg 35.8%

         \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

      6. remove-double-neg 35.8%

         \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

      7. *-commutative 35.8%

         \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   5. Simplified 35.8%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

   6. Taylor expanded in n around inf 41.2%

      \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
   7. Step-by-step derivation

      1. associate-/r* 41.2%

         \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

   8. Simplified 41.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 31.4% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 50.8%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 37.4%

   \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

4. Taylor expanded in n around inf 25.4%

   \[\leadsto 1 - \color{blue}{1} \]
5. Step-by-step derivation

   1. metadata-eval 25.4%

      \[\leadsto \color{blue}{0} \]

6. Applied egg-rr 25.4%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024121 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))