2nthrt (problem 3.4.6)

Percentage Accurate: 53.2% → 85.7%

Time: 28.7s

Alternatives: 20

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + x\right)\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{t_1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{t_0 - \log x}{n} + \left(\frac{-0.16666666666666666 \cdot {\log x}^{3} - -0.16666666666666666 \cdot {t_0}^{3}}{{n}^{3}} + \left(0.041666666666666664 \cdot \frac{{t_0}^{4}}{{n}^{4}} + 0.5 \cdot \frac{{t_0}^{2}}{{n}^{2}}\right)\right)\right) - \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (+ 1.0 x))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-42)
     (/ t_1 (* n x))
     (if (<= (/ 1.0 n) 5e-7)
       (-
        (+
         (/ (- t_0 (log x)) n)
         (+
          (/
           (-
            (* -0.16666666666666666 (pow (log x) 3.0))
            (* -0.16666666666666666 (pow t_0 3.0)))
           (pow n 3.0))
          (+
           (* 0.041666666666666664 (/ (pow t_0 4.0) (pow n 4.0)))
           (* 0.5 (/ (pow t_0 2.0) (pow n 2.0))))))
        (+
         (* 0.041666666666666664 (/ (pow (log x) 4.0) (pow n 4.0)))
         (* 0.5 (/ (pow (log x) 2.0) (pow n 2.0)))))
       (- (exp (/ (log1p x) n)) t_1)))))

C

double code(double x, double n) {
	double t_0 = log((1.0 + x));
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-42) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 5e-7) {
		tmp = (((t_0 - log(x)) / n) + ((((-0.16666666666666666 * pow(log(x), 3.0)) - (-0.16666666666666666 * pow(t_0, 3.0))) / pow(n, 3.0)) + ((0.041666666666666664 * (pow(t_0, 4.0) / pow(n, 4.0))) + (0.5 * (pow(t_0, 2.0) / pow(n, 2.0)))))) - ((0.041666666666666664 * (pow(log(x), 4.0) / pow(n, 4.0))) + (0.5 * (pow(log(x), 2.0) / pow(n, 2.0))));
	} else {
		tmp = exp((log1p(x) / n)) - t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log((1.0 + x));
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-42) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 5e-7) {
		tmp = (((t_0 - Math.log(x)) / n) + ((((-0.16666666666666666 * Math.pow(Math.log(x), 3.0)) - (-0.16666666666666666 * Math.pow(t_0, 3.0))) / Math.pow(n, 3.0)) + ((0.041666666666666664 * (Math.pow(t_0, 4.0) / Math.pow(n, 4.0))) + (0.5 * (Math.pow(t_0, 2.0) / Math.pow(n, 2.0)))))) - ((0.041666666666666664 * (Math.pow(Math.log(x), 4.0) / Math.pow(n, 4.0))) + (0.5 * (Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0))));
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log((1.0 + x))
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-42:
		tmp = t_1 / (n * x)
	elif (1.0 / n) <= 5e-7:
		tmp = (((t_0 - math.log(x)) / n) + ((((-0.16666666666666666 * math.pow(math.log(x), 3.0)) - (-0.16666666666666666 * math.pow(t_0, 3.0))) / math.pow(n, 3.0)) + ((0.041666666666666664 * (math.pow(t_0, 4.0) / math.pow(n, 4.0))) + (0.5 * (math.pow(t_0, 2.0) / math.pow(n, 2.0)))))) - ((0.041666666666666664 * (math.pow(math.log(x), 4.0) / math.pow(n, 4.0))) + (0.5 * (math.pow(math.log(x), 2.0) / math.pow(n, 2.0))))
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_1
	return tmp

Julia

function code(x, n)
	t_0 = log(Float64(1.0 + x))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-42)
		tmp = Float64(t_1 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-7)
		tmp = Float64(Float64(Float64(Float64(t_0 - log(x)) / n) + Float64(Float64(Float64(Float64(-0.16666666666666666 * (log(x) ^ 3.0)) - Float64(-0.16666666666666666 * (t_0 ^ 3.0))) / (n ^ 3.0)) + Float64(Float64(0.041666666666666664 * Float64((t_0 ^ 4.0) / (n ^ 4.0))) + Float64(0.5 * Float64((t_0 ^ 2.0) / (n ^ 2.0)))))) - Float64(Float64(0.041666666666666664 * Float64((log(x) ^ 4.0) / (n ^ 4.0))) + Float64(0.5 * Float64((log(x) ^ 2.0) / (n ^ 2.0)))));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_1);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-42], N[(t$95$1 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-7], N[(N[(N[(N[(t$95$0 - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[(N[(N[(-0.16666666666666666 * N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - N[(-0.16666666666666666 * N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.041666666666666664 * N[(N[Power[t$95$0, 4.0], $MachinePrecision] / N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.041666666666666664 * N[(N[Power[N[Log[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -1.00000000000000004e-42

   1. Initial program 80.3%

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

   3. Taylor expanded in x around inf 33.2%

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

      1. mul-1-neg 33.2%

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

      2. log-rec 33.2%

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

      3. mul-1-neg 33.2%

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

      4. distribute-neg-frac 33.2%

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

      5. mul-1-neg 33.2%

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

      6. remove-double-neg 33.2%

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

      7. *-commutative 33.2%

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

      8. associate-/l* 33.2%

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

      9. associate-/r/ 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in n around inf 46.1%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{x \cdot n} + \color{blue}{\frac{-0.5}{n \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. associate-/r* 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in x around inf 91.4%

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

       1. mul-1-neg 91.4%

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

       2. log-rec 91.4%

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

       3. distribute-frac-neg 91.4%

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

       4. remove-double-neg 91.4%

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

       5. *-rgt-identity 91.4%

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

       6. associate-*r/ 91.4%

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

       7. exp-to-pow 91.4%

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

       8. *-commutative 91.4%

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

   11. Simplified 91.4%

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

   if -1.00000000000000004e-42 < (/.f64 1 n) < 4.99999999999999977e-7

   1. Initial program 27.5%

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

   3. Taylor expanded in n around -inf 83.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \log \left(1 + x\right) - -1 \cdot \log x}{n} + \left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{{n}^{3}} + \left(0.041666666666666664 \cdot \frac{{\log \left(1 + x\right)}^{4}}{{n}^{4}} + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}}\right)\right)\right) - \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right)} \]

   if 4.99999999999999977e-7 < (/.f64 1 n)

   1. Initial program 45.5%

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

   3. Taylor expanded in n around 0 45.5%

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

      1. log1p-def 98.1%

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

   5. Simplified 98.1%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\frac{\log \left(1 + x\right) - \log x}{n} + \left(\frac{-0.16666666666666666 \cdot {\log x}^{3} - -0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3}}{{n}^{3}} + \left(0.041666666666666664 \cdot \frac{{\log \left(1 + x\right)}^{4}}{{n}^{4}} + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}}\right)\right)\right) - \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + x\right)\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{t_1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\left(0.5 \cdot \frac{{t_0}^{2}}{{n}^{2}} + \frac{t_0}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (+ 1.0 x))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-42)
     (/ t_1 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (-
        (+ (* 0.5 (/ (pow t_0 2.0) (pow n 2.0))) (/ t_0 n))
        (+ (* 0.5 (/ (pow (log x) 2.0) (pow n 2.0))) (/ (log x) n)))
       (log (exp (- (exp (/ (log1p x) n)) t_1)))))))

C

double code(double x, double n) {
	double t_0 = log((1.0 + x));
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-42) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = ((0.5 * (pow(t_0, 2.0) / pow(n, 2.0))) + (t_0 / n)) - ((0.5 * (pow(log(x), 2.0) / pow(n, 2.0))) + (log(x) / n));
	} else {
		tmp = log(exp((exp((log1p(x) / n)) - t_1)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log((1.0 + x));
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-42) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = ((0.5 * (Math.pow(t_0, 2.0) / Math.pow(n, 2.0))) + (t_0 / n)) - ((0.5 * (Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0))) + (Math.log(x) / n));
	} else {
		tmp = Math.log(Math.exp((Math.exp((Math.log1p(x) / n)) - t_1)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log((1.0 + x))
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-42:
		tmp = t_1 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = ((0.5 * (math.pow(t_0, 2.0) / math.pow(n, 2.0))) + (t_0 / n)) - ((0.5 * (math.pow(math.log(x), 2.0) / math.pow(n, 2.0))) + (math.log(x) / n))
	else:
		tmp = math.log(math.exp((math.exp((math.log1p(x) / n)) - t_1)))
	return tmp

Julia

function code(x, n)
	t_0 = log(Float64(1.0 + x))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-42)
		tmp = Float64(t_1 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(Float64(Float64(0.5 * Float64((t_0 ^ 2.0) / (n ^ 2.0))) + Float64(t_0 / n)) - Float64(Float64(0.5 * Float64((log(x) ^ 2.0) / (n ^ 2.0))) + Float64(log(x) / n)));
	else
		tmp = log(exp(Float64(exp(Float64(log1p(x) / n)) - t_1)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-42], N[(t$95$1 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[(N[(N[(0.5 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -1.00000000000000004e-42

   1. Initial program 80.3%

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

   3. Taylor expanded in x around inf 33.2%

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

      1. mul-1-neg 33.2%

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

      2. log-rec 33.2%

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

      3. mul-1-neg 33.2%

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

      4. distribute-neg-frac 33.2%

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

      5. mul-1-neg 33.2%

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

      6. remove-double-neg 33.2%

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

      7. *-commutative 33.2%

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

      8. associate-/l* 33.2%

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

      9. associate-/r/ 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in n around inf 46.1%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{x \cdot n} + \color{blue}{\frac{-0.5}{n \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. associate-/r* 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in x around inf 91.4%

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

       1. mul-1-neg 91.4%

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

       2. log-rec 91.4%

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

       3. distribute-frac-neg 91.4%

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

       4. remove-double-neg 91.4%

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

       5. *-rgt-identity 91.4%

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

       6. associate-*r/ 91.4%

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

       7. exp-to-pow 91.4%

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

       8. *-commutative 91.4%

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

   11. Simplified 91.4%

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

   if -1.00000000000000004e-42 < (/.f64 1 n) < 1.99999999999999988e-11

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf 82.9%

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

   if 1.99999999999999988e-11 < (/.f64 1 n)

   1. Initial program 46.3%

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

      1. add-log-exp 46.3%

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

      2. add-exp-log 46.3%

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

      3. log-pow 46.3%

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

      4. +-commutative 46.3%

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

      5. log1p-udef 97.8%

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

      6. *-commutative 97.8%

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

      7. un-div-inv 97.8%

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

   4. Applied egg-rr 97.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right) + \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-42)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (+
        (*
         0.5
         (-
          (/ (pow (log1p x) 2.0) (pow n 2.0))
          (/ (pow (log x) 2.0) (pow n 2.0))))
        (/ (- (log1p x) (log x)) n))
       (log (exp (- (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) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = (0.5 * ((pow(log1p(x), 2.0) / pow(n, 2.0)) - (pow(log(x), 2.0) / pow(n, 2.0)))) + ((log1p(x) - log(x)) / n);
	} else {
		tmp = log(exp((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) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = (0.5 * ((Math.pow(Math.log1p(x), 2.0) / Math.pow(n, 2.0)) - (Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0)))) + ((Math.log1p(x) - Math.log(x)) / n);
	} else {
		tmp = Math.log(Math.exp((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) <= -1e-42:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = (0.5 * ((math.pow(math.log1p(x), 2.0) / math.pow(n, 2.0)) - (math.pow(math.log(x), 2.0) / math.pow(n, 2.0)))) + ((math.log1p(x) - math.log(x)) / n)
	else:
		tmp = math.log(math.exp((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) <= -1e-42)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) / (n ^ 2.0)) - Float64((log(x) ^ 2.0) / (n ^ 2.0)))) + Float64(Float64(log1p(x) - log(x)) / n));
	else
		tmp = log(exp(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], -1e-42], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[(N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -1.00000000000000004e-42

   1. Initial program 80.3%

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

   3. Taylor expanded in x around inf 33.2%

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

      1. mul-1-neg 33.2%

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

      2. log-rec 33.2%

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

      3. mul-1-neg 33.2%

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

      4. distribute-neg-frac 33.2%

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

      5. mul-1-neg 33.2%

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

      6. remove-double-neg 33.2%

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

      7. *-commutative 33.2%

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

      8. associate-/l* 33.2%

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

      9. associate-/r/ 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in n around inf 46.1%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{x \cdot n} + \color{blue}{\frac{-0.5}{n \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. associate-/r* 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in x around inf 91.4%

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

       1. mul-1-neg 91.4%

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

       2. log-rec 91.4%

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

       3. distribute-frac-neg 91.4%

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

       4. remove-double-neg 91.4%

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

       5. *-rgt-identity 91.4%

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

       6. associate-*r/ 91.4%

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

       7. exp-to-pow 91.4%

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

       8. *-commutative 91.4%

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

   11. Simplified 91.4%

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

   if -1.00000000000000004e-42 < (/.f64 1 n) < 1.99999999999999988e-11

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf 82.9%

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

      1. associate--l+ 80.1%

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

      2. +-commutative 80.1%

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

      3. associate--r+ 82.9%

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

      4. div-sub 82.9%

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

      5. remove-double-neg 82.9%

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

      6. mul-1-neg 82.9%

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

      7. distribute-lft-out-- 82.9%

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

      8. mul-1-neg 82.9%

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

      9. associate-*r/ 82.9%

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

   5. Simplified 82.9%

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

   if 1.99999999999999988e-11 < (/.f64 1 n)

   1. Initial program 46.3%

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

      1. add-log-exp 46.3%

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

      2. add-exp-log 46.3%

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

      3. log-pow 46.3%

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

      4. +-commutative 46.3%

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

      5. log1p-udef 97.8%

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

      6. *-commutative 97.8%

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

      7. un-div-inv 97.8%

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

   4. Applied egg-rr 97.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right) + \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n} + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{{n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-42)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (+
        (/ (- (log1p x) (log x)) n)
        (/ (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))) (pow n 2.0)))
       (log (exp (- (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) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = ((log1p(x) - log(x)) / n) + ((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) / pow(n, 2.0));
	} else {
		tmp = log(exp((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) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = ((Math.log1p(x) - Math.log(x)) / n) + ((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) / Math.pow(n, 2.0));
	} else {
		tmp = Math.log(Math.exp((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) <= -1e-42:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = ((math.log1p(x) - math.log(x)) / n) + ((0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0))) / math.pow(n, 2.0))
	else:
		tmp = math.log(math.exp((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) <= -1e-42)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(Float64(Float64(log1p(x) - log(x)) / n) + Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) / (n ^ 2.0)));
	else
		tmp = log(exp(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], -1e-42], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[(N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -1.00000000000000004e-42

   1. Initial program 80.3%

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

   3. Taylor expanded in x around inf 33.2%

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

      1. mul-1-neg 33.2%

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

      2. log-rec 33.2%

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

      3. mul-1-neg 33.2%

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

      4. distribute-neg-frac 33.2%

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

      5. mul-1-neg 33.2%

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

      6. remove-double-neg 33.2%

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

      7. *-commutative 33.2%

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

      8. associate-/l* 33.2%

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

      9. associate-/r/ 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in n around inf 46.1%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{x \cdot n} + \color{blue}{\frac{-0.5}{n \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. associate-/r* 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in x around inf 91.4%

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

       1. mul-1-neg 91.4%

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

       2. log-rec 91.4%

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

       3. distribute-frac-neg 91.4%

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

       4. remove-double-neg 91.4%

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

       5. *-rgt-identity 91.4%

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

       6. associate-*r/ 91.4%

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

       7. exp-to-pow 91.4%

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

       8. *-commutative 91.4%

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

   11. Simplified 91.4%

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

   if -1.00000000000000004e-42 < (/.f64 1 n) < 1.99999999999999988e-11

   1. Initial program 27.1%

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

   3. Taylor expanded in n around 0 27.1%

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

      1. log1p-def 27.1%

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

   5. Simplified 27.1%

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

   6. Taylor expanded in n around -inf 82.9%

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

      1. associate--l+ 82.9%

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

   8. Simplified 82.9%

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

   if 1.99999999999999988e-11 < (/.f64 1 n)

   1. Initial program 46.3%

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

      1. add-log-exp 46.3%

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

      2. add-exp-log 46.3%

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

      3. log-pow 46.3%

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

      4. +-commutative 46.3%

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

      5. log1p-udef 97.8%

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

      6. *-commutative 97.8%

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

      7. un-div-inv 97.8%

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

   4. Applied egg-rr 97.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n} + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{{n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-42)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (log (/ (+ 1.0 x) x)) n)
       (log (exp (- (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) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = log(exp((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) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.log(Math.exp((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) <= -1e-42:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = math.log(math.exp((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) <= -1e-42)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = log(exp(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], -1e-42], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[Log[N[Exp[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -1.00000000000000004e-42

   1. Initial program 80.3%

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

   3. Taylor expanded in x around inf 33.2%

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

      1. mul-1-neg 33.2%

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

      2. log-rec 33.2%

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

      3. mul-1-neg 33.2%

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

      4. distribute-neg-frac 33.2%

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

      5. mul-1-neg 33.2%

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

      6. remove-double-neg 33.2%

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

      7. *-commutative 33.2%

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

      8. associate-/l* 33.2%

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

      9. associate-/r/ 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in n around inf 46.1%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{x \cdot n} + \color{blue}{\frac{-0.5}{n \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. associate-/r* 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in x around inf 91.4%

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

       1. mul-1-neg 91.4%

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

       2. log-rec 91.4%

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

       3. distribute-frac-neg 91.4%

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

       4. remove-double-neg 91.4%

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

       5. *-rgt-identity 91.4%

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

       6. associate-*r/ 91.4%

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

       7. exp-to-pow 91.4%

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

       8. *-commutative 91.4%

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

   11. Simplified 91.4%

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

   if -1.00000000000000004e-42 < (/.f64 1 n) < 1.99999999999999988e-11

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf 82.7%

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

      1. diff-log 82.8%

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

   5. Applied egg-rr 82.8%

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

   if 1.99999999999999988e-11 < (/.f64 1 n)

   1. Initial program 46.3%

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

      1. add-log-exp 46.3%

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

      2. add-exp-log 46.3%

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

      3. log-pow 46.3%

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

      4. +-commutative 46.3%

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

      5. log1p-udef 97.8%

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

      6. *-commutative 97.8%

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

      7. un-div-inv 97.8%

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

   4. Applied egg-rr 97.8%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-42)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (log (/ (+ 1.0 x) 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) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = log(((1.0 + x) / 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) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = Math.log(((1.0 + x) / 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) <= -1e-42:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = math.log(((1.0 + x) / 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) <= -1e-42)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(log(Float64(Float64(1.0 + x) / 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], -1e-42], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $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 1 n) < -1.00000000000000004e-42

   1. Initial program 80.3%

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

   3. Taylor expanded in x around inf 33.2%

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

      1. mul-1-neg 33.2%

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

      2. log-rec 33.2%

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

      3. mul-1-neg 33.2%

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

      4. distribute-neg-frac 33.2%

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

      5. mul-1-neg 33.2%

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

      6. remove-double-neg 33.2%

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

      7. *-commutative 33.2%

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

      8. associate-/l* 33.2%

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

      9. associate-/r/ 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in n around inf 46.1%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{x \cdot n} + \color{blue}{\frac{-0.5}{n \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. associate-/r* 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in x around inf 91.4%

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

       1. mul-1-neg 91.4%

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

       2. log-rec 91.4%

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

       3. distribute-frac-neg 91.4%

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

       4. remove-double-neg 91.4%

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

       5. *-rgt-identity 91.4%

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

       6. associate-*r/ 91.4%

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

       7. exp-to-pow 91.4%

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

       8. *-commutative 91.4%

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

   11. Simplified 91.4%

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

   if -1.00000000000000004e-42 < (/.f64 1 n) < 1.99999999999999988e-11

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf 82.7%

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

      1. diff-log 82.8%

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

   5. Applied egg-rr 82.8%

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

   if 1.99999999999999988e-11 < (/.f64 1 n)

   1. Initial program 46.3%

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

   3. Taylor expanded in n around 0 46.3%

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

      1. log1p-def 97.8%

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

   5. Simplified 97.8%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+173}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-42)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 1e+173)
         (- (+ 1.0 (/ x n)) t_0)
         (sqrt (pow (* n x) -2.0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+173) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = sqrt(pow((n * x), -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-42)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-11) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 1d+173) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = sqrt(((n * x) ** (-2.0d0)))
    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) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+173) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.sqrt(Math.pow((n * x), -2.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-42:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e+173:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.sqrt(math.pow((n * x), -2.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-42)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+173)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = sqrt((Float64(n * x) ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-42)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-11)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 1e+173)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = sqrt(((n * x) ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-42], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+173], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Sqrt[N[Power[N[(n * x), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1.00000000000000004e-42

   1. Initial program 80.3%

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

   3. Taylor expanded in x around inf 33.2%

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

      1. mul-1-neg 33.2%

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

      2. log-rec 33.2%

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

      3. mul-1-neg 33.2%

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

      4. distribute-neg-frac 33.2%

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

      5. mul-1-neg 33.2%

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

      6. remove-double-neg 33.2%

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

      7. *-commutative 33.2%

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

      8. associate-/l* 33.2%

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

      9. associate-/r/ 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in n around inf 46.1%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{x \cdot n} + \color{blue}{\frac{-0.5}{n \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. associate-/r* 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in x around inf 91.4%

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

       1. mul-1-neg 91.4%

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

       2. log-rec 91.4%

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

       3. distribute-frac-neg 91.4%

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

       4. remove-double-neg 91.4%

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

       5. *-rgt-identity 91.4%

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

       6. associate-*r/ 91.4%

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

       7. exp-to-pow 91.4%

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

       8. *-commutative 91.4%

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

   11. Simplified 91.4%

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

   if -1.00000000000000004e-42 < (/.f64 1 n) < 1.99999999999999988e-11

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf 82.7%

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

      1. diff-log 82.8%

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

   5. Applied egg-rr 82.8%

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

   if 1.99999999999999988e-11 < (/.f64 1 n) < 1e173

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0 60.8%

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

   if 1e173 < (/.f64 1 n)

   1. Initial program 18.6%

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

   3. Taylor expanded in n around inf 6.2%

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

   4. Taylor expanded in x around inf 55.7%

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

      1. *-commutative 55.7%

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

   6. Simplified 55.7%

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

      1. add-sqr-sqrt 55.7%

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

      2. sqrt-unprod 86.2%

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

      3. inv-pow 86.2%

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

      4. inv-pow 86.2%

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

      5. pow-prod-up 86.2%

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

      6. metadata-eval 86.2%

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

   8. Applied egg-rr 86.2%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+173}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-42)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (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 tmp;
	if ((1.0 / n) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-42)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-11) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-42:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		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)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-42)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-42)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-11)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-42], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], 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 1 n) < -1.00000000000000004e-42

   1. Initial program 80.3%

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

   3. Taylor expanded in x around inf 33.2%

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

      1. mul-1-neg 33.2%

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

      2. log-rec 33.2%

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

      3. mul-1-neg 33.2%

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

      4. distribute-neg-frac 33.2%

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

      5. mul-1-neg 33.2%

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

      6. remove-double-neg 33.2%

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

      7. *-commutative 33.2%

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

      8. associate-/l* 33.2%

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

      9. associate-/r/ 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in n around inf 46.1%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{x \cdot n} + \color{blue}{\frac{-0.5}{n \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. associate-/r* 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in x around inf 91.4%

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

       1. mul-1-neg 91.4%

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

       2. log-rec 91.4%

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

       3. distribute-frac-neg 91.4%

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

       4. remove-double-neg 91.4%

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

       5. *-rgt-identity 91.4%

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

       6. associate-*r/ 91.4%

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

       7. exp-to-pow 91.4%

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

       8. *-commutative 91.4%

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

   11. Simplified 91.4%

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

   if -1.00000000000000004e-42 < (/.f64 1 n) < 1.99999999999999988e-11

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf 82.7%

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

      1. diff-log 82.8%

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

   5. Applied egg-rr 82.8%

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

   if 1.99999999999999988e-11 < (/.f64 1 n)

   1. Initial program 46.3%

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

   3. Taylor expanded in n around 0 46.3%

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

      1. log1p-def 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in x around 0 97.8%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+173}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-42)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 1e+173) (- (+ 1.0 (/ x n)) t_0) (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+173) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-42)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-11) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 1d+173) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+173) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-42:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e+173:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-42)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+173)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-42)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-11)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 1e+173)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-42], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+173], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1.00000000000000004e-42

   1. Initial program 80.3%

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

   3. Taylor expanded in x around inf 33.2%

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

      1. mul-1-neg 33.2%

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

      2. log-rec 33.2%

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

      3. mul-1-neg 33.2%

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

      4. distribute-neg-frac 33.2%

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

      5. mul-1-neg 33.2%

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

      6. remove-double-neg 33.2%

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

      7. *-commutative 33.2%

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

      8. associate-/l* 33.2%

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

      9. associate-/r/ 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in n around inf 46.1%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{x \cdot n} + \color{blue}{\frac{-0.5}{n \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. associate-/r* 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in x around inf 91.4%

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

       1. mul-1-neg 91.4%

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

       2. log-rec 91.4%

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

       3. distribute-frac-neg 91.4%

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

       4. remove-double-neg 91.4%

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

       5. *-rgt-identity 91.4%

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

       6. associate-*r/ 91.4%

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

       7. exp-to-pow 91.4%

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

       8. *-commutative 91.4%

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

   11. Simplified 91.4%

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

   if -1.00000000000000004e-42 < (/.f64 1 n) < 1.99999999999999988e-11

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf 82.7%

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

      1. diff-log 82.8%

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

   5. Applied egg-rr 82.8%

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

   if 1.99999999999999988e-11 < (/.f64 1 n) < 1e173

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0 60.8%

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

   if 1e173 < (/.f64 1 n)

   1. Initial program 18.6%

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

   3. Taylor expanded in n around inf 6.2%

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

   4. Taylor expanded in x around inf 55.7%

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

      1. *-commutative 55.7%

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

   6. Simplified 55.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+173}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+173}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -5e+43)
     t_0
     (if (<= (/ 1.0 n) 2e-11)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 1e+173) t_0 (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e+43) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-11) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+173) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-5d+43)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d-11) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 1d+173) then
        tmp = t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e+43) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e-11) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+173) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e+43:
		tmp = t_0
	elif (1.0 / n) <= 2e-11:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e+173:
		tmp = t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+43)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+173)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -5e+43)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e-11)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 1e+173)
		tmp = t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+43], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+173], t$95$0, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -5.0000000000000004e43 or 1.99999999999999988e-11 < (/.f64 1 n) < 1e173

   1. Initial program 87.3%

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

   3. Taylor expanded in x around 0 59.8%

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

   if -5.0000000000000004e43 < (/.f64 1 n) < 1.99999999999999988e-11

   1. Initial program 27.9%

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

   3. Taylor expanded in n around inf 77.7%

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

      1. diff-log 77.1%

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

   5. Applied egg-rr 77.1%

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

   if 1e173 < (/.f64 1 n)

   1. Initial program 18.6%

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

   3. Taylor expanded in n around inf 6.2%

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

   4. Taylor expanded in x around inf 55.7%

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

      1. *-commutative 55.7%

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

   6. Simplified 55.7%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+43}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+173}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+173}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-42)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 2e-11)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 1e+173) (- 1.0 t_0) (/ 1.0 (* n x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+173) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1d-42)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 2d-11) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 1d+173) then
        tmp = 1.0d0 - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-42) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 2e-11) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+173) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-42:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 2e-11:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e+173:
		tmp = 1.0 - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-42)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-11)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+173)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1e-42)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 2e-11)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 1e+173)
		tmp = 1.0 - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-42], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-11], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+173], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1.00000000000000004e-42

   1. Initial program 80.3%

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

   3. Taylor expanded in x around inf 33.2%

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

      1. mul-1-neg 33.2%

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

      2. log-rec 33.2%

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

      3. mul-1-neg 33.2%

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

      4. distribute-neg-frac 33.2%

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

      5. mul-1-neg 33.2%

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

      6. remove-double-neg 33.2%

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

      7. *-commutative 33.2%

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

      8. associate-/l* 33.2%

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

      9. associate-/r/ 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in n around inf 46.1%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{x \cdot n} + \color{blue}{\frac{-0.5}{n \cdot {x}^{2}}} \]
   7. Step-by-step derivation

      1. associate-/r* 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in x around inf 91.4%

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

       1. mul-1-neg 91.4%

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

       2. log-rec 91.4%

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

       3. distribute-frac-neg 91.4%

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

       4. remove-double-neg 91.4%

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

       5. *-rgt-identity 91.4%

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

       6. associate-*r/ 91.4%

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

       7. exp-to-pow 91.4%

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

       8. *-commutative 91.4%

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

   11. Simplified 91.4%

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

   if -1.00000000000000004e-42 < (/.f64 1 n) < 1.99999999999999988e-11

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf 82.7%

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

      1. diff-log 82.8%

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

   5. Applied egg-rr 82.8%

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

   if 1.99999999999999988e-11 < (/.f64 1 n) < 1e173

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0 59.2%

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

   if 1e173 < (/.f64 1 n)

   1. Initial program 18.6%

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

   3. Taylor expanded in n around inf 6.2%

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

   4. Taylor expanded in x around inf 55.7%

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

      1. *-commutative 55.7%

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

   6. Simplified 55.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+173}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-236}:\\ \;\;\;\;\frac{1}{\frac{-n}{\log x}}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-217}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.0132:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6.8e-236)
   (/ 1.0 (/ (- n) (log x)))
   (if (<= x 4.6e-217)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.0132) (- (/ x n) (/ (log x) n)) (/ (/ 1.0 n) x)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6.8e-236) {
		tmp = 1.0 / (-n / log(x));
	} else if (x <= 4.6e-217) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.0132) {
		tmp = (x / n) - (log(x) / n);
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 6.8d-236) then
        tmp = 1.0d0 / (-n / log(x))
    else if (x <= 4.6d-217) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.0132d0) then
        tmp = (x / n) - (log(x) / n)
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 6.8e-236) {
		tmp = 1.0 / (-n / Math.log(x));
	} else if (x <= 4.6e-217) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.0132) {
		tmp = (x / n) - (Math.log(x) / n);
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 6.8e-236:
		tmp = 1.0 / (-n / math.log(x))
	elif x <= 4.6e-217:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.0132:
		tmp = (x / n) - (math.log(x) / n)
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6.8e-236)
		tmp = Float64(1.0 / Float64(Float64(-n) / log(x)));
	elseif (x <= 4.6e-217)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.0132)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 6.8e-236)
		tmp = 1.0 / (-n / log(x));
	elseif (x <= 4.6e-217)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.0132)
		tmp = (x / n) - (log(x) / n);
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6.8e-236], N[(1.0 / N[((-n) / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.6e-217], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0132], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 6.79999999999999961e-236

   1. Initial program 45.9%

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

   3. Taylor expanded in n around inf 57.6%

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

      1. clear-num 57.6%

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

      2. inv-pow 57.6%

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

      3. log1p-udef 57.6%

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

   5. Applied egg-rr 57.6%

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

      1. unpow-1 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in x around 0 57.6%

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

      1. associate-*r/ 57.6%

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

      2. neg-mul-1 57.6%

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

   10. Simplified 57.6%

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

   if 6.79999999999999961e-236 < x < 4.6000000000000001e-217

   1. Initial program 71.7%

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

   3. Taylor expanded in x around 0 71.7%

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

   if 4.6000000000000001e-217 < x < 0.0132

   1. Initial program 29.5%

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

   3. Taylor expanded in n around inf 56.1%

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

   4. Taylor expanded in x around 0 56.1%

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

      1. neg-mul-1 56.1%

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

      2. +-commutative 56.1%

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

      3. unsub-neg 56.1%

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

   6. Simplified 56.1%

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

   if 0.0132 < x

   1. Initial program 63.1%

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

   3. Taylor expanded in n around inf 62.7%

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

   4. Taylor expanded in x around inf 67.1%

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

      1. *-commutative 67.1%

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

   6. Simplified 67.1%

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

   7. Taylor expanded in x around 0 67.1%

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

      1. associate-/r* 67.4%

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

   9. Simplified 67.4%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-236}:\\ \;\;\;\;\frac{1}{\frac{-n}{\log x}}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-217}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.0132:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.1 \cdot 10^{-236}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-218}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.0132:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 7.1e-236)
   (/ (- (log x)) n)
   (if (<= x 1.6e-218)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.0132) (/ (- x (log x)) n) (/ (/ 1.0 n) x)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 7.1e-236) {
		tmp = -log(x) / n;
	} else if (x <= 1.6e-218) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.0132) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 7.1d-236) then
        tmp = -log(x) / n
    else if (x <= 1.6d-218) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.0132d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 7.1e-236) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.6e-218) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.0132) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 7.1e-236:
		tmp = -math.log(x) / n
	elif x <= 1.6e-218:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.0132:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 7.1e-236)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.6e-218)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.0132)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 7.1e-236)
		tmp = -log(x) / n;
	elseif (x <= 1.6e-218)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.0132)
		tmp = (x - log(x)) / n;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 7.1e-236], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.6e-218], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0132], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 7.10000000000000002e-236

   1. Initial program 45.9%

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

   3. Taylor expanded in n around inf 57.6%

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

   4. Taylor expanded in x around 0 57.6%

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

      1. neg-mul-1 57.6%

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

   6. Simplified 57.6%

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

   if 7.10000000000000002e-236 < x < 1.6000000000000001e-218

   1. Initial program 71.7%

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

   3. Taylor expanded in x around 0 71.7%

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

   if 1.6000000000000001e-218 < x < 0.0132

   1. Initial program 29.5%

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

   3. Taylor expanded in n around inf 56.1%

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

   4. Taylor expanded in x around 0 56.1%

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

   if 0.0132 < x

   1. Initial program 63.1%

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

   3. Taylor expanded in n around inf 62.7%

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

   4. Taylor expanded in x around inf 67.1%

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

      1. *-commutative 67.1%

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

   6. Simplified 67.1%

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

   7. Taylor expanded in x around 0 67.1%

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

      1. associate-/r* 67.4%

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

   9. Simplified 67.4%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.1 \cdot 10^{-236}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-218}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.0132:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-238}:\\ \;\;\;\;\frac{1}{\frac{-n}{\log x}}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-220}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.0132:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.8e-238)
   (/ 1.0 (/ (- n) (log x)))
   (if (<= x 4e-220)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.0132) (/ (- x (log x)) n) (/ (/ 1.0 n) x)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.8e-238) {
		tmp = 1.0 / (-n / log(x));
	} else if (x <= 4e-220) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.0132) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.8d-238) then
        tmp = 1.0d0 / (-n / log(x))
    else if (x <= 4d-220) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.0132d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.8e-238) {
		tmp = 1.0 / (-n / Math.log(x));
	} else if (x <= 4e-220) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.0132) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.8e-238:
		tmp = 1.0 / (-n / math.log(x))
	elif x <= 4e-220:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.0132:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.8e-238)
		tmp = Float64(1.0 / Float64(Float64(-n) / log(x)));
	elseif (x <= 4e-220)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.0132)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.8e-238)
		tmp = 1.0 / (-n / log(x));
	elseif (x <= 4e-220)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.0132)
		tmp = (x - log(x)) / n;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.8e-238], N[(1.0 / N[((-n) / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e-220], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0132], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.80000000000000004e-238

   1. Initial program 45.9%

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

   3. Taylor expanded in n around inf 57.6%

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

      1. clear-num 57.6%

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

      2. inv-pow 57.6%

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

      3. log1p-udef 57.6%

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

   5. Applied egg-rr 57.6%

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

      1. unpow-1 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in x around 0 57.6%

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

      1. associate-*r/ 57.6%

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

      2. neg-mul-1 57.6%

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

   10. Simplified 57.6%

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

   if 2.80000000000000004e-238 < x < 3.99999999999999997e-220

   1. Initial program 71.7%

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

   3. Taylor expanded in x around 0 71.7%

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

   if 3.99999999999999997e-220 < x < 0.0132

   1. Initial program 29.5%

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

   3. Taylor expanded in n around inf 56.1%

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

   4. Taylor expanded in x around 0 56.1%

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

   if 0.0132 < x

   1. Initial program 63.1%

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

   3. Taylor expanded in n around inf 62.7%

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

   4. Taylor expanded in x around inf 67.1%

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

      1. *-commutative 67.1%

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

   6. Simplified 67.1%

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

   7. Taylor expanded in x around 0 67.1%

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

      1. associate-/r* 67.4%

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

   9. Simplified 67.4%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-238}:\\ \;\;\;\;\frac{1}{\frac{-n}{\log x}}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-220}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.0132:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0132:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.0132) (/ (- x (log x)) n) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.0132) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.0132d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.0132) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.0132:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.0132)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.0132)
		tmp = (x - log(x)) / n;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.0132], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0132

   1. Initial program 36.4%

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

   3. Taylor expanded in n around inf 53.7%

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

   4. Taylor expanded in x around 0 53.7%

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

   if 0.0132 < x

   1. Initial program 63.1%

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

   3. Taylor expanded in n around inf 62.7%

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

   4. Taylor expanded in x around inf 67.1%

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

      1. *-commutative 67.1%

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

   6. Simplified 67.1%

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

   7. Taylor expanded in x around 0 67.1%

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

      1. associate-/r* 67.4%

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

   9. Simplified 67.4%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0132:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0132:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.0132) (/ (- (log x)) n) (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.0132) {
		tmp = -log(x) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.0132d0) then
        tmp = -log(x) / n
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.0132) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.0132:
		tmp = -math.log(x) / n
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.0132)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.0132)
		tmp = -log(x) / n;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.0132], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0132

   1. Initial program 36.4%

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

   3. Taylor expanded in n around inf 53.7%

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

   4. Taylor expanded in x around 0 53.4%

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

      1. neg-mul-1 53.4%

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

   6. Simplified 53.4%

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

   if 0.0132 < x

   1. Initial program 63.1%

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

   3. Taylor expanded in n around inf 62.7%

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

   4. Taylor expanded in x around inf 67.1%

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

      1. *-commutative 67.1%

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

   6. Simplified 67.1%

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

   7. Taylor expanded in x around 0 67.1%

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

      1. associate-/r* 67.4%

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

   9. Simplified 67.4%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0132:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.5% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -400:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -400.0) (/ 1.0 (* n (+ x 0.5))) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if (n <= -400.0) {
		tmp = 1.0 / (n * (x + 0.5));
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-400.0d0)) then
        tmp = 1.0d0 / (n * (x + 0.5d0))
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -400.0) {
		tmp = 1.0 / (n * (x + 0.5));
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -400.0:
		tmp = 1.0 / (n * (x + 0.5))
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -400.0)
		tmp = Float64(1.0 / Float64(n * Float64(x + 0.5)));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -400.0)
		tmp = 1.0 / (n * (x + 0.5));
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -400.0], N[(1.0 / N[(n * N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -400

   1. Initial program 28.3%

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

   3. Taylor expanded in n around inf 77.8%

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

      1. clear-num 77.8%

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

      2. inv-pow 77.8%

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

      3. log1p-udef 77.8%

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

   5. Applied egg-rr 77.8%

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

      1. unpow-1 77.8%

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

   7. Simplified 77.8%

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

   8. Taylor expanded in x around inf 48.6%

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

      1. +-commutative 48.6%

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

      2. *-commutative 48.6%

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

      3. distribute-lft-out 48.6%

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

   10. Simplified 48.6%

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

   if -400 < n

   1. Initial program 54.7%

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

   3. Taylor expanded in n around inf 46.5%

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

   4. Taylor expanded in x around inf 36.5%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -400:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 40.8% 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 45.9%

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

3. Taylor expanded in n around inf 56.9%

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

4. Taylor expanded in x around inf 38.7%

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

   1. *-commutative 38.7%

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

6. Simplified 38.7%

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

7. Final simplification 38.7%

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

Alternative 19: 41.3% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.9%

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

3. Taylor expanded in n around inf 56.9%

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

4. Taylor expanded in x around inf 38.7%

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

   1. *-commutative 38.7%

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

6. Simplified 38.7%

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

7. Taylor expanded in x around 0 38.7%

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

   1. associate-/r* 38.8%

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

9. Simplified 38.8%

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

10. Final simplification 38.8%

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

Alternative 20: 3.2% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.9%

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

3. Taylor expanded in n around inf 56.9%

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

4. Taylor expanded in x around inf 38.7%

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

   1. *-commutative 38.7%

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

6. Simplified 38.7%

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

7. Taylor expanded in x around 0 38.7%

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

   1. associate-/r* 38.8%

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

9. Simplified 38.8%

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

    1. associate-/l/ 38.7%

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

    2. add-exp-log 15.2%

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

    3. exp-neg 15.2%

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

    4. add-sqr-sqrt 5.3%

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

    5. sqrt-unprod 5.8%

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

    6. sqr-neg 5.8%

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

    7. sqrt-unprod 0.5%

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

    8. add-sqr-sqrt 1.6%

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

    9. *-un-lft-identity 1.6%

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

    10. add-exp-log 3.3%

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

    11. *-commutative 3.3%

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

11. Applied egg-rr 3.3%

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

12. Final simplification 3.3%

    \[\leadsto n \cdot x \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024022 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))