2nthrt (problem 3.4.6)

Percentage Accurate: 53.3% → 85.4%

Time: 23.7s

Alternatives: 18

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.3% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-102)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5000.0)
       (-
        (fma 0.5 (/ (pow (log1p x) 2.0) (* n n)) (/ (log1p x) n))
        (fma 0.5 (/ (pow (log x) 2.0) (* n n)) (/ (log x) n)))
       (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-102) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5000.0) {
		tmp = fma(0.5, (pow(log1p(x), 2.0) / (n * n)), (log1p(x) / n)) - fma(0.5, (pow(log(x), 2.0) / (n * n)), (log(x) / n));
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-102)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5000.0)
		tmp = Float64(fma(0.5, Float64((log1p(x) ^ 2.0) / Float64(n * n)), Float64(log1p(x) / n)) - fma(0.5, Float64((log(x) ^ 2.0) / Float64(n * n)), Float64(log(x) / n)));
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-102], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5000.0], N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -1.99999999999999987e-102

   1. Initial program 76.9%

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

   2. Taylor expanded in x around inf 91.1%

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

      1. mul-1-neg 91.1%

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

      2. log-rec 91.1%

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

      3. distribute-frac-neg 91.1%

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

      4. remove-double-neg 91.1%

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

      5. *-commutative 91.1%

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

   4. Simplified 91.1%

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

   5. Taylor expanded in x around 0 91.1%

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

      1. *-rgt-identity 91.1%

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

      2. associate-*r/ 91.1%

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

      3. exp-to-pow 91.1%

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

      4. *-commutative 91.1%

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

   7. Simplified 91.1%

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

   if -1.99999999999999987e-102 < (/.f64 1 n) < 5e3

   1. Initial program 32.7%

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

   2. Taylor expanded in n around inf 88.1%

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

      1. fma-def 88.1%

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

      2. log1p-def 88.1%

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

      3. unpow2 88.1%

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

      4. log1p-def 88.1%

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

      5. fma-def 88.1%

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

      6. unpow2 88.1%

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

   4. Simplified 88.1%

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

   if 5e3 < (/.f64 1 n)

   1. Initial program 62.8%

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

   2. Taylor expanded in n around 0 62.8%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right)}{n}\right) - \mathsf{fma}\left(0.5, \frac{{\log x}^{2}}{n \cdot n}, \frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+218}:\\ \;\;\;\;\left|1 - t_0\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-102)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-21)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1e+218) (fabs (- 1.0 t_0)) (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-102) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1e+218) {
		tmp = fabs((1.0 - t_0));
	} else {
		tmp = 1.0 / (n * x);
	}
	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) <= -2e-102) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1e+218) {
		tmp = Math.abs((1.0 - t_0));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-102:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-21:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1e+218:
		tmp = math.fabs((1.0 - t_0))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-102)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e+218)
		tmp = abs(Float64(1.0 - t_0));
	else
		tmp = Float64(1.0 / Float64(n * x));
	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], -2e-102], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+218], N[Abs[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1.99999999999999987e-102

   1. Initial program 76.9%

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

   2. Taylor expanded in x around inf 91.1%

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

      1. mul-1-neg 91.1%

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

      2. log-rec 91.1%

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

      3. distribute-frac-neg 91.1%

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

      4. remove-double-neg 91.1%

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

      5. *-commutative 91.1%

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

   4. Simplified 91.1%

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

   5. Taylor expanded in x around 0 91.1%

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

      1. *-rgt-identity 91.1%

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

      2. associate-*r/ 91.1%

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

      3. exp-to-pow 91.1%

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

      4. *-commutative 91.1%

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

   7. Simplified 91.1%

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

   if -1.99999999999999987e-102 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 32.7%

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

   2. Taylor expanded in n around inf 88.9%

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

      1. +-rgt-identity 88.9%

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

      2. +-rgt-identity 88.9%

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

      3. log1p-def 88.9%

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

   4. Simplified 88.9%

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

   if 4.99999999999999973e-21 < (/.f64 1 n) < 1.00000000000000008e218

   1. Initial program 79.5%

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

   2. Taylor expanded in x around 0 76.1%

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

      1. add-sqr-sqrt 76.1%

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

      2. sqrt-unprod 79.6%

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

      3. pow2 79.6%

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

   4. Applied egg-rr 79.6%

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

      1. unpow2 79.6%

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

      2. rem-sqrt-square 79.6%

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

   6. Simplified 79.6%

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

   if 1.00000000000000008e218 < (/.f64 1 n)

   1. Initial program 19.3%

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

   2. Taylor expanded in n around inf 15.8%

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

      1. +-rgt-identity 15.8%

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

      2. +-rgt-identity 15.8%

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

      3. log1p-def 15.8%

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

   4. Simplified 15.8%

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

   5. Taylor expanded in x around inf 83.9%

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

      1. *-commutative 83.9%

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

   7. Simplified 83.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+218}:\\ \;\;\;\;\left|1 - {x}^{\left(\frac{1}{n}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 3: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+218}:\\ \;\;\;\;\left|1 - t_0\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-102)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-21)
       (- (/ (log1p x) n) (/ (log x) n))
       (if (<= (/ 1.0 n) 1e+218) (fabs (- 1.0 t_0)) (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-102) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (log1p(x) / n) - (log(x) / n);
	} else if ((1.0 / n) <= 1e+218) {
		tmp = fabs((1.0 - t_0));
	} else {
		tmp = 1.0 / (n * x);
	}
	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) <= -2e-102) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (Math.log1p(x) / n) - (Math.log(x) / n);
	} else if ((1.0 / n) <= 1e+218) {
		tmp = Math.abs((1.0 - t_0));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-102:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-21:
		tmp = (math.log1p(x) / n) - (math.log(x) / n)
	elif (1.0 / n) <= 1e+218:
		tmp = math.fabs((1.0 - t_0))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-102)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(log1p(x) / n) - Float64(log(x) / n));
	elseif (Float64(1.0 / n) <= 1e+218)
		tmp = abs(Float64(1.0 - t_0));
	else
		tmp = Float64(1.0 / Float64(n * x));
	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], -2e-102], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+218], N[Abs[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1.99999999999999987e-102

   1. Initial program 76.9%

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

   2. Taylor expanded in x around inf 91.1%

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

      1. mul-1-neg 91.1%

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

      2. log-rec 91.1%

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

      3. distribute-frac-neg 91.1%

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

      4. remove-double-neg 91.1%

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

      5. *-commutative 91.1%

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

   4. Simplified 91.1%

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

   5. Taylor expanded in x around 0 91.1%

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

      1. *-rgt-identity 91.1%

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

      2. associate-*r/ 91.1%

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

      3. exp-to-pow 91.1%

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

      4. *-commutative 91.1%

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

   7. Simplified 91.1%

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

   if -1.99999999999999987e-102 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 32.7%

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

   2. Taylor expanded in n around inf 88.9%

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

      1. +-rgt-identity 88.9%

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

      2. +-rgt-identity 88.9%

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

      3. log1p-def 88.9%

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

   4. Simplified 88.9%

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

      1. div-sub 88.9%

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

   6. Applied egg-rr 88.9%

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

   if 4.99999999999999973e-21 < (/.f64 1 n) < 1.00000000000000008e218

   1. Initial program 79.5%

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

   2. Taylor expanded in x around 0 76.1%

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

      1. add-sqr-sqrt 76.1%

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

      2. sqrt-unprod 79.6%

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

      3. pow2 79.6%

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

   4. Applied egg-rr 79.6%

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

      1. unpow2 79.6%

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

      2. rem-sqrt-square 79.6%

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

   6. Simplified 79.6%

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

   if 1.00000000000000008e218 < (/.f64 1 n)

   1. Initial program 19.3%

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

   2. Taylor expanded in n around inf 15.8%

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

      1. +-rgt-identity 15.8%

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

      2. +-rgt-identity 15.8%

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

      3. log1p-def 15.8%

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

   4. Simplified 15.8%

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

   5. Taylor expanded in x around inf 83.9%

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

      1. *-commutative 83.9%

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

   7. Simplified 83.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+218}:\\ \;\;\;\;\left|1 - {x}^{\left(\frac{1}{n}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 4: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-102)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-21)
       (- (/ (log1p x) n) (/ (log x) n))
       (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-102) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (log1p(x) / n) - (log(x) / n);
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-102) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (Math.log1p(x) / n) - (Math.log(x) / n);
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-102:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-21:
		tmp = (math.log1p(x) / n) - (math.log(x) / n)
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-102)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(log1p(x) / n) - Float64(log(x) / n));
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-102], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -1.99999999999999987e-102

   1. Initial program 76.9%

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

   2. Taylor expanded in x around inf 91.1%

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

      1. mul-1-neg 91.1%

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

      2. log-rec 91.1%

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

      3. distribute-frac-neg 91.1%

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

      4. remove-double-neg 91.1%

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

      5. *-commutative 91.1%

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

   4. Simplified 91.1%

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

   5. Taylor expanded in x around 0 91.1%

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

      1. *-rgt-identity 91.1%

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

      2. associate-*r/ 91.1%

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

      3. exp-to-pow 91.1%

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

      4. *-commutative 91.1%

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

   7. Simplified 91.1%

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

   if -1.99999999999999987e-102 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 32.7%

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

   2. Taylor expanded in n around inf 88.9%

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

      1. +-rgt-identity 88.9%

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

      2. +-rgt-identity 88.9%

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

      3. log1p-def 88.9%

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

   4. Simplified 88.9%

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

      1. div-sub 88.9%

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

   6. Applied egg-rr 88.9%

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

   if 4.99999999999999973e-21 < (/.f64 1 n)

   1. Initial program 61.5%

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

   2. Taylor expanded in n around 0 61.5%

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

      1. log1p-def 96.8%

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

   4. Simplified 96.8%

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

   5. Taylor expanded in x around 0 96.8%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right)}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 5: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+218}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-102)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-21)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 1e+218) (- (+ 1.0 (/ x n)) t_0) (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-102) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 1e+218) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	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) <= -2e-102) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1e+218) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-102:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-21:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 1e+218:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-102)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e+218)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	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], -2e-102], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+218], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1.99999999999999987e-102

   1. Initial program 76.9%

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

   2. Taylor expanded in x around inf 91.1%

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

      1. mul-1-neg 91.1%

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

      2. log-rec 91.1%

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

      3. distribute-frac-neg 91.1%

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

      4. remove-double-neg 91.1%

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

      5. *-commutative 91.1%

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

   4. Simplified 91.1%

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

   5. Taylor expanded in x around 0 91.1%

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

      1. *-rgt-identity 91.1%

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

      2. associate-*r/ 91.1%

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

      3. exp-to-pow 91.1%

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

      4. *-commutative 91.1%

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

   7. Simplified 91.1%

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

   if -1.99999999999999987e-102 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 32.7%

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

   2. Taylor expanded in n around inf 88.9%

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

      1. +-rgt-identity 88.9%

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

      2. +-rgt-identity 88.9%

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

      3. log1p-def 88.9%

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

   4. Simplified 88.9%

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

   if 4.99999999999999973e-21 < (/.f64 1 n) < 1.00000000000000008e218

   1. Initial program 79.5%

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

   2. Taylor expanded in x around 0 77.2%

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

   if 1.00000000000000008e218 < (/.f64 1 n)

   1. Initial program 19.3%

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

   2. Taylor expanded in n around inf 15.8%

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

      1. +-rgt-identity 15.8%

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

      2. +-rgt-identity 15.8%

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

      3. log1p-def 15.8%

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

   4. Simplified 15.8%

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

   5. Taylor expanded in x around inf 83.9%

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

      1. *-commutative 83.9%

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

   7. Simplified 83.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-102}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+218}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 6: 52.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -200000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-140}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1.6 \cdot 10^{-258}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+218}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -200000.0)
     t_0
     (if (<= (/ 1.0 n) -2e-140)
       (* (/ 1.0 n) (/ 1.0 x))
       (if (<= (/ 1.0 n) -1.6e-258)
         (/ (- (log x)) n)
         (if (<= (/ 1.0 n) 5e-285)
           t_0
           (if (<= (/ 1.0 n) 5e-22)
             (/ (- x (log x)) n)
             (if (<= (/ 1.0 n) 1e+218) t_0 (/ 1.0 (* n x))))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -200000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e-140) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if ((1.0 / n) <= -1.6e-258) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= 5e-285) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-22) {
		tmp = (x - log(x)) / n;
	} else if ((1.0 / n) <= 1e+218) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-200000.0d0)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-2d-140)) then
        tmp = (1.0d0 / n) * (1.0d0 / x)
    else if ((1.0d0 / n) <= (-1.6d-258)) then
        tmp = -log(x) / n
    else if ((1.0d0 / n) <= 5d-285) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-22) then
        tmp = (x - log(x)) / n
    else if ((1.0d0 / n) <= 1d+218) then
        tmp = t_0
    else
        tmp = 1.0d0 / (n * 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) <= -200000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e-140) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if ((1.0 / n) <= -1.6e-258) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= 5e-285) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-22) {
		tmp = (x - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1e+218) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -200000.0:
		tmp = t_0
	elif (1.0 / n) <= -2e-140:
		tmp = (1.0 / n) * (1.0 / x)
	elif (1.0 / n) <= -1.6e-258:
		tmp = -math.log(x) / n
	elif (1.0 / n) <= 5e-285:
		tmp = t_0
	elif (1.0 / n) <= 5e-22:
		tmp = (x - math.log(x)) / n
	elif (1.0 / n) <= 1e+218:
		tmp = t_0
	else:
		tmp = 1.0 / (n * 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) <= -200000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -2e-140)
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	elseif (Float64(1.0 / n) <= -1.6e-258)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-285)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-22)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e+218)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(n * 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) <= -200000.0)
		tmp = t_0;
	elseif ((1.0 / n) <= -2e-140)
		tmp = (1.0 / n) * (1.0 / x);
	elseif ((1.0 / n) <= -1.6e-258)
		tmp = -log(x) / n;
	elseif ((1.0 / n) <= 5e-285)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-22)
		tmp = (x - log(x)) / n;
	elseif ((1.0 / n) <= 1e+218)
		tmp = t_0;
	else
		tmp = 1.0 / (n * 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], -200000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-140], N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.6e-258], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-285], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-22], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+218], t$95$0, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2e5 or -1.6000000000000001e-258 < (/.f64 1 n) < 5.00000000000000018e-285 or 4.99999999999999954e-22 < (/.f64 1 n) < 1.00000000000000008e218

   1. Initial program 90.7%

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

   2. Taylor expanded in x around 0 67.8%

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

   if -2e5 < (/.f64 1 n) < -2e-140

   1. Initial program 9.5%

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

   2. Taylor expanded in n around inf 45.6%

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

      1. +-rgt-identity 45.6%

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

      2. +-rgt-identity 45.6%

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

      3. log1p-def 45.5%

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

   4. Simplified 45.5%

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

   5. Taylor expanded in x around inf 62.0%

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

      1. *-commutative 62.0%

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

   7. Simplified 62.0%

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

      1. inv-pow 62.0%

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

      2. unpow-prod-down 62.1%

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

      3. inv-pow 62.1%

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

      4. inv-pow 62.1%

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

   9. Applied egg-rr 62.1%

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

   if -2e-140 < (/.f64 1 n) < -1.6000000000000001e-258

   1. Initial program 21.8%

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

   2. Taylor expanded in x around 0 21.8%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. associate-*r/ 78.9%

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

      2. mul-1-neg 78.9%

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

   5. Simplified 78.9%

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

   if 5.00000000000000018e-285 < (/.f64 1 n) < 4.99999999999999954e-22

   1. Initial program 24.0%

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

   2. Taylor expanded in n around inf 83.3%

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

      1. +-rgt-identity 83.3%

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

      2. +-rgt-identity 83.3%

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

      3. log1p-def 83.3%

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

   4. Simplified 83.3%

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

   5. Taylor expanded in x around 0 62.4%

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

   if 1.00000000000000008e218 < (/.f64 1 n)

   1. Initial program 19.3%

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

   2. Taylor expanded in n around inf 15.8%

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

      1. +-rgt-identity 15.8%

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

      2. +-rgt-identity 15.8%

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

      3. log1p-def 15.8%

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

   4. Simplified 15.8%

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

   5. Taylor expanded in x around inf 83.9%

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

      1. *-commutative 83.9%

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

   7. Simplified 83.9%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -200000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-140}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1.6 \cdot 10^{-258}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+218}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 7: 66.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-140}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1.6 \cdot 10^{-258}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+218}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-140)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) -1.6e-258)
       (/ (- (log x)) n)
       (if (<= (/ 1.0 n) 5e-285)
         (- 1.0 t_0)
         (if (<= (/ 1.0 n) 5e-21)
           (- (/ x n) (/ (log x) n))
           (if (<= (/ 1.0 n) 1e+218)
             (- (+ 1.0 (/ x n)) t_0)
             (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-140) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= -1.6e-258) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= 5e-285) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (x / n) - (log(x) / n);
	} else if ((1.0 / n) <= 1e+218) {
		tmp = (1.0 + (x / n)) - t_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) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-2d-140)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= (-1.6d-258)) then
        tmp = -log(x) / n
    else if ((1.0d0 / n) <= 5d-285) then
        tmp = 1.0d0 - t_0
    else if ((1.0d0 / n) <= 5d-21) then
        tmp = (x / n) - (log(x) / n)
    else if ((1.0d0 / n) <= 1d+218) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-140) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= -1.6e-258) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= 5e-285) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = (x / n) - (Math.log(x) / n);
	} else if ((1.0 / n) <= 1e+218) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-140:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= -1.6e-258:
		tmp = -math.log(x) / n
	elif (1.0 / n) <= 5e-285:
		tmp = 1.0 - t_0
	elif (1.0 / n) <= 5e-21:
		tmp = (x / n) - (math.log(x) / n)
	elif (1.0 / n) <= 1e+218:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-140)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= -1.6e-258)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-285)
		tmp = Float64(1.0 - t_0);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	elseif (Float64(1.0 / n) <= 1e+218)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * 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) <= -2e-140)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= -1.6e-258)
		tmp = -log(x) / n;
	elseif ((1.0 / n) <= 5e-285)
		tmp = 1.0 - t_0;
	elseif ((1.0 / n) <= 5e-21)
		tmp = (x / n) - (log(x) / n);
	elseif ((1.0 / n) <= 1e+218)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-140], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.6e-258], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-285], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+218], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -2e-140

   1. Initial program 75.2%

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

   2. Taylor expanded in x around inf 89.9%

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

      1. mul-1-neg 89.9%

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

      2. log-rec 89.9%

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

      3. distribute-frac-neg 89.9%

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

      4. remove-double-neg 89.9%

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

      5. *-commutative 89.9%

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

   4. Simplified 89.9%

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

   5. Taylor expanded in x around 0 89.9%

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

      1. *-rgt-identity 89.9%

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

      2. associate-*r/ 89.9%

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

      3. exp-to-pow 89.9%

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

      4. *-commutative 89.9%

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

   7. Simplified 89.9%

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

   if -2e-140 < (/.f64 1 n) < -1.6000000000000001e-258

   1. Initial program 21.8%

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

   2. Taylor expanded in x around 0 21.8%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. associate-*r/ 78.9%

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

      2. mul-1-neg 78.9%

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

   5. Simplified 78.9%

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

   if -1.6000000000000001e-258 < (/.f64 1 n) < 5.00000000000000018e-285

   1. Initial program 71.4%

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

   2. Taylor expanded in x around 0 71.4%

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

   if 5.00000000000000018e-285 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 24.9%

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

   2. Taylor expanded in n around inf 83.8%

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

      1. +-rgt-identity 83.8%

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

      2. +-rgt-identity 83.8%

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

      3. log1p-def 83.8%

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

   4. Simplified 83.8%

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

   5. Taylor expanded in x around 0 62.0%

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

      1. +-commutative 62.0%

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

      2. mul-1-neg 62.0%

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

      3. unsub-neg 62.0%

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

   7. Simplified 62.0%

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

   if 4.99999999999999973e-21 < (/.f64 1 n) < 1.00000000000000008e218

   1. Initial program 79.5%

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

   2. Taylor expanded in x around 0 77.2%

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

   if 1.00000000000000008e218 < (/.f64 1 n)

   1. Initial program 19.3%

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

   2. Taylor expanded in n around inf 15.8%

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

      1. +-rgt-identity 15.8%

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

      2. +-rgt-identity 15.8%

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

      3. log1p-def 15.8%

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

   4. Simplified 15.8%

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

   5. Taylor expanded in x around inf 83.9%

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

      1. *-commutative 83.9%

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

   7. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-140}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1.6 \cdot 10^{-258}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+218}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 8: 66.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-140}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1.6 \cdot 10^{-258}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\left(x + \left(x \cdot x\right) \cdot -0.5\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+218}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-140)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) -1.6e-258)
       (/ (- (log x)) n)
       (if (<= (/ 1.0 n) 5e-285)
         (- 1.0 t_0)
         (if (<= (/ 1.0 n) 5e-21)
           (/ (- (+ x (* (* x x) -0.5)) (log x)) n)
           (if (<= (/ 1.0 n) 1e+218)
             (- (+ 1.0 (/ x n)) t_0)
             (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-140) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= -1.6e-258) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= 5e-285) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = ((x + ((x * x) * -0.5)) - log(x)) / n;
	} else if ((1.0 / n) <= 1e+218) {
		tmp = (1.0 + (x / n)) - t_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) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-2d-140)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= (-1.6d-258)) then
        tmp = -log(x) / n
    else if ((1.0d0 / n) <= 5d-285) then
        tmp = 1.0d0 - t_0
    else if ((1.0d0 / n) <= 5d-21) then
        tmp = ((x + ((x * x) * (-0.5d0))) - log(x)) / n
    else if ((1.0d0 / n) <= 1d+218) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-140) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= -1.6e-258) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= 5e-285) {
		tmp = 1.0 - t_0;
	} else if ((1.0 / n) <= 5e-21) {
		tmp = ((x + ((x * x) * -0.5)) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1e+218) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-140:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= -1.6e-258:
		tmp = -math.log(x) / n
	elif (1.0 / n) <= 5e-285:
		tmp = 1.0 - t_0
	elif (1.0 / n) <= 5e-21:
		tmp = ((x + ((x * x) * -0.5)) - math.log(x)) / n
	elif (1.0 / n) <= 1e+218:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-140)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= -1.6e-258)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-285)
		tmp = Float64(1.0 - t_0);
	elseif (Float64(1.0 / n) <= 5e-21)
		tmp = Float64(Float64(Float64(x + Float64(Float64(x * x) * -0.5)) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e+218)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * 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) <= -2e-140)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= -1.6e-258)
		tmp = -log(x) / n;
	elseif ((1.0 / n) <= 5e-285)
		tmp = 1.0 - t_0;
	elseif ((1.0 / n) <= 5e-21)
		tmp = ((x + ((x * x) * -0.5)) - log(x)) / n;
	elseif ((1.0 / n) <= 1e+218)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-140], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.6e-258], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-285], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-21], N[(N[(N[(x + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+218], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -2e-140

   1. Initial program 75.2%

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

   2. Taylor expanded in x around inf 89.9%

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

      1. mul-1-neg 89.9%

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

      2. log-rec 89.9%

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

      3. distribute-frac-neg 89.9%

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

      4. remove-double-neg 89.9%

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

      5. *-commutative 89.9%

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

   4. Simplified 89.9%

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

   5. Taylor expanded in x around 0 89.9%

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

      1. *-rgt-identity 89.9%

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

      2. associate-*r/ 89.9%

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

      3. exp-to-pow 89.9%

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

      4. *-commutative 89.9%

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

   7. Simplified 89.9%

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

   if -2e-140 < (/.f64 1 n) < -1.6000000000000001e-258

   1. Initial program 21.8%

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

   2. Taylor expanded in x around 0 21.8%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. associate-*r/ 78.9%

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

      2. mul-1-neg 78.9%

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

   5. Simplified 78.9%

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

   if -1.6000000000000001e-258 < (/.f64 1 n) < 5.00000000000000018e-285

   1. Initial program 71.4%

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

   2. Taylor expanded in x around 0 71.4%

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

   if 5.00000000000000018e-285 < (/.f64 1 n) < 4.99999999999999973e-21

   1. Initial program 24.9%

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

   2. Taylor expanded in n around inf 83.8%

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

      1. +-rgt-identity 83.8%

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

      2. +-rgt-identity 83.8%

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

      3. log1p-def 83.8%

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

   4. Simplified 83.8%

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

   5. Taylor expanded in x around 0 62.3%

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

      1. *-commutative 62.3%

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

      2. unpow2 62.3%

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

   7. Simplified 62.3%

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

   if 4.99999999999999973e-21 < (/.f64 1 n) < 1.00000000000000008e218

   1. Initial program 79.5%

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

   2. Taylor expanded in x around 0 77.2%

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

   if 1.00000000000000008e218 < (/.f64 1 n)

   1. Initial program 19.3%

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

   2. Taylor expanded in n around inf 15.8%

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

      1. +-rgt-identity 15.8%

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

      2. +-rgt-identity 15.8%

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

      3. log1p-def 15.8%

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

   4. Simplified 15.8%

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

   5. Taylor expanded in x around inf 83.9%

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

      1. *-commutative 83.9%

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

   7. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-140}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1.6 \cdot 10^{-258}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\left(x + \left(x \cdot x\right) \cdot -0.5\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+218}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 9: 66.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := 1 - t_0\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-140}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1.6 \cdot 10^{-258}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+218}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- 1.0 t_0)))
   (if (<= (/ 1.0 n) -2e-140)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) -1.6e-258)
       (/ (- (log x)) n)
       (if (<= (/ 1.0 n) 5e-285)
         t_1
         (if (<= (/ 1.0 n) 5e-22)
           (- (/ x n) (/ (log x) n))
           (if (<= (/ 1.0 n) 1e+218) t_1 (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = 1.0 - t_0;
	double tmp;
	if ((1.0 / n) <= -2e-140) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= -1.6e-258) {
		tmp = -log(x) / n;
	} else if ((1.0 / n) <= 5e-285) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-22) {
		tmp = (x / n) - (log(x) / n);
	} else if ((1.0 / n) <= 1e+218) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = 1.0d0 - t_0
    if ((1.0d0 / n) <= (-2d-140)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= (-1.6d-258)) then
        tmp = -log(x) / n
    else if ((1.0d0 / n) <= 5d-285) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-22) then
        tmp = (x / n) - (log(x) / n)
    else if ((1.0d0 / n) <= 1d+218) then
        tmp = t_1
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = 1.0 - t_0;
	double tmp;
	if ((1.0 / n) <= -2e-140) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= -1.6e-258) {
		tmp = -Math.log(x) / n;
	} else if ((1.0 / n) <= 5e-285) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-22) {
		tmp = (x / n) - (Math.log(x) / n);
	} else if ((1.0 / n) <= 1e+218) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = 1.0 - t_0
	tmp = 0
	if (1.0 / n) <= -2e-140:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= -1.6e-258:
		tmp = -math.log(x) / n
	elif (1.0 / n) <= 5e-285:
		tmp = t_1
	elif (1.0 / n) <= 5e-22:
		tmp = (x / n) - (math.log(x) / n)
	elif (1.0 / n) <= 1e+218:
		tmp = t_1
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(1.0 - t_0)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-140)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= -1.6e-258)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e-285)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-22)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	elseif (Float64(1.0 / n) <= 1e+218)
		tmp = t_1;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = 1.0 - t_0;
	tmp = 0.0;
	if ((1.0 / n) <= -2e-140)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= -1.6e-258)
		tmp = -log(x) / n;
	elseif ((1.0 / n) <= 5e-285)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-22)
		tmp = (x / n) - (log(x) / n);
	elseif ((1.0 / n) <= 1e+218)
		tmp = t_1;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-140], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.6e-258], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-285], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-22], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+218], t$95$1, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -2e-140

   1. Initial program 75.2%

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

   2. Taylor expanded in x around inf 89.9%

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

      1. mul-1-neg 89.9%

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

      2. log-rec 89.9%

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

      3. distribute-frac-neg 89.9%

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

      4. remove-double-neg 89.9%

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

      5. *-commutative 89.9%

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

   4. Simplified 89.9%

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

   5. Taylor expanded in x around 0 89.9%

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

      1. *-rgt-identity 89.9%

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

      2. associate-*r/ 89.9%

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

      3. exp-to-pow 89.9%

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

      4. *-commutative 89.9%

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

   7. Simplified 89.9%

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

   if -2e-140 < (/.f64 1 n) < -1.6000000000000001e-258

   1. Initial program 21.8%

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

   2. Taylor expanded in x around 0 21.8%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. associate-*r/ 78.9%

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

      2. mul-1-neg 78.9%

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

   5. Simplified 78.9%

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

   if -1.6000000000000001e-258 < (/.f64 1 n) < 5.00000000000000018e-285 or 4.99999999999999954e-22 < (/.f64 1 n) < 1.00000000000000008e218

   1. Initial program 75.3%

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

   2. Taylor expanded in x around 0 73.3%

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

   if 5.00000000000000018e-285 < (/.f64 1 n) < 4.99999999999999954e-22

   1. Initial program 24.0%

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

   2. Taylor expanded in n around inf 83.3%

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

      1. +-rgt-identity 83.3%

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

      2. +-rgt-identity 83.3%

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

      3. log1p-def 83.3%

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

   4. Simplified 83.3%

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

   5. Taylor expanded in x around 0 62.4%

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

      1. +-commutative 62.4%

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

      2. mul-1-neg 62.4%

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

      3. unsub-neg 62.4%

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

   7. Simplified 62.4%

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

   if 1.00000000000000008e218 < (/.f64 1 n)

   1. Initial program 19.3%

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

   2. Taylor expanded in n around inf 15.8%

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

      1. +-rgt-identity 15.8%

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

      2. +-rgt-identity 15.8%

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

      3. log1p-def 15.8%

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

   4. Simplified 15.8%

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

   5. Taylor expanded in x around inf 83.9%

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

      1. *-commutative 83.9%

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

   7. Simplified 83.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-140}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1.6 \cdot 10^{-258}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-285}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+218}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 10: 60.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{+139}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;n \leq -14.5:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-219}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{+20} \lor \neg \left(n \leq 3.9 \cdot 10^{+284}\right):\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \left(x - \log x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -3.1e+139)
   (/ (- (log x)) n)
   (if (<= n -14.5)
     (* (/ 1.0 n) (/ 1.0 x))
     (if (<= n 8e-219)
       (/ 0.3333333333333333 (* n (pow x 3.0)))
       (if (or (<= n 5e+20) (not (<= n 3.9e+284)))
         (- 1.0 (pow x (/ 1.0 n)))
         (* (/ 1.0 n) (- x (log x))))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -3.1e+139) {
		tmp = -log(x) / n;
	} else if (n <= -14.5) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if (n <= 8e-219) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if ((n <= 5e+20) || !(n <= 3.9e+284)) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (1.0 / n) * (x - log(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 <= (-3.1d+139)) then
        tmp = -log(x) / n
    else if (n <= (-14.5d0)) then
        tmp = (1.0d0 / n) * (1.0d0 / x)
    else if (n <= 8d-219) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if ((n <= 5d+20) .or. (.not. (n <= 3.9d+284))) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (1.0d0 / n) * (x - log(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -3.1e+139) {
		tmp = -Math.log(x) / n;
	} else if (n <= -14.5) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if (n <= 8e-219) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if ((n <= 5e+20) || !(n <= 3.9e+284)) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (1.0 / n) * (x - Math.log(x));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -3.1e+139:
		tmp = -math.log(x) / n
	elif n <= -14.5:
		tmp = (1.0 / n) * (1.0 / x)
	elif n <= 8e-219:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif (n <= 5e+20) or not (n <= 3.9e+284):
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (1.0 / n) * (x - math.log(x))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -3.1e+139)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (n <= -14.5)
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	elseif (n <= 8e-219)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif ((n <= 5e+20) || !(n <= 3.9e+284))
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(1.0 / n) * Float64(x - log(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -3.1e+139)
		tmp = -log(x) / n;
	elseif (n <= -14.5)
		tmp = (1.0 / n) * (1.0 / x);
	elseif (n <= 8e-219)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif ((n <= 5e+20) || ~((n <= 3.9e+284)))
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (1.0 / n) * (x - log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -3.1e+139], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[n, -14.5], N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8e-219], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[n, 5e+20], N[Not[LessEqual[n, 3.9e+284]], $MachinePrecision]], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] * N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -3.1e139

   1. Initial program 33.7%

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

   2. Taylor expanded in x around 0 33.7%

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

   3. Taylor expanded in n around inf 69.6%

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

      1. associate-*r/ 69.6%

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

      2. mul-1-neg 69.6%

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

   5. Simplified 69.6%

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

   if -3.1e139 < n < -14.5

   1. Initial program 9.5%

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

   2. Taylor expanded in n around inf 45.6%

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

      1. +-rgt-identity 45.6%

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

      2. +-rgt-identity 45.6%

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

      3. log1p-def 45.5%

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

   4. Simplified 45.5%

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

   5. Taylor expanded in x around inf 62.0%

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

      1. *-commutative 62.0%

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

   7. Simplified 62.0%

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

      1. inv-pow 62.0%

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

      2. unpow-prod-down 62.1%

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

      3. inv-pow 62.1%

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

      4. inv-pow 62.1%

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

   9. Applied egg-rr 62.1%

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

   if -14.5 < n < 8.0000000000000003e-219

   1. Initial program 89.7%

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

   2. Taylor expanded in n around inf 34.3%

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

      1. +-rgt-identity 34.3%

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

      2. +-rgt-identity 34.3%

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

      3. log1p-def 34.3%

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

   4. Simplified 34.3%

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

   5. Taylor expanded in x around inf 29.2%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. sub-neg 29.2%

         \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. +-commutative 29.2%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. associate-+l+ 29.2%

         \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}}{n} \]

      4. associate-*r/ 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      5. metadata-eval 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      6. associate-*r/ 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)\right)}{n} \]

      7. metadata-eval 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\frac{\color{blue}{0.5}}{{x}^{2}}\right)\right)}{n} \]

      8. distribute-neg-frac 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \color{blue}{\frac{-0.5}{{x}^{2}}}\right)}{n} \]

      9. metadata-eval 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{\color{blue}{-0.5}}{{x}^{2}}\right)}{n} \]

      10. unpow2 29.2%

          \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{\color{blue}{x \cdot x}}\right)}{n} \]

   7. Simplified 29.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{x \cdot x}\right)}}{n} \]

   8. Taylor expanded in x around 0 79.7%

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

   if 8.0000000000000003e-219 < n < 5e20 or 3.89999999999999976e284 < n

   1. Initial program 80.1%

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

   2. Taylor expanded in x around 0 77.5%

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

   if 5e20 < n < 3.89999999999999976e284

   1. Initial program 24.0%

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

   2. Taylor expanded in n around inf 83.3%

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

      1. +-rgt-identity 83.3%

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

      2. +-rgt-identity 83.3%

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

      3. log1p-def 83.3%

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

   4. Simplified 83.3%

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

   5. Taylor expanded in x around 0 62.4%

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

      1. div-inv 62.4%

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

   7. Applied egg-rr 62.4%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{+139}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;n \leq -14.5:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-219}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{+20} \lor \neg \left(n \leq 3.9 \cdot 10^{+284}\right):\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \left(x - \log x\right)\\ \end{array} \]

Alternative 11: 60.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{+141}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;n \leq -15.2:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;n \leq 10^{-218}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{+20} \lor \neg \left(n \leq 2.5 \cdot 10^{+284}\right):\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -1.15e+141)
   (/ (- (log x)) n)
   (if (<= n -15.2)
     (* (/ 1.0 n) (/ 1.0 x))
     (if (<= n 1e-218)
       (/ 0.3333333333333333 (* n (pow x 3.0)))
       (if (or (<= n 5e+20) (not (<= n 2.5e+284)))
         (- 1.0 (pow x (/ 1.0 n)))
         (- (/ x n) (/ (log x) n)))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -1.15e+141) {
		tmp = -log(x) / n;
	} else if (n <= -15.2) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if (n <= 1e-218) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if ((n <= 5e+20) || !(n <= 2.5e+284)) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (x / n) - (log(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 (n <= (-1.15d+141)) then
        tmp = -log(x) / n
    else if (n <= (-15.2d0)) then
        tmp = (1.0d0 / n) * (1.0d0 / x)
    else if (n <= 1d-218) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if ((n <= 5d+20) .or. (.not. (n <= 2.5d+284))) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (x / n) - (log(x) / n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -1.15e+141) {
		tmp = -Math.log(x) / n;
	} else if (n <= -15.2) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if (n <= 1e-218) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if ((n <= 5e+20) || !(n <= 2.5e+284)) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (x / n) - (Math.log(x) / n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -1.15e+141:
		tmp = -math.log(x) / n
	elif n <= -15.2:
		tmp = (1.0 / n) * (1.0 / x)
	elif n <= 1e-218:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif (n <= 5e+20) or not (n <= 2.5e+284):
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (x / n) - (math.log(x) / n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -1.15e+141)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (n <= -15.2)
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	elseif (n <= 1e-218)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif ((n <= 5e+20) || !(n <= 2.5e+284))
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -1.15e+141)
		tmp = -log(x) / n;
	elseif (n <= -15.2)
		tmp = (1.0 / n) * (1.0 / x);
	elseif (n <= 1e-218)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif ((n <= 5e+20) || ~((n <= 2.5e+284)))
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (x / n) - (log(x) / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -1.15e+141], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[n, -15.2], N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1e-218], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[n, 5e+20], N[Not[LessEqual[n, 2.5e+284]], $MachinePrecision]], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -1.1500000000000001e141

   1. Initial program 33.7%

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

   2. Taylor expanded in x around 0 33.7%

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

   3. Taylor expanded in n around inf 69.6%

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

      1. associate-*r/ 69.6%

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

      2. mul-1-neg 69.6%

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

   5. Simplified 69.6%

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

   if -1.1500000000000001e141 < n < -15.199999999999999

   1. Initial program 9.5%

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

   2. Taylor expanded in n around inf 45.6%

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

      1. +-rgt-identity 45.6%

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

      2. +-rgt-identity 45.6%

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

      3. log1p-def 45.5%

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

   4. Simplified 45.5%

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

   5. Taylor expanded in x around inf 62.0%

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

      1. *-commutative 62.0%

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

   7. Simplified 62.0%

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

      1. inv-pow 62.0%

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

      2. unpow-prod-down 62.1%

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

      3. inv-pow 62.1%

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

      4. inv-pow 62.1%

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

   9. Applied egg-rr 62.1%

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

   if -15.199999999999999 < n < 1e-218

   1. Initial program 89.7%

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

   2. Taylor expanded in n around inf 34.3%

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

      1. +-rgt-identity 34.3%

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

      2. +-rgt-identity 34.3%

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

      3. log1p-def 34.3%

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

   4. Simplified 34.3%

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

   5. Taylor expanded in x around inf 29.2%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. sub-neg 29.2%

         \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. +-commutative 29.2%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. associate-+l+ 29.2%

         \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}}{n} \]

      4. associate-*r/ 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      5. metadata-eval 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      6. associate-*r/ 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)\right)}{n} \]

      7. metadata-eval 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\frac{\color{blue}{0.5}}{{x}^{2}}\right)\right)}{n} \]

      8. distribute-neg-frac 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \color{blue}{\frac{-0.5}{{x}^{2}}}\right)}{n} \]

      9. metadata-eval 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{\color{blue}{-0.5}}{{x}^{2}}\right)}{n} \]

      10. unpow2 29.2%

          \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{\color{blue}{x \cdot x}}\right)}{n} \]

   7. Simplified 29.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{x \cdot x}\right)}}{n} \]

   8. Taylor expanded in x around 0 79.7%

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

   if 1e-218 < n < 5e20 or 2.5e284 < n

   1. Initial program 80.1%

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

   2. Taylor expanded in x around 0 77.5%

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

   if 5e20 < n < 2.5e284

   1. Initial program 24.0%

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

   2. Taylor expanded in n around inf 83.3%

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

      1. +-rgt-identity 83.3%

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

      2. +-rgt-identity 83.3%

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

      3. log1p-def 83.3%

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

   4. Simplified 83.3%

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

   5. Taylor expanded in x around 0 62.4%

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

      1. +-commutative 62.4%

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

      2. mul-1-neg 62.4%

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

      3. unsub-neg 62.4%

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

   7. Simplified 62.4%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{+141}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;n \leq -15.2:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;n \leq 10^{-218}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{+20} \lor \neg \left(n \leq 2.5 \cdot 10^{+284}\right):\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \end{array} \]

Alternative 12: 60.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;n \leq -5.4:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-219}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{+20} \lor \neg \left(n \leq 3.3 \cdot 10^{+284}\right):\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -2.8e+141)
   (/ (- (log x)) n)
   (if (<= n -5.4)
     (* (/ 1.0 n) (/ 1.0 x))
     (if (<= n 1.35e-219)
       (/ 0.3333333333333333 (* n (pow x 3.0)))
       (if (or (<= n 5e+20) (not (<= n 3.3e+284)))
         (- 1.0 (pow x (/ 1.0 n)))
         (/ (- x (log x)) n))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -2.8e+141) {
		tmp = -log(x) / n;
	} else if (n <= -5.4) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if (n <= 1.35e-219) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if ((n <= 5e+20) || !(n <= 3.3e+284)) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (x - log(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 (n <= (-2.8d+141)) then
        tmp = -log(x) / n
    else if (n <= (-5.4d0)) then
        tmp = (1.0d0 / n) * (1.0d0 / x)
    else if (n <= 1.35d-219) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if ((n <= 5d+20) .or. (.not. (n <= 3.3d+284))) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (x - log(x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -2.8e+141) {
		tmp = -Math.log(x) / n;
	} else if (n <= -5.4) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if (n <= 1.35e-219) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if ((n <= 5e+20) || !(n <= 3.3e+284)) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (x - Math.log(x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -2.8e+141:
		tmp = -math.log(x) / n
	elif n <= -5.4:
		tmp = (1.0 / n) * (1.0 / x)
	elif n <= 1.35e-219:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif (n <= 5e+20) or not (n <= 3.3e+284):
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (x - math.log(x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -2.8e+141)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (n <= -5.4)
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	elseif (n <= 1.35e-219)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif ((n <= 5e+20) || !(n <= 3.3e+284))
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(x - log(x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -2.8e+141)
		tmp = -log(x) / n;
	elseif (n <= -5.4)
		tmp = (1.0 / n) * (1.0 / x);
	elseif (n <= 1.35e-219)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif ((n <= 5e+20) || ~((n <= 3.3e+284)))
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (x - log(x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -2.8e+141], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[n, -5.4], N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.35e-219], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[n, 5e+20], N[Not[LessEqual[n, 3.3e+284]], $MachinePrecision]], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -2.79999999999999991e141

   1. Initial program 33.7%

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

   2. Taylor expanded in x around 0 33.7%

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

   3. Taylor expanded in n around inf 69.6%

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

      1. associate-*r/ 69.6%

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

      2. mul-1-neg 69.6%

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

   5. Simplified 69.6%

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

   if -2.79999999999999991e141 < n < -5.4000000000000004

   1. Initial program 9.5%

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

   2. Taylor expanded in n around inf 45.6%

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

      1. +-rgt-identity 45.6%

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

      2. +-rgt-identity 45.6%

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

      3. log1p-def 45.5%

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

   4. Simplified 45.5%

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

   5. Taylor expanded in x around inf 62.0%

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

      1. *-commutative 62.0%

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

   7. Simplified 62.0%

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

      1. inv-pow 62.0%

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

      2. unpow-prod-down 62.1%

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

      3. inv-pow 62.1%

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

      4. inv-pow 62.1%

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

   9. Applied egg-rr 62.1%

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

   if -5.4000000000000004 < n < 1.35e-219

   1. Initial program 89.7%

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

   2. Taylor expanded in n around inf 34.3%

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

      1. +-rgt-identity 34.3%

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

      2. +-rgt-identity 34.3%

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

      3. log1p-def 34.3%

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

   4. Simplified 34.3%

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

   5. Taylor expanded in x around inf 29.2%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. sub-neg 29.2%

         \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} \]

      2. +-commutative 29.2%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} \]

      3. associate-+l+ 29.2%

         \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}}{n} \]

      4. associate-*r/ 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      5. metadata-eval 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{\color{blue}{0.3333333333333333}}{{x}^{3}} + \left(-0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{n} \]

      6. associate-*r/ 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)\right)}{n} \]

      7. metadata-eval 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \left(-\frac{\color{blue}{0.5}}{{x}^{2}}\right)\right)}{n} \]

      8. distribute-neg-frac 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \color{blue}{\frac{-0.5}{{x}^{2}}}\right)}{n} \]

      9. metadata-eval 29.2%

         \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{\color{blue}{-0.5}}{{x}^{2}}\right)}{n} \]

      10. unpow2 29.2%

          \[\leadsto \frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{\color{blue}{x \cdot x}}\right)}{n} \]

   7. Simplified 29.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{x \cdot x}\right)}}{n} \]

   8. Taylor expanded in x around 0 79.7%

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

   if 1.35e-219 < n < 5e20 or 3.2999999999999997e284 < n

   1. Initial program 80.1%

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

   2. Taylor expanded in x around 0 77.5%

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

   if 5e20 < n < 3.2999999999999997e284

   1. Initial program 24.0%

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

   2. Taylor expanded in n around inf 83.3%

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

      1. +-rgt-identity 83.3%

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

      2. +-rgt-identity 83.3%

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

      3. log1p-def 83.3%

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

   4. Simplified 83.3%

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

   5. Taylor expanded in x around 0 62.4%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;n \leq -5.4:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{-219}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 5 \cdot 10^{+20} \lor \neg \left(n \leq 3.3 \cdot 10^{+284}\right):\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \end{array} \]

Alternative 13: 55.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-278}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-225}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.5e-278)
   (/ (- (log x)) n)
   (if (<= x 1.4e-225)
     (/ 1.0 (* n x))
     (if (<= x 0.95)
       (/ (- x (log x)) n)
       (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.5e-278) {
		tmp = -log(x) / n;
	} else if (x <= 1.4e-225) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.95) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.5d-278) then
        tmp = -log(x) / n
    else if (x <= 1.4d-225) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 0.95d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.5e-278) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.4e-225) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.95) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.5e-278:
		tmp = -math.log(x) / n
	elif x <= 1.4e-225:
		tmp = 1.0 / (n * x)
	elif x <= 0.95:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.5e-278)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.4e-225)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 0.95)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.5e-278)
		tmp = -log(x) / n;
	elseif (x <= 1.4e-225)
		tmp = 1.0 / (n * x);
	elseif (x <= 0.95)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.5e-278], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.4e-225], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.95], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.5e-278

   1. Initial program 45.3%

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

   2. Taylor expanded in x around 0 45.3%

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

   3. Taylor expanded in n around inf 59.6%

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

      1. associate-*r/ 59.6%

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

      2. mul-1-neg 59.6%

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

   5. Simplified 59.6%

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

   if 1.5e-278 < x < 1.4e-225

   1. Initial program 89.9%

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

   2. Taylor expanded in n around inf 14.7%

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

      1. +-rgt-identity 14.7%

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

      2. +-rgt-identity 14.7%

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

      3. log1p-def 14.7%

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

   4. Simplified 14.7%

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

   5. Taylor expanded in x around inf 59.6%

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

      1. *-commutative 59.6%

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

   7. Simplified 59.6%

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

   if 1.4e-225 < x < 0.94999999999999996

   1. Initial program 44.2%

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

   2. Taylor expanded in n around inf 51.6%

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

      1. +-rgt-identity 51.6%

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

      2. +-rgt-identity 51.6%

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

      3. log1p-def 51.6%

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

   4. Simplified 51.6%

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

   5. Taylor expanded in x around 0 51.0%

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

   if 0.94999999999999996 < x

   1. Initial program 67.2%

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

   2. Taylor expanded in n around inf 66.5%

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

      1. +-rgt-identity 66.5%

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

      2. +-rgt-identity 66.5%

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

      3. log1p-def 66.5%

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

   4. Simplified 66.5%

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

   5. Taylor expanded in x around inf 65.6%

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

      1. associate-*r/ 65.6%

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

      2. metadata-eval 65.6%

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

      3. unpow2 65.6%

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

   7. Simplified 65.6%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-278}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-225}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \]

Alternative 14: 54.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 8.2 \cdot 10^{-279}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-226}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)))
   (if (<= x 8.2e-279)
     t_0
     (if (<= x 2.6e-226)
       (/ 1.0 (* n x))
       (if (<= x 0.68) t_0 (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double tmp;
	if (x <= 8.2e-279) {
		tmp = t_0;
	} else if (x <= 2.6e-226) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.68) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -log(x) / n
    if (x <= 8.2d-279) then
        tmp = t_0
    else if (x <= 2.6d-226) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 0.68d0) then
        tmp = t_0
    else
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double tmp;
	if (x <= 8.2e-279) {
		tmp = t_0;
	} else if (x <= 2.6e-226) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.68) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	tmp = 0
	if x <= 8.2e-279:
		tmp = t_0
	elif x <= 2.6e-226:
		tmp = 1.0 / (n * x)
	elif x <= 0.68:
		tmp = t_0
	else:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 8.2e-279)
		tmp = t_0;
	elseif (x <= 2.6e-226)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 0.68)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log(x) / n;
	tmp = 0.0;
	if (x <= 8.2e-279)
		tmp = t_0;
	elseif (x <= 2.6e-226)
		tmp = 1.0 / (n * x);
	elseif (x <= 0.68)
		tmp = t_0;
	else
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 8.2e-279], t$95$0, If[LessEqual[x, 2.6e-226], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.68], t$95$0, N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.20000000000000034e-279 or 2.5999999999999998e-226 < x < 0.680000000000000049

   1. Initial program 44.3%

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

   2. Taylor expanded in x around 0 43.6%

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

   3. Taylor expanded in n around inf 51.6%

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

      1. associate-*r/ 51.6%

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

      2. mul-1-neg 51.6%

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

   5. Simplified 51.6%

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

   if 8.20000000000000034e-279 < x < 2.5999999999999998e-226

   1. Initial program 89.9%

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

   2. Taylor expanded in n around inf 14.7%

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

      1. +-rgt-identity 14.7%

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

      2. +-rgt-identity 14.7%

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

      3. log1p-def 14.7%

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

   4. Simplified 14.7%

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

   5. Taylor expanded in x around inf 59.6%

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

      1. *-commutative 59.6%

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

   7. Simplified 59.6%

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

   if 0.680000000000000049 < x

   1. Initial program 67.2%

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

   2. Taylor expanded in n around inf 66.5%

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

      1. +-rgt-identity 66.5%

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

      2. +-rgt-identity 66.5%

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

      3. log1p-def 66.5%

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

   4. Simplified 66.5%

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

   5. Taylor expanded in x around inf 65.6%

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

      1. associate-*r/ 65.6%

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

      2. metadata-eval 65.6%

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

      3. unpow2 65.6%

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

   7. Simplified 65.6%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-279}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-226}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \]

Alternative 15: 40.8% accurate, 30.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

2. Taylor expanded in n around inf 54.9%

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

   1. +-rgt-identity 54.9%

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

   2. +-rgt-identity 54.9%

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

   3. log1p-def 54.9%

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

4. Simplified 54.9%

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

5. Taylor expanded in x around inf 41.7%

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

   1. *-commutative 41.7%

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

7. Simplified 41.7%

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

   1. inv-pow 41.7%

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

   2. unpow-prod-down 41.9%

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

   3. inv-pow 41.9%

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

   4. inv-pow 41.9%

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

9. Applied egg-rr 41.9%

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

10. Final simplification 41.9%

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

Alternative 16: 40.3% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

2. Taylor expanded in n around inf 54.9%

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

   1. +-rgt-identity 54.9%

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

   2. +-rgt-identity 54.9%

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

   3. log1p-def 54.9%

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

4. Simplified 54.9%

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

5. Taylor expanded in x around inf 41.7%

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

   1. *-commutative 41.7%

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

7. Simplified 41.7%

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

8. Final simplification 41.7%

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

Alternative 17: 40.8% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

2. Taylor expanded in n around inf 54.9%

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

   1. +-rgt-identity 54.9%

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

   2. +-rgt-identity 54.9%

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

   3. log1p-def 54.9%

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

4. Simplified 54.9%

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

5. Taylor expanded in x around inf 41.9%

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

6. Final simplification 41.9%

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

Alternative 18: 4.6% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

2. Taylor expanded in n around inf 54.9%

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

   1. +-rgt-identity 54.9%

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

   2. +-rgt-identity 54.9%

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

   3. log1p-def 54.9%

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

4. Simplified 54.9%

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

5. Taylor expanded in x around 0 31.7%

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

6. Taylor expanded in x around inf 4.8%

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

7. Final simplification 4.8%

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

Reproduce

herbie shell --seed 2023283 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))