2nthrt (problem 3.4.6)

Percentage Accurate: 53.9% → 86.0%

Time: 41.3s

Alternatives: 20

Speedup: 7.5×

• 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: 53.9% 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: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -33000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} \cdot 0.5 - \log x\right)}{n}\\ \mathbf{elif}\;n \leq 310000:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -33000000.0)
   (/
    (+
     (log1p x)
     (- (* (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n) 0.5) (log x)))
    n)
   (if (<= n 310000.0)
     (- (exp (/ x n)) (pow x (/ 1.0 n)))
     (/ (log (/ (+ 1.0 x) x)) n))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -33000000.0) {
		tmp = (log1p(x) + ((((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n) * 0.5) - log(x))) / n;
	} else if (n <= 310000.0) {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = log(((1.0 + x) / x)) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -33000000.0) {
		tmp = (Math.log1p(x) + ((((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) / n) * 0.5) - Math.log(x))) / n;
	} else if (n <= 310000.0) {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log(((1.0 + x) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -33000000.0:
		tmp = (math.log1p(x) + ((((math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)) / n) * 0.5) - math.log(x))) / n
	elif n <= 310000.0:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.log(((1.0 + x) / x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -33000000.0)
		tmp = Float64(Float64(log1p(x) + Float64(Float64(Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n) * 0.5) - log(x))) / n);
	elseif (n <= 310000.0)
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[n, -33000000.0], N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] * 0.5), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 310000.0], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.3e7

   1. Initial program 36.2%

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

   3. Taylor expanded in n around inf 87.8%

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

      1. associate--l+ 87.8%

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

      2. log1p-define 87.8%

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

      3. +-commutative 87.8%

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

      4. associate--r+ 87.8%

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

      5. distribute-lft-out-- 87.8%

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

      6. div-sub 87.8%

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

      7. log1p-define 87.8%

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

   5. Simplified 87.8%

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

   if -3.3e7 < n < 3.1e5

   1. Initial program 83.5%

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

   3. Taylor expanded in n around 0 83.5%

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

      1. log1p-define 98.1%

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

      2. *-rgt-identity 98.1%

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

      3. associate-/l* 98.1%

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

      4. exp-to-pow 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in x around 0 98.1%

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

   if 3.1e5 < n

   1. Initial program 30.3%

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

   3. Taylor expanded in n around inf 78.3%

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

      1. log1p-define 78.3%

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

   5. Simplified 78.3%

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

      1. log1p-undefine 78.3%

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

      2. diff-log 78.8%

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

   7. Applied egg-rr 78.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -33000000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} \cdot 0.5 - \log x\right)}{n}\\ \mathbf{elif}\;n \leq 310000:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(1 + x\right)}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 5e-32)
   (/
    (log
     (/
      (*
       (sqrt (exp (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n)))
       (+ 1.0 x))
      x))
    n)
   (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 5e-32) {
		tmp = log(((sqrt(exp(((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n))) * (1.0 + x)) / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 5e-32) {
		tmp = Math.log(((Math.sqrt(Math.exp(((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) / n))) * (1.0 + x)) / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 5e-32:
		tmp = math.log(((math.sqrt(math.exp(((math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)) / n))) * (1.0 + x)) / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 5e-32)
		tmp = Float64(log(Float64(Float64(sqrt(exp(Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n))) * Float64(1.0 + x)) / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-32], N[(N[Log[N[(N[(N[Sqrt[N[Exp[N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < 5e-32

   1. Initial program 54.5%

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

   3. Taylor expanded in n around inf 78.4%

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

      1. associate--l+ 66.1%

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

      2. log1p-define 66.1%

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

      3. +-commutative 66.1%

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

      4. associate--r+ 78.4%

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

      5. distribute-lft-out-- 78.4%

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

      6. div-sub 78.4%

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

      7. log1p-define 78.4%

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

   5. Simplified 78.4%

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

      1. associate-+r- 78.4%

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

      2. add-log-exp 87.6%

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

      3. exp-diff 87.6%

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

      4. add-exp-log 59.3%

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

   7. Applied egg-rr 59.3%

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

      1. +-commutative 59.3%

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

      2. exp-sum 59.3%

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

      3. *-commutative 59.3%

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

      4. exp-prod 59.3%

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

      5. unpow1/2 59.3%

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

      6. log1p-define 59.3%

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

      7. rem-exp-log 87.9%

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

   9. Simplified 87.9%

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

   if 5e-32 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 57.0%

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

   3. Taylor expanded in n around 0 57.0%

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

      1. log1p-define 95.5%

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

      2. *-rgt-identity 95.5%

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

      3. associate-/l* 95.5%

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

      4. exp-to-pow 95.5%

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

   5. Simplified 95.5%

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

Alternative 3: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt[3]{1 - t\_0}\right)}^{3}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (- (pow (+ 1.0 x) (/ 1.0 n)) t_0)))
   (if (<= t_1 -2e-6)
     (pow (cbrt (- 1.0 t_0)) 3.0)
     (if (<= t_1 0.0) (/ (log (/ (+ 1.0 x) x)) n) (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = pow((1.0 + x), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -2e-6) {
		tmp = pow(cbrt((1.0 - t_0)), 3.0);
	} else if (t_1 <= 0.0) {
		tmp = log(((1.0 + x) / 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 t_1 = Math.pow((1.0 + x), (1.0 / n)) - t_0;
	double tmp;
	if (t_1 <= -2e-6) {
		tmp = Math.pow(Math.cbrt((1.0 - t_0)), 3.0);
	} else if (t_1 <= 0.0) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - t_0)
	tmp = 0.0
	if (t_1 <= -2e-6)
		tmp = cbrt(Float64(1.0 - t_0)) ^ 3.0;
	elseif (t_1 <= 0.0)
		tmp = Float64(log(Float64(Float64(1.0 + x) / 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]}, Block[{t$95$1 = N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-6], N[Power[N[Power[N[(1.0 - t$95$0), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < -1.99999999999999991e-6

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. associate-/l* 98.9%

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

      3. exp-to-pow 98.9%

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

   5. Simplified 98.9%

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

      1. add-cube-cbrt 98.9%

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

      2. pow3 99.0%

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

   7. Applied egg-rr 99.0%

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

   if -1.99999999999999991e-6 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 0.0

   1. Initial program 42.9%

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

   3. Taylor expanded in n around inf 84.3%

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

      1. log1p-define 84.3%

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

   5. Simplified 84.3%

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

      1. log1p-undefine 84.3%

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

      2. diff-log 84.7%

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

   7. Applied egg-rr 84.7%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 59.6%

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

   3. Taylor expanded in n around 0 59.6%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in 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 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\sqrt[3]{1 - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\\ \mathbf{elif}\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 0:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-8}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\frac{\log \left(1 + \left(\frac{1}{x} + \frac{\log x}{n \cdot x}\right)\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-8)
     (- (exp (/ x n)) t_0)
     (if (<= (/ 1.0 n) 5e-32)
       (/ (log (+ 1.0 (+ (/ 1.0 x) (/ (log x) (* n x))))) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-8) {
		tmp = exp((x / n)) - t_0;
	} else if ((1.0 / n) <= 5e-32) {
		tmp = log((1.0 + ((1.0 / x) + (log(x) / (n * x))))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-8) {
		tmp = Math.exp((x / n)) - t_0;
	} else if ((1.0 / n) <= 5e-32) {
		tmp = Math.log((1.0 + ((1.0 / x) + (Math.log(x) / (n * x))))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-8:
		tmp = math.exp((x / n)) - t_0
	elif (1.0 / n) <= 5e-32:
		tmp = math.log((1.0 + ((1.0 / x) + (math.log(x) / (n * x))))) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-8)
		tmp = Float64(exp(Float64(x / n)) - t_0);
	elseif (Float64(1.0 / n) <= 5e-32)
		tmp = Float64(log(Float64(1.0 + Float64(Float64(1.0 / x) + Float64(log(x) / Float64(n * x))))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-8], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-32], N[(N[Log[N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-8

   1. Initial program 98.0%

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

   3. Taylor expanded in n around 0 98.0%

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

      1. log1p-define 98.0%

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

      2. *-rgt-identity 98.0%

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

      3. associate-/l* 98.0%

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

      4. exp-to-pow 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around 0 98.0%

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

   if -2e-8 < (/.f64 #s(literal 1 binary64) n) < 5e-32

   1. Initial program 32.9%

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

   3. Taylor expanded in n around inf 83.3%

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

      1. associate--l+ 83.3%

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

      2. log1p-define 83.3%

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

      3. +-commutative 83.3%

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

      4. associate--r+ 83.3%

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

      5. distribute-lft-out-- 83.3%

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

      6. div-sub 83.3%

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

      7. log1p-define 83.3%

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

   5. Simplified 83.3%

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

      1. associate-+r- 83.3%

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

      2. add-log-exp 83.3%

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

      3. exp-diff 83.3%

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

      4. add-exp-log 58.6%

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

   7. Applied egg-rr 58.6%

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

      1. +-commutative 58.6%

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

      2. exp-sum 58.6%

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

      3. *-commutative 58.6%

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

      4. exp-prod 58.6%

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

      5. unpow1/2 58.6%

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

      6. log1p-define 58.6%

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

      7. rem-exp-log 83.7%

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

   9. Simplified 83.7%

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

   10. Taylor expanded in x around inf 83.4%

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

       1. +-commutative 83.4%

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

       2. mul-1-neg 83.4%

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

       3. log-rec 83.4%

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

       4. distribute-frac-neg 83.4%

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

       5. remove-double-neg 83.4%

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

       6. *-commutative 83.4%

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

   12. Simplified 83.4%

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

   if 5e-32 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 57.0%

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

   3. Taylor expanded in n around 0 57.0%

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

      1. log1p-define 95.5%

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

      2. *-rgt-identity 95.5%

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

      3. associate-/l* 95.5%

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

      4. exp-to-pow 95.5%

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

   5. Simplified 95.5%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-8}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\frac{\log \left(1 + \left(\frac{1}{x} + \frac{\log x}{n \cdot x}\right)\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + x}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -0.1:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+115}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left({t\_0}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 x) x)))
   (if (<= (/ 1.0 n) -0.1)
     (/ (exp (/ (log x) n)) (* n x))
     (if (<= (/ 1.0 n) 5e-6)
       (/ (log t_0) n)
       (if (<= (/ 1.0 n) 1e+115)
         (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
         (log (pow t_0 (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = (1.0 + x) / x;
	double tmp;
	if ((1.0 / n) <= -0.1) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-6) {
		tmp = log(t_0) / n;
	} else if ((1.0 / n) <= 1e+115) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = log(pow(t_0, (1.0 / n)));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + x) / x
    if ((1.0d0 / n) <= (-0.1d0)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 5d-6) then
        tmp = log(t_0) / n
    else if ((1.0d0 / n) <= 1d+115) then
        tmp = (1.0d0 + (x / n)) - (x ** (1.0d0 / n))
    else
        tmp = log((t_0 ** (1.0d0 / n)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 + x) / x;
	double tmp;
	if ((1.0 / n) <= -0.1) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-6) {
		tmp = Math.log(t_0) / n;
	} else if ((1.0 / n) <= 1e+115) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log(Math.pow(t_0, (1.0 / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 + x) / x
	tmp = 0
	if (1.0 / n) <= -0.1:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 5e-6:
		tmp = math.log(t_0) / n
	elif (1.0 / n) <= 1e+115:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.log(math.pow(t_0, (1.0 / n)))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 + x) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.1)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-6)
		tmp = Float64(log(t_0) / n);
	elseif (Float64(1.0 / n) <= 1e+115)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = log((t_0 ^ Float64(1.0 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 + x) / x;
	tmp = 0.0;
	if ((1.0 / n) <= -0.1)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 5e-6)
		tmp = log(t_0) / n;
	elseif ((1.0 / n) <= 1e+115)
		tmp = (1.0 + (x / n)) - (x ^ (1.0 / n));
	else
		tmp = log((t_0 ^ (1.0 / n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.1], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-6], N[(N[Log[t$95$0], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+115], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[Power[t$95$0, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.10000000000000001

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -0.10000000000000001 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000041e-6

   1. Initial program 33.8%

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

   3. Taylor expanded in n around inf 81.5%

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

      1. log1p-define 81.5%

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

   5. Simplified 81.5%

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

      1. log1p-undefine 81.5%

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

      2. diff-log 81.9%

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

   7. Applied egg-rr 81.9%

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

   if 5.00000000000000041e-6 < (/.f64 #s(literal 1 binary64) n) < 1e115

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0 86.2%

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

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

   1. Initial program 24.0%

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

   3. Taylor expanded in n around inf 6.4%

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

      1. log1p-define 6.4%

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

   5. Simplified 6.4%

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

      1. add-log-exp 80.6%

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

      2. div-inv 80.6%

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

      3. exp-prod 80.6%

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

      4. exp-diff 80.6%

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

      5. add-exp-log 80.6%

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

      6. log1p-undefine 80.6%

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

      7. rem-exp-log 80.6%

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

   7. Applied egg-rr 80.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.1:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+115}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\frac{1 + x}{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8500000000 \lor \neg \left(n \leq 310000\right):\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -8500000000.0) (not (<= n 310000.0)))
   (/ (log (/ (+ 1.0 x) x)) n)
   (- (exp (/ x n)) (pow x (/ 1.0 n)))))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -8500000000.0) || !(n <= 310000.0)) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-8500000000.0d0)) .or. (.not. (n <= 310000.0d0))) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = exp((x / n)) - (x ** (1.0d0 / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -8500000000.0) || !(n <= 310000.0)) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -8500000000.0) or not (n <= 310000.0):
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -8500000000.0) || !(n <= 310000.0))
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -8500000000.0) || ~((n <= 310000.0)))
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = exp((x / n)) - (x ^ (1.0 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -8500000000.0], N[Not[LessEqual[n, 310000.0]], $MachinePrecision]], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -8.5e9 or 3.1e5 < n

   1. Initial program 32.8%

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

   3. Taylor expanded in n around inf 82.5%

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

      1. log1p-define 82.5%

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

   5. Simplified 82.5%

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

      1. log1p-undefine 82.5%

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

      2. diff-log 82.9%

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

   7. Applied egg-rr 82.9%

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

   if -8.5e9 < n < 3.1e5

   1. Initial program 83.4%

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

   3. Taylor expanded in n around 0 83.4%

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

      1. log1p-define 97.9%

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

      2. *-rgt-identity 97.9%

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

      3. associate-/l* 97.9%

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

      4. exp-to-pow 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0 97.9%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8500000000 \lor \neg \left(n \leq 310000\right):\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -10000000000:\\ \;\;\;\;\frac{\log \left(1 + \left(\frac{1}{x} + \frac{\log x}{n \cdot x}\right)\right)}{n}\\ \mathbf{elif}\;n \leq 310000:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -10000000000.0)
   (/ (log (+ 1.0 (+ (/ 1.0 x) (/ (log x) (* n x))))) n)
   (if (<= n 310000.0)
     (- (exp (/ x n)) (pow x (/ 1.0 n)))
     (/ (log (/ (+ 1.0 x) x)) n))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -10000000000.0) {
		tmp = log((1.0 + ((1.0 / x) + (log(x) / (n * x))))) / n;
	} else if (n <= 310000.0) {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = log(((1.0 + 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 (n <= (-10000000000.0d0)) then
        tmp = log((1.0d0 + ((1.0d0 / x) + (log(x) / (n * x))))) / n
    else if (n <= 310000.0d0) then
        tmp = exp((x / n)) - (x ** (1.0d0 / n))
    else
        tmp = log(((1.0d0 + x) / x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -10000000000.0) {
		tmp = Math.log((1.0 + ((1.0 / x) + (Math.log(x) / (n * x))))) / n;
	} else if (n <= 310000.0) {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log(((1.0 + x) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -10000000000.0:
		tmp = math.log((1.0 + ((1.0 / x) + (math.log(x) / (n * x))))) / n
	elif n <= 310000.0:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.log(((1.0 + x) / x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -10000000000.0)
		tmp = Float64(log(Float64(1.0 + Float64(Float64(1.0 / x) + Float64(log(x) / Float64(n * x))))) / n);
	elseif (n <= 310000.0)
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -10000000000.0)
		tmp = log((1.0 + ((1.0 / x) + (log(x) / (n * x))))) / n;
	elseif (n <= 310000.0)
		tmp = exp((x / n)) - (x ^ (1.0 / n));
	else
		tmp = log(((1.0 + x) / x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -10000000000.0], N[(N[Log[N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 310000.0], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1e10

   1. Initial program 35.6%

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

   3. Taylor expanded in n around inf 87.9%

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

      1. associate--l+ 87.9%

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

      2. log1p-define 87.9%

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

      3. +-commutative 87.9%

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

      4. associate--r+ 87.9%

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

      5. distribute-lft-out-- 87.9%

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

      6. div-sub 87.9%

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

      7. log1p-define 87.9%

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

   5. Simplified 87.9%

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

      1. associate-+r- 87.9%

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

      2. add-log-exp 87.9%

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

      3. exp-diff 87.9%

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

      4. add-exp-log 61.3%

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

   7. Applied egg-rr 61.3%

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

      1. +-commutative 61.3%

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

      2. exp-sum 61.3%

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

      3. *-commutative 61.3%

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

      4. exp-prod 61.3%

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

      5. unpow1/2 61.3%

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

      6. log1p-define 61.3%

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

      7. rem-exp-log 88.2%

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

   9. Simplified 88.2%

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

   10. Taylor expanded in x around inf 87.7%

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

       1. +-commutative 87.7%

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

       2. mul-1-neg 87.7%

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

       3. log-rec 87.7%

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

       4. distribute-frac-neg 87.7%

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

       5. remove-double-neg 87.7%

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

       6. *-commutative 87.7%

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

   12. Simplified 87.7%

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

   if -1e10 < n < 3.1e5

   1. Initial program 83.4%

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

   3. Taylor expanded in n around 0 83.4%

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

      1. log1p-define 97.9%

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

      2. *-rgt-identity 97.9%

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

      3. associate-/l* 97.9%

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

      4. exp-to-pow 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0 97.9%

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

   if 3.1e5 < n

   1. Initial program 30.3%

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

   3. Taylor expanded in n around inf 78.3%

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

      1. log1p-define 78.3%

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

   5. Simplified 78.3%

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

      1. log1p-undefine 78.3%

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

      2. diff-log 78.8%

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

   7. Applied egg-rr 78.8%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -10000000000:\\ \;\;\;\;\frac{\log \left(1 + \left(\frac{1}{x} + \frac{\log x}{n \cdot x}\right)\right)}{n}\\ \mathbf{elif}\;n \leq 310000:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.1:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+115}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.1)
     (/ (exp (/ (log x) n)) (* n x))
     (if (<= (/ 1.0 n) 5e-6)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 1e+115)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (/ 1.0 t_0) (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.1) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-6) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+115) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (1.0 / t_0) / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-0.1d0)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 5d-6) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 1d+115) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (1.0d0 / t_0) / (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) <= -0.1) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-6) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+115) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (1.0 / t_0) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -0.1:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 5e-6:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e+115:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (1.0 / t_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) <= -0.1)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-6)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+115)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(1.0 / t_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) <= -0.1)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 5e-6)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 1e+115)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = (1.0 / t_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], -0.1], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-6], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+115], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.10000000000000001

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -0.10000000000000001 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000041e-6

   1. Initial program 33.8%

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

   3. Taylor expanded in n around inf 81.5%

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

      1. log1p-define 81.5%

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

   5. Simplified 81.5%

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

      1. log1p-undefine 81.5%

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

      2. diff-log 81.9%

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

   7. Applied egg-rr 81.9%

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

   if 5.00000000000000041e-6 < (/.f64 #s(literal 1 binary64) n) < 1e115

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0 86.2%

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

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

   1. Initial program 24.0%

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

   3. Taylor expanded in x around inf 0.8%

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

      1. mul-1-neg 0.8%

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

      2. log-rec 0.8%

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

      3. mul-1-neg 0.8%

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

      4. distribute-neg-frac 0.8%

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

      5. mul-1-neg 0.8%

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

      6. remove-double-neg 0.8%

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

      7. *-commutative 0.8%

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

   5. Simplified 0.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 80.6%

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

      3. sqr-neg 80.6%

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

      4. distribute-frac-neg 80.6%

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

      5. distribute-frac-neg 80.6%

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

      6. sqrt-unprod 80.6%

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

      7. add-sqr-sqrt 80.6%

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

      8. distribute-frac-neg 80.6%

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

      9. exp-neg 80.6%

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

      10. div-inv 80.6%

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

      11. exp-to-pow 80.6%

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

   7. Applied egg-rr 80.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.1:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+115}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.1:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+115}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.1)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-6)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 1e+115)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (/ 1.0 t_0) (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.1) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-6) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+115) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (1.0 / t_0) / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-0.1d0)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 5d-6) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 1d+115) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (1.0d0 / t_0) / (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) <= -0.1) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-6) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+115) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (1.0 / t_0) / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -0.1:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-6:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e+115:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (1.0 / t_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) <= -0.1)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-6)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+115)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(1.0 / t_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) <= -0.1)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 5e-6)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 1e+115)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = (1.0 / t_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], -0.1], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-6], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+115], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.10000000000000001

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. associate-/l* 99.9%

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

      9. exp-to-pow 99.9%

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

      10. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -0.10000000000000001 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000041e-6

   1. Initial program 33.8%

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

   3. Taylor expanded in n around inf 81.5%

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

      1. log1p-define 81.5%

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

   5. Simplified 81.5%

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

      1. log1p-undefine 81.5%

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

      2. diff-log 81.9%

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

   7. Applied egg-rr 81.9%

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

   if 5.00000000000000041e-6 < (/.f64 #s(literal 1 binary64) n) < 1e115

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0 86.2%

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

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

   1. Initial program 24.0%

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

   3. Taylor expanded in x around inf 0.8%

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

      1. mul-1-neg 0.8%

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

      2. log-rec 0.8%

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

      3. mul-1-neg 0.8%

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

      4. distribute-neg-frac 0.8%

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

      5. mul-1-neg 0.8%

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

      6. remove-double-neg 0.8%

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

      7. *-commutative 0.8%

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

   5. Simplified 0.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 80.6%

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

      3. sqr-neg 80.6%

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

      4. distribute-frac-neg 80.6%

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

      5. distribute-frac-neg 80.6%

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

      6. sqrt-unprod 80.6%

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

      7. add-sqr-sqrt 80.6%

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

      8. distribute-frac-neg 80.6%

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

      9. exp-neg 80.6%

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

      10. div-inv 80.6%

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

      11. exp-to-pow 80.6%

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

   7. Applied egg-rr 80.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.1:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+115}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.1:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+115}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.1)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-6)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 1e+115)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (log1p (+ x -1.0)) (- n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.1) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-6) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+115) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log1p((x + -1.0)) / -n;
	}
	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) <= -0.1) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-6) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+115) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -0.1:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-6:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e+115:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log1p((x + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.1)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-6)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+115)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(log1p(Float64(x + -1.0)) / Float64(-n));
	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], -0.1], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-6], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+115], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.10000000000000001

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. associate-/l* 99.9%

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

      9. exp-to-pow 99.9%

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

      10. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -0.10000000000000001 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000041e-6

   1. Initial program 33.8%

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

   3. Taylor expanded in n around inf 81.5%

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

      1. log1p-define 81.5%

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

   5. Simplified 81.5%

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

      1. log1p-undefine 81.5%

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

      2. diff-log 81.9%

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

   7. Applied egg-rr 81.9%

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

   if 5.00000000000000041e-6 < (/.f64 #s(literal 1 binary64) n) < 1e115

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0 86.2%

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

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

   1. Initial program 24.0%

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

   3. Taylor expanded in n around inf 6.4%

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

      1. log1p-define 6.4%

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

   5. Simplified 6.4%

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

   6. Taylor expanded in x around 0 6.4%

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

      1. neg-mul-1 6.4%

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

   8. Simplified 6.4%

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

      1. log1p-expm1-u 75.9%

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

      2. expm1-undefine 75.9%

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

      3. add-exp-log 75.9%

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

   10. Applied egg-rr 75.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.1:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+115}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.1:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+115}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.1)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-6)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 1e+115) (- 1.0 t_0) (/ (log1p (+ x -1.0)) (- n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.1) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-6) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+115) {
		tmp = 1.0 - t_0;
	} else {
		tmp = log1p((x + -1.0)) / -n;
	}
	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) <= -0.1) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-6) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+115) {
		tmp = 1.0 - t_0;
	} else {
		tmp = Math.log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -0.1:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-6:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e+115:
		tmp = 1.0 - t_0
	else:
		tmp = math.log1p((x + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.1)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-6)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+115)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(log1p(Float64(x + -1.0)) / Float64(-n));
	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], -0.1], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-6], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+115], N[(1.0 - t$95$0), $MachinePrecision], N[(N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.10000000000000001

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. associate-/l* 99.9%

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

      9. exp-to-pow 99.9%

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

      10. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if -0.10000000000000001 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000041e-6

   1. Initial program 33.8%

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

   3. Taylor expanded in n around inf 81.5%

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

      1. log1p-define 81.5%

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

   5. Simplified 81.5%

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

      1. log1p-undefine 81.5%

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

      2. diff-log 81.9%

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

   7. Applied egg-rr 81.9%

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

   if 5.00000000000000041e-6 < (/.f64 #s(literal 1 binary64) n) < 1e115

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0 86.1%

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

      1. *-rgt-identity 86.1%

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

      2. associate-/l* 86.1%

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

      3. exp-to-pow 86.1%

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

   5. Simplified 86.1%

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

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

   1. Initial program 24.0%

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

   3. Taylor expanded in n around inf 6.4%

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

      1. log1p-define 6.4%

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

   5. Simplified 6.4%

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

   6. Taylor expanded in x around 0 6.4%

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

      1. neg-mul-1 6.4%

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

   8. Simplified 6.4%

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

      1. log1p-expm1-u 75.9%

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

      2. expm1-undefine 75.9%

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

      3. add-exp-log 75.9%

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

   10. Applied egg-rr 75.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.1:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+115}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{\frac{x \cdot \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} - x}{x \cdot x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+115}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -500.0)
   (/
    (/
     (- (* x (/ (+ (/ (+ (/ 0.25 x) -0.3333333333333333) x) -0.5) x)) x)
     (* x x))
    n)
   (if (<= (/ 1.0 n) 5e-6)
     (/ (log (/ (+ 1.0 x) x)) n)
     (if (<= (/ 1.0 n) 1e+115)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (log1p (+ x -1.0)) (- n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = (((x * (((((0.25 / x) + -0.3333333333333333) / x) + -0.5) / x)) - x) / (x * x)) / n;
	} else if ((1.0 / n) <= 5e-6) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+115) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = (((x * (((((0.25 / x) + -0.3333333333333333) / x) + -0.5) / x)) - x) / (x * x)) / n;
	} else if ((1.0 / n) <= 5e-6) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+115) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -500.0:
		tmp = (((x * (((((0.25 / x) + -0.3333333333333333) / x) + -0.5) / x)) - x) / (x * x)) / n
	elif (1.0 / n) <= 5e-6:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e+115:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = math.log1p((x + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500.0)
		tmp = Float64(Float64(Float64(Float64(x * Float64(Float64(Float64(Float64(Float64(0.25 / x) + -0.3333333333333333) / x) + -0.5) / x)) - x) / Float64(x * x)) / n);
	elseif (Float64(1.0 / n) <= 5e-6)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+115)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(log1p(Float64(x + -1.0)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -500.0], N[(N[(N[(N[(x * N[(N[(N[(N[(N[(0.25 / x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / x), $MachinePrecision] + -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-6], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+115], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -500

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 43.5%

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

      1. log1p-define 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in x around -inf 1.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   8. Applied egg-rr 71.3%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} \cdot \left(-x\right) - x \cdot -1}{x \cdot \left(-x\right)}}}{n} \]

   if -500 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000041e-6

   1. Initial program 33.6%

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

   3. Taylor expanded in n around inf 81.0%

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

      1. log1p-define 81.0%

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

   5. Simplified 81.0%

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

      1. log1p-undefine 81.0%

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

      2. diff-log 81.3%

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

   7. Applied egg-rr 81.3%

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

   if 5.00000000000000041e-6 < (/.f64 #s(literal 1 binary64) n) < 1e115

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0 86.1%

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

      1. *-rgt-identity 86.1%

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

      2. associate-/l* 86.1%

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

      3. exp-to-pow 86.1%

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

   5. Simplified 86.1%

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

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

   1. Initial program 24.0%

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

   3. Taylor expanded in n around inf 6.4%

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

      1. log1p-define 6.4%

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

   5. Simplified 6.4%

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

   6. Taylor expanded in x around 0 6.4%

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

      1. neg-mul-1 6.4%

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

   8. Simplified 6.4%

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

      1. log1p-expm1-u 75.9%

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

      2. expm1-undefine 75.9%

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

      3. add-exp-log 75.9%

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

   10. Applied egg-rr 75.9%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{\frac{x \cdot \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} - x}{x \cdot x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+115}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-140}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - \frac{1}{x} \cdot 0.25}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.85e-140)
   (/ (log x) (- n))
   (if (<= x 9.8e-99)
     (/ (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* n x)) (/ 0.5 n)) x)) x)
     (if (<= x 0.9)
       (/ (- x (log x)) n)
       (if (<= x 5.2e+73)
         (/
          (/
           (+
            1.0
            (/ (- (/ (- 0.3333333333333333 (* (/ 1.0 x) 0.25)) x) 0.5) x))
           x)
          n)
         0.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.85e-140) {
		tmp = log(x) / -n;
	} else if (x <= 9.8e-99) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.9) {
		tmp = (x - log(x)) / n;
	} else if (x <= 5.2e+73) {
		tmp = ((1.0 + ((((0.3333333333333333 - ((1.0 / x) * 0.25)) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.85d-140) then
        tmp = log(x) / -n
    else if (x <= 9.8d-99) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) - (0.5d0 / n)) / x)) / x
    else if (x <= 0.9d0) then
        tmp = (x - log(x)) / n
    else if (x <= 5.2d+73) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 - ((1.0d0 / x) * 0.25d0)) / x) - 0.5d0) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.85e-140) {
		tmp = Math.log(x) / -n;
	} else if (x <= 9.8e-99) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.9) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 5.2e+73) {
		tmp = ((1.0 + ((((0.3333333333333333 - ((1.0 / x) * 0.25)) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.85e-140:
		tmp = math.log(x) / -n
	elif x <= 9.8e-99:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x
	elif x <= 0.9:
		tmp = (x - math.log(x)) / n
	elif x <= 5.2e+73:
		tmp = ((1.0 + ((((0.3333333333333333 - ((1.0 / x) * 0.25)) / x) - 0.5) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.85e-140)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 9.8e-99)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) - Float64(0.5 / n)) / x)) / x);
	elseif (x <= 0.9)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 5.2e+73)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(Float64(1.0 / x) * 0.25)) / x) - 0.5) / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.85e-140)
		tmp = log(x) / -n;
	elseif (x <= 9.8e-99)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	elseif (x <= 0.9)
		tmp = (x - log(x)) / n;
	elseif (x <= 5.2e+73)
		tmp = ((1.0 + ((((0.3333333333333333 - ((1.0 / x) * 0.25)) / x) - 0.5) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.85e-140], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 9.8e-99], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.9], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 5.2e+73], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 - N[(N[(1.0 / x), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.84999999999999989e-140

   1. Initial program 45.3%

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

   3. Taylor expanded in n around inf 56.3%

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

      1. log1p-define 56.3%

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

   5. Simplified 56.3%

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

   6. Taylor expanded in x around 0 56.3%

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

      1. neg-mul-1 56.3%

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

   8. Simplified 56.3%

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

   if 1.84999999999999989e-140 < x < 9.8000000000000006e-99

   1. Initial program 59.6%

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

   3. Taylor expanded in n around inf 23.2%

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

      1. log1p-define 23.2%

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

   5. Simplified 23.2%

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

   6. Taylor expanded in x around -inf 73.2%

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

      1. associate-*r/ 73.2%

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

      2. mul-1-neg 73.2%

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

      3. associate-*r/ 73.2%

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

      4. mul-1-neg 73.2%

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

      5. associate-*r/ 73.2%

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

      6. metadata-eval 73.2%

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

      7. *-commutative 73.2%

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

      8. associate-*r/ 73.2%

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

      9. metadata-eval 73.2%

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

   8. Simplified 73.2%

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

   if 9.8000000000000006e-99 < x < 0.900000000000000022

   1. Initial program 35.7%

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

   3. Taylor expanded in n around inf 59.2%

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

      1. log1p-define 59.2%

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

   5. Simplified 59.2%

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

   6. Taylor expanded in x around 0 58.6%

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

   if 0.900000000000000022 < x < 5.2000000000000001e73

   1. Initial program 33.6%

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

   3. Taylor expanded in n around inf 39.1%

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

      1. log1p-define 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in x around -inf 79.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   if 5.2000000000000001e73 < x

   1. Initial program 82.0%

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

      1. add-log-exp 82.0%

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

      2. pow-to-exp 82.0%

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

      3. un-div-inv 82.0%

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

      4. +-commutative 82.0%

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

      5. log1p-define 82.0%

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

   4. Applied egg-rr 82.0%

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

   5. Taylor expanded in x around inf 82.0%

      \[\leadsto \log \color{blue}{1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-140}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - \frac{1}{x} \cdot 0.25}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 1.85 \cdot 10^{-140}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - \frac{1}{x} \cdot 0.25}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 1.85e-140)
     t_0
     (if (<= x 1.45e-98)
       (/ (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* n x)) (/ 0.5 n)) x)) x)
       (if (<= x 0.7)
         t_0
         (if (<= x 1.6e+75)
           (/
            (/
             (+
              1.0
              (/ (- (/ (- 0.3333333333333333 (* (/ 1.0 x) 0.25)) x) 0.5) x))
             x)
            n)
           0.0))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 1.85e-140) {
		tmp = t_0;
	} else if (x <= 1.45e-98) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.7) {
		tmp = t_0;
	} else if (x <= 1.6e+75) {
		tmp = ((1.0 + ((((0.3333333333333333 - ((1.0 / x) * 0.25)) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 1.85d-140) then
        tmp = t_0
    else if (x <= 1.45d-98) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (n * x)) - (0.5d0 / n)) / x)) / x
    else if (x <= 0.7d0) then
        tmp = t_0
    else if (x <= 1.6d+75) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 - ((1.0d0 / x) * 0.25d0)) / x) - 0.5d0) / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 1.85e-140) {
		tmp = t_0;
	} else if (x <= 1.45e-98) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.7) {
		tmp = t_0;
	} else if (x <= 1.6e+75) {
		tmp = ((1.0 + ((((0.3333333333333333 - ((1.0 / x) * 0.25)) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 1.85e-140:
		tmp = t_0
	elif x <= 1.45e-98:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x
	elif x <= 0.7:
		tmp = t_0
	elif x <= 1.6e+75:
		tmp = ((1.0 + ((((0.3333333333333333 - ((1.0 / x) * 0.25)) / x) - 0.5) / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 1.85e-140)
		tmp = t_0;
	elseif (x <= 1.45e-98)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(n * x)) - Float64(0.5 / n)) / x)) / x);
	elseif (x <= 0.7)
		tmp = t_0;
	elseif (x <= 1.6e+75)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(Float64(1.0 / x) * 0.25)) / x) - 0.5) / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 1.85e-140)
		tmp = t_0;
	elseif (x <= 1.45e-98)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (n * x)) - (0.5 / n)) / x)) / x;
	elseif (x <= 0.7)
		tmp = t_0;
	elseif (x <= 1.6e+75)
		tmp = ((1.0 + ((((0.3333333333333333 - ((1.0 / x) * 0.25)) / x) - 0.5) / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 1.85e-140], t$95$0, If[LessEqual[x, 1.45e-98], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(n * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.7], t$95$0, If[LessEqual[x, 1.6e+75], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 - N[(N[(1.0 / x), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.84999999999999989e-140 or 1.45e-98 < x < 0.69999999999999996

   1. Initial program 41.5%

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

   3. Taylor expanded in n around inf 57.4%

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

      1. log1p-define 57.4%

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

   5. Simplified 57.4%

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

   6. Taylor expanded in x around 0 56.4%

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

      1. neg-mul-1 56.4%

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

   8. Simplified 56.4%

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

   if 1.84999999999999989e-140 < x < 1.45e-98

   1. Initial program 59.6%

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

   3. Taylor expanded in n around inf 23.2%

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

      1. log1p-define 23.2%

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

   5. Simplified 23.2%

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

   6. Taylor expanded in x around -inf 73.2%

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

      1. associate-*r/ 73.2%

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

      2. mul-1-neg 73.2%

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

      3. associate-*r/ 73.2%

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

      4. mul-1-neg 73.2%

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

      5. associate-*r/ 73.2%

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

      6. metadata-eval 73.2%

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

      7. *-commutative 73.2%

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

      8. associate-*r/ 73.2%

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

      9. metadata-eval 73.2%

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

   8. Simplified 73.2%

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

   if 0.69999999999999996 < x < 1.59999999999999992e75

   1. Initial program 33.6%

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

   3. Taylor expanded in n around inf 39.1%

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

      1. log1p-define 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in x around -inf 79.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   if 1.59999999999999992e75 < x

   1. Initial program 82.0%

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

      1. add-log-exp 82.0%

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

      2. pow-to-exp 82.0%

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

      3. un-div-inv 82.0%

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

      4. +-commutative 82.0%

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

      5. log1p-define 82.0%

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

   4. Applied egg-rr 82.0%

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

   5. Taylor expanded in x around inf 82.0%

      \[\leadsto \log \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-140}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{n \cdot x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.7:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 - \frac{1}{x} \cdot 0.25}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} - x}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{-x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+18)
   (/
    (/
     (- (* x (/ (+ (/ (+ (/ 0.25 x) -0.3333333333333333) x) -0.5) x)) x)
     (* x x))
    n)
   (/ (/ (+ -1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (- x)) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+18) {
		tmp = (((x * (((((0.25 / x) + -0.3333333333333333) / x) + -0.5) / x)) - x) / (x * x)) / n;
	} else {
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d+18)) then
        tmp = (((x * (((((0.25d0 / x) + (-0.3333333333333333d0)) / x) + (-0.5d0)) / x)) - x) / (x * x)) / n
    else
        tmp = (((-1.0d0) + ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / -x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+18) {
		tmp = (((x * (((((0.25 / x) + -0.3333333333333333) / x) + -0.5) / x)) - x) / (x * x)) / n;
	} else {
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+18:
		tmp = (((x * (((((0.25 / x) + -0.3333333333333333) / x) + -0.5) / x)) - x) / (x * x)) / n
	else:
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+18)
		tmp = Float64(Float64(Float64(Float64(x * Float64(Float64(Float64(Float64(Float64(0.25 / x) + -0.3333333333333333) / x) + -0.5) / x)) - x) / Float64(x * x)) / n);
	else
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(-x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e+18)
		tmp = (((x * (((((0.25 / x) + -0.3333333333333333) / x) + -0.5) / x)) - x) / (x * x)) / n;
	else
		tmp = ((-1.0 + ((0.5 + (-0.3333333333333333 / x)) / x)) / -x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+18], N[(N[(N[(N[(x * N[(N[(N[(N[(N[(0.25 / x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / x), $MachinePrecision] + -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(-1.0 + N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e18

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 45.2%

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

      1. log1p-define 45.2%

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

   5. Simplified 45.2%

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

   6. Taylor expanded in x around -inf 1.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   8. Applied egg-rr 70.5%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} \cdot \left(-x\right) - x \cdot -1}{x \cdot \left(-x\right)}}}{n} \]

   if -1e18 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 40.5%

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

   3. Taylor expanded in n around inf 63.9%

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

      1. log1p-define 63.9%

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

   5. Simplified 63.9%

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

   6. Taylor expanded in x around -inf 35.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in x around -inf 45.7%

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

      1. mul-1-neg 45.7%

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

      2. distribute-neg-frac2 45.7%

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

      3. sub-neg 45.7%

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

      4. associate-*r/ 45.7%

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

      5. sub-neg 45.7%

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

      6. metadata-eval 45.7%

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

      7. distribute-lft-in 45.7%

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

      8. neg-mul-1 45.7%

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

      9. associate-*r/ 45.7%

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

      10. metadata-eval 45.7%

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

      11. distribute-neg-frac 45.7%

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

      12. metadata-eval 45.7%

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

      13. metadata-eval 45.7%

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

      14. metadata-eval 45.7%

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

   9. Simplified 45.7%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} - x}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{-x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.3% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1 + \frac{0.5 + \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}{-n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/
  (/ (+ -1.0 (/ (+ 0.5 (/ (+ 0.3333333333333333 (/ -0.25 x)) x)) x)) x)
  (- n)))

C

double code(double x, double n) {
	return ((-1.0 + ((0.5 + ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / -n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (((-1.0d0) + ((0.5d0 + ((0.3333333333333333d0 + ((-0.25d0) / x)) / x)) / x)) / x) / -n
end function

Java

public static double code(double x, double n) {
	return ((-1.0 + ((0.5 + ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / -n;
}

Python

def code(x, n):
	return ((-1.0 + ((0.5 + ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / -n

Julia

function code(x, n)
	return Float64(Float64(Float64(-1.0 + Float64(Float64(0.5 + Float64(Float64(0.3333333333333333 + Float64(-0.25 / x)) / x)) / x)) / x) / Float64(-n))
end

MATLAB

function tmp = code(x, n)
	tmp = ((-1.0 + ((0.5 + ((0.3333333333333333 + (-0.25 / x)) / x)) / x)) / x) / -n;
end

Wolfram

code[x_, n_] := N[(N[(N[(-1.0 + N[(N[(0.5 + N[(N[(0.3333333333333333 + N[(-0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / (-n)), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in n around inf 59.4%

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

   1. log1p-define 59.4%

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

5. Simplified 59.4%

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

6. Taylor expanded in x around -inf 27.0%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
7. Step-by-step derivation

   1. fma-neg 27.0%

      \[\leadsto \frac{-1 \cdot \frac{\color{blue}{\mathsf{fma}\left(-1, \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x}, -1\right)}}{x}}{n} \]

8. Applied egg-rr 46.7%

   \[\leadsto \frac{-1 \cdot \frac{\color{blue}{\mathsf{fma}\left(-1, \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}, -1\right)}}{x}}{n} \]
9. Step-by-step derivation

   1. fma-undefine 46.7%

      \[\leadsto \frac{-1 \cdot \frac{\color{blue}{-1 \cdot \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + -1}}{x}}{n} \]

   2. metadata-eval 46.7%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + \color{blue}{-1 \cdot 1}}{x}}{n} \]

   3. distribute-lft-in 46.7%

      \[\leadsto \frac{-1 \cdot \frac{\color{blue}{-1 \cdot \left(\frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x} + 1\right)}}{x}}{n} \]

   4. +-commutative 46.7%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \color{blue}{\left(1 + \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}\right)}}{x}}{n} \]

   5. distribute-lft-in 46.7%

      \[\leadsto \frac{-1 \cdot \frac{\color{blue}{-1 \cdot 1 + -1 \cdot \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}}}{x}}{n} \]

   6. metadata-eval 46.7%

      \[\leadsto \frac{-1 \cdot \frac{\color{blue}{-1} + -1 \cdot \frac{\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5}{x}}{x}}{n} \]

   7. associate-*r/ 46.7%

      \[\leadsto \frac{-1 \cdot \frac{-1 + \color{blue}{\frac{-1 \cdot \left(\frac{\frac{0.25}{x} + -0.3333333333333333}{x} + -0.5\right)}{x}}}{x}}{n} \]

10. Simplified 46.7%

    \[\leadsto \frac{-1 \cdot \frac{\color{blue}{-1 + \frac{0.5 + \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}}{x}}{n} \]

11. Final simplification 46.7%

    \[\leadsto \frac{\frac{-1 + \frac{0.5 + \frac{0.3333333333333333 + \frac{-0.25}{x}}{x}}{x}}{x}}{-n} \]
12. Add Preprocessing

Alternative 17: 46.7% accurate, 15.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in n around inf 59.4%

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

   1. log1p-define 59.4%

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

5. Simplified 59.4%

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

6. Taylor expanded in x around -inf 27.0%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

7. Taylor expanded in x around -inf 46.4%

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

   1. mul-1-neg 46.4%

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

   2. distribute-neg-frac2 46.4%

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

   3. sub-neg 46.4%

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

   4. associate-*r/ 46.4%

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

   5. sub-neg 46.4%

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

   6. metadata-eval 46.4%

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

   7. distribute-lft-in 46.4%

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

   8. neg-mul-1 46.4%

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

   9. associate-*r/ 46.4%

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

   10. metadata-eval 46.4%

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

   11. distribute-neg-frac 46.4%

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

   12. metadata-eval 46.4%

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

   13. metadata-eval 46.4%

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

   14. metadata-eval 46.4%

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

9. Simplified 46.4%

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

10. Final simplification 46.4%

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

Alternative 18: 41.1% 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 54.9%

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

3. Taylor expanded in n around inf 59.4%

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

   1. log1p-define 59.4%

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

5. Simplified 59.4%

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

6. Taylor expanded in x around inf 38.6%

   \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
7. Add Preprocessing

Alternative 19: 40.6% 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 54.9%

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

3. Taylor expanded in n around inf 59.4%

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

   1. log1p-define 59.4%

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

5. Simplified 59.4%

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

6. Taylor expanded in x around inf 37.7%

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

   1. *-commutative 37.7%

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

8. Simplified 37.7%

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

9. Final simplification 37.7%

   \[\leadsto \frac{1}{n \cdot x} \]
10. Add Preprocessing

Alternative 20: 4.5% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

3. Taylor expanded in n around inf 59.4%

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

   1. log1p-define 59.4%

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

5. Simplified 59.4%

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

6. Taylor expanded in x around inf 37.7%

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

   1. *-commutative 37.7%

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

8. Simplified 37.7%

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

   1. associate-/r* 38.6%

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

   2. div-inv 38.6%

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

   3. rem-exp-log 38.0%

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

   4. neg-log 38.0%

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

   5. add-sqr-sqrt 12.3%

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

   6. sqrt-unprod 13.6%

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

   7. sqr-neg 13.6%

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

   8. unpow2 13.6%

      \[\leadsto e^{\sqrt{\color{blue}{{\log x}^{2}}}} \cdot \frac{1}{n} \]

   9. sqrt-pow1 4.6%

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

   10. metadata-eval 4.6%

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

   11. pow1 4.6%

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

   12. add-exp-log 4.6%

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

10. Applied egg-rr 4.6%

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

    1. associate-*r/ 4.6%

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

    2. *-rgt-identity 4.6%

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

12. Simplified 4.6%

    \[\leadsto \color{blue}{\frac{x}{n}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))