2nthrt (problem 3.4.6)

Percentage Accurate: 53.1% → 91.7%

Time: 1.2min

Alternatives: 15

Speedup: 1.8×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.1% 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: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1500:\\ \;\;\;\;\frac{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.5, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1500.0)
   (/
    (log
     (/
      x
      (exp
       (+
        (log1p x)
        (/
         (fma
          (- (pow (log1p x) 2.0) (pow (log x) 2.0))
          0.5
          (/
           (* 0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
           n))
         n)))))
    (- n))
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1500.0) {
		tmp = log((x / exp((log1p(x) + (fma((pow(log1p(x), 2.0) - pow(log(x), 2.0)), 0.5, ((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / n)) / n))))) / -n;
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1500.0)
		tmp = Float64(log(Float64(x / exp(Float64(log1p(x) + Float64(fma(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)), 0.5, Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / n)) / n))))) / Float64(-n));
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1500.0], N[(N[Log[N[(x / N[Exp[N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1500

   1. Initial program 41.5%

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

   3. Taylor expanded in n around -inf 76.4%

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

   5. Simplified 76.4%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}}{n}\right)}{-n}} \]
   6. Step-by-step derivation

      1. add-log-exp 84.3%

         \[\leadsto \frac{\log x - \color{blue}{\log \left(e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}}{n}}\right)}}{-n} \]

      2. diff-log 84.3%

         \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}}{n}}}\right)}}{-n} \]

   7. Applied egg-rr 84.3%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.5, \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}\right)}{n}}}\right)}}{-n} \]

   if 1500 < x

   1. Initial program 66.3%

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

   3. Taylor expanded in x around inf 95.7%

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

      1. associate-/r* 98.0%

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

      2. mul-1-neg 98.0%

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

      3. log-rec 98.0%

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

      4. mul-1-neg 98.0%

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

      5. distribute-neg-frac 98.0%

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

      6. mul-1-neg 98.0%

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

      7. remove-double-neg 98.0%

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

      8. *-rgt-identity 98.0%

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

      9. associate-/l* 98.0%

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

      10. exp-to-pow 98.0%

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

   5. Simplified 98.0%

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

Alternative 2: 87.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 14500:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n} - 0.5 \cdot \left({\log x}^{2} - {\left(\mathsf{log1p}\left(x\right)\right)}^{2}\right)}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 14500.0)
   (/
    (-
     (+
      (log1p x)
      (/
       (-
        (* 0.16666666666666666 (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n))
        (* 0.5 (- (pow (log x) 2.0) (pow (log1p x) 2.0))))
       n))
     (log x))
    n)
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 14500.0) {
		tmp = ((log1p(x) + (((0.16666666666666666 * ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n)) - (0.5 * (pow(log(x), 2.0) - pow(log1p(x), 2.0)))) / n)) - log(x)) / n;
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 14500.0) {
		tmp = ((Math.log1p(x) + (((0.16666666666666666 * ((Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0)) / n)) - (0.5 * (Math.pow(Math.log(x), 2.0) - Math.pow(Math.log1p(x), 2.0)))) / n)) - Math.log(x)) / n;
	} else {
		tmp = (Math.pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 14500.0:
		tmp = ((math.log1p(x) + (((0.16666666666666666 * ((math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0)) / n)) - (0.5 * (math.pow(math.log(x), 2.0) - math.pow(math.log1p(x), 2.0)))) / n)) - math.log(x)) / n
	else:
		tmp = (math.pow(x, (1.0 / n)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 14500.0)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(Float64(Float64(0.16666666666666666 * Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n)) - Float64(0.5 * Float64((log(x) ^ 2.0) - (log1p(x) ^ 2.0)))) / n)) - log(x)) / n);
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 14500.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(0.16666666666666666 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 14500

   1. Initial program 41.5%

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

   3. Taylor expanded in n around -inf 76.4%

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

   5. Simplified 76.4%

      \[\leadsto \color{blue}{\frac{\log x - \left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}}{n}\right)}{-n}} \]

   if 14500 < x

   1. Initial program 66.3%

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

   3. Taylor expanded in x around inf 95.7%

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

      1. associate-/r* 98.0%

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

      2. mul-1-neg 98.0%

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

      3. log-rec 98.0%

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

      4. mul-1-neg 98.0%

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

      5. distribute-neg-frac 98.0%

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

      6. mul-1-neg 98.0%

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

      7. remove-double-neg 98.0%

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

      8. *-rgt-identity 98.0%

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

      9. associate-/l* 98.0%

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

      10. exp-to-pow 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 14500:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n} - 0.5 \cdot \left({\log x}^{2} - {\left(\mathsf{log1p}\left(x\right)\right)}^{2}\right)}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-24}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-125)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-24)
       (/ (- (log1p x) (log x)) n)
       (exp (log (- (exp (/ (log1p x) n)) t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-125) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-24) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp(log((exp((log1p(x) / n)) - t_0)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-125) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-24) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp(Math.log((Math.exp((Math.log1p(x) / n)) - t_0)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-125:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-24:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp(math.log((math.exp((math.log1p(x) / n)) - t_0)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-125)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-24)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = exp(log(Float64(exp(Float64(log1p(x) / n)) - t_0)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-125], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-24], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[Exp[N[Log[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 #s(literal 1 binary64) n) < -2.00000000000000002e-125

   1. Initial program 75.9%

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

   3. Taylor expanded in x around inf 88.9%

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

      1. associate-/r* 88.9%

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

      2. mul-1-neg 88.9%

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

      3. log-rec 88.9%

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

      4. mul-1-neg 88.9%

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

      5. distribute-neg-frac 88.9%

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

      6. mul-1-neg 88.9%

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

      7. remove-double-neg 88.9%

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

      8. *-rgt-identity 88.9%

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

      9. associate-/l* 88.9%

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

      10. exp-to-pow 88.9%

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

   5. Simplified 88.9%

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

   if -2.00000000000000002e-125 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999924e-25

   1. Initial program 33.0%

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

   3. Taylor expanded in n around inf 81.0%

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

      1. log1p-define 81.0%

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

   5. Simplified 81.0%

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

   if 9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.2%

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

      1. add-exp-log 52.2%

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

      2. pow-to-exp 52.2%

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

      3. un-div-inv 52.2%

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

      4. +-commutative 52.2%

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

      5. log1p-define 98.9%

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

   4. Applied egg-rr 98.9%

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

Alternative 4: 84.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-24}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-125)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-24)
       (/ (- (log1p x) (log x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-125) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-24) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-125) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-24) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-125:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-24:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-125)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-24)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-125], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-24], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[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 #s(literal 1 binary64) n) < -2.00000000000000002e-125

   1. Initial program 75.9%

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

   3. Taylor expanded in x around inf 88.9%

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

      1. associate-/r* 88.9%

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

      2. mul-1-neg 88.9%

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

      3. log-rec 88.9%

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

      4. mul-1-neg 88.9%

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

      5. distribute-neg-frac 88.9%

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

      6. mul-1-neg 88.9%

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

      7. remove-double-neg 88.9%

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

      8. *-rgt-identity 88.9%

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

      9. associate-/l* 88.9%

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

      10. exp-to-pow 88.9%

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

   5. Simplified 88.9%

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

   if -2.00000000000000002e-125 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999924e-25

   1. Initial program 33.0%

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

   3. Taylor expanded in n around inf 81.0%

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

      1. log1p-define 81.0%

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

   5. Simplified 81.0%

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

   if 9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.2%

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

   3. Taylor expanded in n around 0 52.2%

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

      1. log1p-define 98.9%

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

      2. *-rgt-identity 98.9%

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

      3. associate-*l/ 98.9%

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

      4. associate-/l* 98.9%

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

      5. exp-to-pow 98.9%

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

   5. Simplified 98.9%

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

Alternative 5: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-24}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-125)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-24)
       (/ (- (log1p x) (log x)) n)
       (-
        (+
         1.0
         (*
          x
          (+
           (/ 1.0 n)
           (* x (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))))
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-125) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-24) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-125) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-24) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / Math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-125:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-24:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / math.pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-125)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-24)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(0.5 * Float64(1.0 / (n ^ 2.0))) + Float64(0.5 * Float64(-1.0 / n))))))) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-125], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-24], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(x * N[(N[(0.5 * N[(1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000002e-125

   1. Initial program 75.9%

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

   3. Taylor expanded in x around inf 88.9%

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

      1. associate-/r* 88.9%

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

      2. mul-1-neg 88.9%

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

      3. log-rec 88.9%

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

      4. mul-1-neg 88.9%

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

      5. distribute-neg-frac 88.9%

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

      6. mul-1-neg 88.9%

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

      7. remove-double-neg 88.9%

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

      8. *-rgt-identity 88.9%

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

      9. associate-/l* 88.9%

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

      10. exp-to-pow 88.9%

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

   5. Simplified 88.9%

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

   if -2.00000000000000002e-125 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999924e-25

   1. Initial program 33.0%

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

   3. Taylor expanded in n around inf 81.0%

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

      1. log1p-define 81.0%

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

   5. Simplified 81.0%

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

   if 9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.2%

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

   3. Taylor expanded in x around 0 91.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-24}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log x}{-n}\\ t_2 := \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \mathbf{if}\;x \leq 1.045 \cdot 10^{-286}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-209}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-117}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-26}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (/ (log x) (- n)))
        (t_2 (log1p (expm1 (/ 1.0 (* x n))))))
   (if (<= x 1.045e-286)
     t_1
     (if (<= x 8.2e-209)
       (- (+ 1.0 (/ x n)) t_0)
       (if (<= x 6.2e-149)
         t_1
         (if (<= x 2.7e-117)
           t_2
           (if (<= x 1.7e-85)
             t_1
             (if (<= x 1.72e-26)
               t_2
               (if (<= x 0.0135) t_1 (/ (/ t_0 n) x))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(x) / -n;
	double t_2 = log1p(expm1((1.0 / (x * n))));
	double tmp;
	if (x <= 1.045e-286) {
		tmp = t_1;
	} else if (x <= 8.2e-209) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 6.2e-149) {
		tmp = t_1;
	} else if (x <= 2.7e-117) {
		tmp = t_2;
	} else if (x <= 1.7e-85) {
		tmp = t_1;
	} else if (x <= 1.72e-26) {
		tmp = t_2;
	} else if (x <= 0.0135) {
		tmp = t_1;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(x) / -n;
	double t_2 = Math.log1p(Math.expm1((1.0 / (x * n))));
	double tmp;
	if (x <= 1.045e-286) {
		tmp = t_1;
	} else if (x <= 8.2e-209) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 6.2e-149) {
		tmp = t_1;
	} else if (x <= 2.7e-117) {
		tmp = t_2;
	} else if (x <= 1.7e-85) {
		tmp = t_1;
	} else if (x <= 1.72e-26) {
		tmp = t_2;
	} else if (x <= 0.0135) {
		tmp = t_1;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(x) / -n
	t_2 = math.log1p(math.expm1((1.0 / (x * n))))
	tmp = 0
	if x <= 1.045e-286:
		tmp = t_1
	elif x <= 8.2e-209:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 6.2e-149:
		tmp = t_1
	elif x <= 2.7e-117:
		tmp = t_2
	elif x <= 1.7e-85:
		tmp = t_1
	elif x <= 1.72e-26:
		tmp = t_2
	elif x <= 0.0135:
		tmp = t_1
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(x) / Float64(-n))
	t_2 = log1p(expm1(Float64(1.0 / Float64(x * n))))
	tmp = 0.0
	if (x <= 1.045e-286)
		tmp = t_1;
	elseif (x <= 8.2e-209)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 6.2e-149)
		tmp = t_1;
	elseif (x <= 2.7e-117)
		tmp = t_2;
	elseif (x <= 1.7e-85)
		tmp = t_1;
	elseif (x <= 1.72e-26)
		tmp = t_2;
	elseif (x <= 0.0135)
		tmp = t_1;
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$2 = N[Log[1 + N[(Exp[N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.045e-286], t$95$1, If[LessEqual[x, 8.2e-209], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 6.2e-149], t$95$1, If[LessEqual[x, 2.7e-117], t$95$2, If[LessEqual[x, 1.7e-85], t$95$1, If[LessEqual[x, 1.72e-26], t$95$2, If[LessEqual[x, 0.0135], t$95$1, N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.04500000000000002e-286 or 8.19999999999999955e-209 < x < 6.19999999999999974e-149 or 2.70000000000000003e-117 < x < 1.7e-85 or 1.72000000000000001e-26 < x < 0.0134999999999999998

   1. Initial program 26.7%

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

   3. Taylor expanded in x around 0 26.7%

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

      1. *-rgt-identity 26.7%

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

      2. associate-*l/ 26.7%

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

      3. associate-/l* 26.7%

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

      4. exp-to-pow 26.7%

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

   5. Simplified 26.7%

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

   6. Taylor expanded in n around inf 71.6%

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

      1. associate-*r/ 71.6%

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

      2. neg-mul-1 71.6%

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

   8. Simplified 71.6%

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

   if 1.04500000000000002e-286 < x < 8.19999999999999955e-209

   1. Initial program 61.4%

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

   3. Taylor expanded in x around 0 61.5%

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

   if 6.19999999999999974e-149 < x < 2.70000000000000003e-117 or 1.7e-85 < x < 1.72000000000000001e-26

   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 x around inf 34.9%

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

      1. mul-1-neg 34.9%

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

      2. log-rec 34.9%

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

      3. mul-1-neg 34.9%

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

      4. distribute-neg-frac 34.9%

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

      5. mul-1-neg 34.9%

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

      6. remove-double-neg 34.9%

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

      7. *-commutative 34.9%

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

   5. Simplified 34.9%

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

   6. Taylor expanded in n around inf 22.9%

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

      1. log1p-expm1-u 64.5%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]

   8. Applied egg-rr 64.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]

   if 0.0134999999999999998 < x

   1. Initial program 66.3%

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

   3. Taylor expanded in x around inf 95.7%

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

      1. associate-/r* 98.0%

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

      2. mul-1-neg 98.0%

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

      3. log-rec 98.0%

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

      4. mul-1-neg 98.0%

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

      5. distribute-neg-frac 98.0%

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

      6. mul-1-neg 98.0%

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

      7. remove-double-neg 98.0%

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

      8. *-rgt-identity 98.0%

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

      9. associate-/l* 98.0%

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

      10. exp-to-pow 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.045 \cdot 10^{-286}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-209}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-149}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-85}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-24}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-125)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-24)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 5e+141)
         (- (+ 1.0 (/ x n)) t_0)
         (log1p (expm1 (/ 1.0 (* x n)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-125) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-24) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 5e+141) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log1p(expm1((1.0 / (x * n))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-125) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-24) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 5e+141) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log1p(Math.expm1((1.0 / (x * n))));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-125:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-24:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 5e+141:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log1p(math.expm1((1.0 / (x * n))))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-125)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-24)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 5e+141)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = log1p(expm1(Float64(1.0 / Float64(x * n))));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-125], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-24], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+141], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000002e-125

   1. Initial program 75.9%

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

   3. Taylor expanded in x around inf 88.9%

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

      1. associate-/r* 88.9%

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

      2. mul-1-neg 88.9%

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

      3. log-rec 88.9%

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

      4. mul-1-neg 88.9%

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

      5. distribute-neg-frac 88.9%

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

      6. mul-1-neg 88.9%

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

      7. remove-double-neg 88.9%

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

      8. *-rgt-identity 88.9%

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

      9. associate-/l* 88.9%

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

      10. exp-to-pow 88.9%

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

   5. Simplified 88.9%

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

   if -2.00000000000000002e-125 < (/.f64 #s(literal 1 binary64) n) < 9.99999999999999924e-25

   1. Initial program 33.0%

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

   3. Taylor expanded in n around inf 81.0%

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

      1. log1p-define 81.0%

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

   5. Simplified 81.0%

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

   if 9.99999999999999924e-25 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000025e141

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0 91.0%

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

   if 5.00000000000000025e141 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 8.5%

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

   3. Taylor expanded in x around inf 0.7%

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

      1. mul-1-neg 0.7%

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

      2. log-rec 0.7%

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

      3. mul-1-neg 0.7%

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

      4. distribute-neg-frac 0.7%

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

      5. mul-1-neg 0.7%

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

      6. remove-double-neg 0.7%

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

      7. *-commutative 0.7%

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

   5. Simplified 0.7%

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

   6. Taylor expanded in n around inf 51.3%

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

      1. log1p-expm1-u 89.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]

   8. Applied egg-rr 89.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 70.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log x}{-n}\\ t_2 := \left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{if}\;x \leq 1.045 \cdot 10^{-286}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-209}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1 + \left(\frac{0.3333333333333333}{{x}^{2}} + \frac{-0.5}{x}\right)}{x \cdot n}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-30}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.013:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1 (/ (log x) (- n)))
        (t_2 (- (+ 1.0 (/ x n)) t_0)))
   (if (<= x 1.045e-286)
     t_1
     (if (<= x 2.4e-209)
       t_2
       (if (<= x 7.5e-148)
         t_1
         (if (<= x 5.2e-114)
           (/
            (+ 1.0 (+ (/ 0.3333333333333333 (pow x 2.0)) (/ -0.5 x)))
            (* x n))
           (if (<= x 3.4e-44)
             t_1
             (if (<= x 2.25e-30)
               t_2
               (if (<= x 0.013) t_1 (/ (/ t_0 n) x))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(x) / -n;
	double t_2 = (1.0 + (x / n)) - t_0;
	double tmp;
	if (x <= 1.045e-286) {
		tmp = t_1;
	} else if (x <= 2.4e-209) {
		tmp = t_2;
	} else if (x <= 7.5e-148) {
		tmp = t_1;
	} else if (x <= 5.2e-114) {
		tmp = (1.0 + ((0.3333333333333333 / pow(x, 2.0)) + (-0.5 / x))) / (x * n);
	} else if (x <= 3.4e-44) {
		tmp = t_1;
	} else if (x <= 2.25e-30) {
		tmp = t_2;
	} else if (x <= 0.013) {
		tmp = t_1;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(x) / -n
    t_2 = (1.0d0 + (x / n)) - t_0
    if (x <= 1.045d-286) then
        tmp = t_1
    else if (x <= 2.4d-209) then
        tmp = t_2
    else if (x <= 7.5d-148) then
        tmp = t_1
    else if (x <= 5.2d-114) then
        tmp = (1.0d0 + ((0.3333333333333333d0 / (x ** 2.0d0)) + ((-0.5d0) / x))) / (x * n)
    else if (x <= 3.4d-44) then
        tmp = t_1
    else if (x <= 2.25d-30) then
        tmp = t_2
    else if (x <= 0.013d0) then
        tmp = t_1
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(x) / -n;
	double t_2 = (1.0 + (x / n)) - t_0;
	double tmp;
	if (x <= 1.045e-286) {
		tmp = t_1;
	} else if (x <= 2.4e-209) {
		tmp = t_2;
	} else if (x <= 7.5e-148) {
		tmp = t_1;
	} else if (x <= 5.2e-114) {
		tmp = (1.0 + ((0.3333333333333333 / Math.pow(x, 2.0)) + (-0.5 / x))) / (x * n);
	} else if (x <= 3.4e-44) {
		tmp = t_1;
	} else if (x <= 2.25e-30) {
		tmp = t_2;
	} else if (x <= 0.013) {
		tmp = t_1;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(x) / -n
	t_2 = (1.0 + (x / n)) - t_0
	tmp = 0
	if x <= 1.045e-286:
		tmp = t_1
	elif x <= 2.4e-209:
		tmp = t_2
	elif x <= 7.5e-148:
		tmp = t_1
	elif x <= 5.2e-114:
		tmp = (1.0 + ((0.3333333333333333 / math.pow(x, 2.0)) + (-0.5 / x))) / (x * n)
	elif x <= 3.4e-44:
		tmp = t_1
	elif x <= 2.25e-30:
		tmp = t_2
	elif x <= 0.013:
		tmp = t_1
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(x) / Float64(-n))
	t_2 = Float64(Float64(1.0 + Float64(x / n)) - t_0)
	tmp = 0.0
	if (x <= 1.045e-286)
		tmp = t_1;
	elseif (x <= 2.4e-209)
		tmp = t_2;
	elseif (x <= 7.5e-148)
		tmp = t_1;
	elseif (x <= 5.2e-114)
		tmp = Float64(Float64(1.0 + Float64(Float64(0.3333333333333333 / (x ^ 2.0)) + Float64(-0.5 / x))) / Float64(x * n));
	elseif (x <= 3.4e-44)
		tmp = t_1;
	elseif (x <= 2.25e-30)
		tmp = t_2;
	elseif (x <= 0.013)
		tmp = t_1;
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(x) / -n;
	t_2 = (1.0 + (x / n)) - t_0;
	tmp = 0.0;
	if (x <= 1.045e-286)
		tmp = t_1;
	elseif (x <= 2.4e-209)
		tmp = t_2;
	elseif (x <= 7.5e-148)
		tmp = t_1;
	elseif (x <= 5.2e-114)
		tmp = (1.0 + ((0.3333333333333333 / (x ^ 2.0)) + (-0.5 / x))) / (x * n);
	elseif (x <= 3.4e-44)
		tmp = t_1;
	elseif (x <= 2.25e-30)
		tmp = t_2;
	elseif (x <= 0.013)
		tmp = t_1;
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[x, 1.045e-286], t$95$1, If[LessEqual[x, 2.4e-209], t$95$2, If[LessEqual[x, 7.5e-148], t$95$1, If[LessEqual[x, 5.2e-114], N[(N[(1.0 + N[(N[(0.3333333333333333 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.4e-44], t$95$1, If[LessEqual[x, 2.25e-30], t$95$2, If[LessEqual[x, 0.013], t$95$1, N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.04500000000000002e-286 or 2.4000000000000001e-209 < x < 7.5000000000000005e-148 or 5.20000000000000026e-114 < x < 3.40000000000000016e-44 or 2.24999999999999984e-30 < x < 0.0129999999999999994

   1. Initial program 28.5%

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

   3. Taylor expanded in x around 0 28.5%

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

      1. *-rgt-identity 28.5%

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

      2. associate-*l/ 28.5%

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

      3. associate-/l* 28.5%

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

      4. exp-to-pow 28.5%

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

   5. Simplified 28.5%

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

   6. Taylor expanded in n around inf 64.6%

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

      1. associate-*r/ 64.6%

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

      2. neg-mul-1 64.6%

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

   8. Simplified 64.6%

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

   if 1.04500000000000002e-286 < x < 2.4000000000000001e-209 or 3.40000000000000016e-44 < x < 2.24999999999999984e-30

   1. Initial program 60.9%

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

   3. Taylor expanded in x around 0 61.3%

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

   if 7.5000000000000005e-148 < x < 5.20000000000000026e-114

   1. Initial program 49.3%

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

   3. Taylor expanded in x around inf 0.8%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(0.16666666666666666 \cdot \frac{1}{{n}^{3}} + 0.3333333333333333 \cdot \frac{1}{n}\right) - 0.5 \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 23.7%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + {x}^{\left(\frac{1}{n}\right)} \cdot \left(\frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x} + \frac{\frac{0.16666666666666666}{{n}^{3}} + \left(\frac{0.3333333333333333}{n} - \frac{0.5}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   6. Taylor expanded in n around inf 70.1%

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

      1. associate-*r/ 70.1%

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

      2. metadata-eval 70.1%

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

      3. associate-*r/ 70.1%

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

      4. metadata-eval 70.1%

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

   8. Simplified 70.1%

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

   9. Taylor expanded in n around inf 70.1%

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

       1. associate--l+ 70.1%

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

       2. associate-*r/ 70.1%

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

       3. metadata-eval 70.1%

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

       4. sub-neg 70.1%

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

       5. associate-*r/ 70.1%

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

       6. metadata-eval 70.1%

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

       7. distribute-neg-frac 70.1%

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

       8. metadata-eval 70.1%

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

       9. *-commutative 70.1%

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

   11. Simplified 70.1%

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

   if 0.0129999999999999994 < x

   1. Initial program 66.3%

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

   3. Taylor expanded in x around inf 95.7%

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

      1. associate-/r* 98.0%

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

      2. mul-1-neg 98.0%

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

      3. log-rec 98.0%

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

      4. mul-1-neg 98.0%

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

      5. distribute-neg-frac 98.0%

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

      6. mul-1-neg 98.0%

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

      7. remove-double-neg 98.0%

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

      8. *-rgt-identity 98.0%

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

      9. associate-/l* 98.0%

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

      10. exp-to-pow 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.045 \cdot 10^{-286}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-209}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-148}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1 + \left(\frac{0.3333333333333333}{{x}^{2}} + \frac{-0.5}{x}\right)}{x \cdot n}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-30}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.013:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 1.045 \cdot 10^{-286}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-209}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;x \leq 0.0145:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log x) (- n))))
   (if (<= x 1.045e-286)
     t_1
     (if (<= x 2.25e-209)
       (- (+ 1.0 (/ x n)) t_0)
       (if (<= x 1.5e-147)
         t_1
         (if (<= x 5.8e-119)
           (/ 0.3333333333333333 (* n (pow x 3.0)))
           (if (<= x 0.0145) t_1 (/ (/ t_0 n) x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(x) / -n;
	double tmp;
	if (x <= 1.045e-286) {
		tmp = t_1;
	} else if (x <= 2.25e-209) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 1.5e-147) {
		tmp = t_1;
	} else if (x <= 5.8e-119) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (x <= 0.0145) {
		tmp = t_1;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(x) / -n
    if (x <= 1.045d-286) then
        tmp = t_1
    else if (x <= 2.25d-209) then
        tmp = (1.0d0 + (x / n)) - t_0
    else if (x <= 1.5d-147) then
        tmp = t_1
    else if (x <= 5.8d-119) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (x <= 0.0145d0) then
        tmp = t_1
    else
        tmp = (t_0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(x) / -n;
	double tmp;
	if (x <= 1.045e-286) {
		tmp = t_1;
	} else if (x <= 2.25e-209) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 1.5e-147) {
		tmp = t_1;
	} else if (x <= 5.8e-119) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (x <= 0.0145) {
		tmp = t_1;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(x) / -n
	tmp = 0
	if x <= 1.045e-286:
		tmp = t_1
	elif x <= 2.25e-209:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 1.5e-147:
		tmp = t_1
	elif x <= 5.8e-119:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif x <= 0.0145:
		tmp = t_1
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 1.045e-286)
		tmp = t_1;
	elseif (x <= 2.25e-209)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 1.5e-147)
		tmp = t_1;
	elseif (x <= 5.8e-119)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (x <= 0.0145)
		tmp = t_1;
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(x) / -n;
	tmp = 0.0;
	if (x <= 1.045e-286)
		tmp = t_1;
	elseif (x <= 2.25e-209)
		tmp = (1.0 + (x / n)) - t_0;
	elseif (x <= 1.5e-147)
		tmp = t_1;
	elseif (x <= 5.8e-119)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (x <= 0.0145)
		tmp = t_1;
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 1.045e-286], t$95$1, If[LessEqual[x, 2.25e-209], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 1.5e-147], t$95$1, If[LessEqual[x, 5.8e-119], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0145], t$95$1, N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.04500000000000002e-286 or 2.2499999999999999e-209 < x < 1.5000000000000001e-147 or 5.8e-119 < x < 0.0145000000000000007

   1. Initial program 31.9%

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

   3. Taylor expanded in x around 0 31.9%

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

      1. *-rgt-identity 31.9%

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

      2. associate-*l/ 31.9%

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

      3. associate-/l* 31.9%

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

      4. exp-to-pow 31.9%

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

   5. Simplified 31.9%

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

   6. Taylor expanded in n around inf 59.4%

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

      1. associate-*r/ 59.4%

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

      2. neg-mul-1 59.4%

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

   8. Simplified 59.4%

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

   if 1.04500000000000002e-286 < x < 2.2499999999999999e-209

   1. Initial program 61.4%

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

   3. Taylor expanded in x around 0 61.5%

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

   if 1.5000000000000001e-147 < x < 5.8e-119

   1. Initial program 49.3%

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

   3. Taylor expanded in x around inf 0.8%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(0.16666666666666666 \cdot \frac{1}{{n}^{3}} + 0.3333333333333333 \cdot \frac{1}{n}\right) - 0.5 \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 23.7%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + {x}^{\left(\frac{1}{n}\right)} \cdot \left(\frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x} + \frac{\frac{0.16666666666666666}{{n}^{3}} + \left(\frac{0.3333333333333333}{n} - \frac{0.5}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   6. Taylor expanded in n around inf 70.1%

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

      1. associate-*r/ 70.1%

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

      2. metadata-eval 70.1%

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

      3. associate-*r/ 70.1%

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

      4. metadata-eval 70.1%

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

   8. Simplified 70.1%

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

   9. Taylor expanded in x around 0 69.9%

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

   if 0.0145000000000000007 < x

   1. Initial program 66.3%

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

   3. Taylor expanded in x around inf 95.7%

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

      1. associate-/r* 98.0%

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

      2. mul-1-neg 98.0%

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

      3. log-rec 98.0%

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

      4. mul-1-neg 98.0%

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

      5. distribute-neg-frac 98.0%

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

      6. mul-1-neg 98.0%

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

      7. remove-double-neg 98.0%

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

      8. *-rgt-identity 98.0%

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

      9. associate-/l* 98.0%

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

      10. exp-to-pow 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.045 \cdot 10^{-286}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-209}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;x \leq 0.0145:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.045 \cdot 10^{-286}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-209}:\\ \;\;\;\;1 - t\_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-119}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;x \leq 0.013:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (pow x (/ 1.0 n))))
   (if (<= x 1.045e-286)
     t_0
     (if (<= x 6.6e-209)
       (- 1.0 t_1)
       (if (<= x 1.5e-147)
         t_0
         (if (<= x 6e-119)
           (/ 0.3333333333333333 (* n (pow x 3.0)))
           (if (<= x 0.013) t_0 (/ (/ t_1 n) x))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.045e-286) {
		tmp = t_0;
	} else if (x <= 6.6e-209) {
		tmp = 1.0 - t_1;
	} else if (x <= 1.5e-147) {
		tmp = t_0;
	} else if (x <= 6e-119) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (x <= 0.013) {
		tmp = t_0;
	} else {
		tmp = (t_1 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = x ** (1.0d0 / n)
    if (x <= 1.045d-286) then
        tmp = t_0
    else if (x <= 6.6d-209) then
        tmp = 1.0d0 - t_1
    else if (x <= 1.5d-147) then
        tmp = t_0
    else if (x <= 6d-119) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (x <= 0.013d0) then
        tmp = t_0
    else
        tmp = (t_1 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.045e-286) {
		tmp = t_0;
	} else if (x <= 6.6e-209) {
		tmp = 1.0 - t_1;
	} else if (x <= 1.5e-147) {
		tmp = t_0;
	} else if (x <= 6e-119) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (x <= 0.013) {
		tmp = t_0;
	} else {
		tmp = (t_1 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.045e-286:
		tmp = t_0
	elif x <= 6.6e-209:
		tmp = 1.0 - t_1
	elif x <= 1.5e-147:
		tmp = t_0
	elif x <= 6e-119:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif x <= 0.013:
		tmp = t_0
	else:
		tmp = (t_1 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 1.045e-286)
		tmp = t_0;
	elseif (x <= 6.6e-209)
		tmp = Float64(1.0 - t_1);
	elseif (x <= 1.5e-147)
		tmp = t_0;
	elseif (x <= 6e-119)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (x <= 0.013)
		tmp = t_0;
	else
		tmp = Float64(Float64(t_1 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 1.045e-286)
		tmp = t_0;
	elseif (x <= 6.6e-209)
		tmp = 1.0 - t_1;
	elseif (x <= 1.5e-147)
		tmp = t_0;
	elseif (x <= 6e-119)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (x <= 0.013)
		tmp = t_0;
	else
		tmp = (t_1 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.045e-286], t$95$0, If[LessEqual[x, 6.6e-209], N[(1.0 - t$95$1), $MachinePrecision], If[LessEqual[x, 1.5e-147], t$95$0, If[LessEqual[x, 6e-119], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.013], t$95$0, N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.04500000000000002e-286 or 6.59999999999999948e-209 < x < 1.5000000000000001e-147 or 6.0000000000000004e-119 < x < 0.0129999999999999994

   1. Initial program 31.9%

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

   3. Taylor expanded in x around 0 31.9%

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

      1. *-rgt-identity 31.9%

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

      2. associate-*l/ 31.9%

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

      3. associate-/l* 31.9%

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

      4. exp-to-pow 31.9%

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

   5. Simplified 31.9%

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

   6. Taylor expanded in n around inf 59.4%

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

      1. associate-*r/ 59.4%

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

      2. neg-mul-1 59.4%

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

   8. Simplified 59.4%

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

   if 1.04500000000000002e-286 < x < 6.59999999999999948e-209

   1. Initial program 61.4%

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

   3. Taylor expanded in x around 0 61.4%

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

      1. *-rgt-identity 61.4%

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

      2. associate-*l/ 61.4%

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

      3. associate-/l* 61.4%

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

      4. exp-to-pow 61.4%

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

   5. Simplified 61.4%

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

   if 1.5000000000000001e-147 < x < 6.0000000000000004e-119

   1. Initial program 49.3%

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

   3. Taylor expanded in x around inf 0.8%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(0.16666666666666666 \cdot \frac{1}{{n}^{3}} + 0.3333333333333333 \cdot \frac{1}{n}\right) - 0.5 \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 23.7%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + {x}^{\left(\frac{1}{n}\right)} \cdot \left(\frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x} + \frac{\frac{0.16666666666666666}{{n}^{3}} + \left(\frac{0.3333333333333333}{n} - \frac{0.5}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   6. Taylor expanded in n around inf 70.1%

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

      1. associate-*r/ 70.1%

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

      2. metadata-eval 70.1%

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

      3. associate-*r/ 70.1%

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

      4. metadata-eval 70.1%

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

   8. Simplified 70.1%

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

   9. Taylor expanded in x around 0 69.9%

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

   if 0.0129999999999999994 < x

   1. Initial program 66.3%

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

   3. Taylor expanded in x around inf 95.7%

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

      1. associate-/r* 98.0%

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

      2. mul-1-neg 98.0%

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

      3. log-rec 98.0%

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

      4. mul-1-neg 98.0%

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

      5. distribute-neg-frac 98.0%

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

      6. mul-1-neg 98.0%

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

      7. remove-double-neg 98.0%

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

      8. *-rgt-identity 98.0%

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

      9. associate-/l* 98.0%

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

      10. exp-to-pow 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.045 \cdot 10^{-286}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-209}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-119}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;x \leq 0.013:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.045 \cdot 10^{-286}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-209}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 1.045e-286)
     t_0
     (if (<= x 2.35e-209)
       t_1
       (if (<= x 3.4e-44)
         t_0
         (if (<= x 1.02e-30)
           t_1
           (if (<= x 0.68) t_0 (/ (/ (- 1.0 (/ 0.5 x)) n) x))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.045e-286) {
		tmp = t_0;
	} else if (x <= 2.35e-209) {
		tmp = t_1;
	} else if (x <= 3.4e-44) {
		tmp = t_0;
	} else if (x <= 1.02e-30) {
		tmp = t_1;
	} else if (x <= 0.68) {
		tmp = t_0;
	} else {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 1.045d-286) then
        tmp = t_0
    else if (x <= 2.35d-209) then
        tmp = t_1
    else if (x <= 3.4d-44) then
        tmp = t_0
    else if (x <= 1.02d-30) then
        tmp = t_1
    else if (x <= 0.68d0) then
        tmp = t_0
    else
        tmp = ((1.0d0 - (0.5d0 / x)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.045e-286) {
		tmp = t_0;
	} else if (x <= 2.35e-209) {
		tmp = t_1;
	} else if (x <= 3.4e-44) {
		tmp = t_0;
	} else if (x <= 1.02e-30) {
		tmp = t_1;
	} else if (x <= 0.68) {
		tmp = t_0;
	} else {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.045e-286:
		tmp = t_0
	elif x <= 2.35e-209:
		tmp = t_1
	elif x <= 3.4e-44:
		tmp = t_0
	elif x <= 1.02e-30:
		tmp = t_1
	elif x <= 0.68:
		tmp = t_0
	else:
		tmp = ((1.0 - (0.5 / x)) / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 1.045e-286)
		tmp = t_0;
	elseif (x <= 2.35e-209)
		tmp = t_1;
	elseif (x <= 3.4e-44)
		tmp = t_0;
	elseif (x <= 1.02e-30)
		tmp = t_1;
	elseif (x <= 0.68)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 1.045e-286)
		tmp = t_0;
	elseif (x <= 2.35e-209)
		tmp = t_1;
	elseif (x <= 3.4e-44)
		tmp = t_0;
	elseif (x <= 1.02e-30)
		tmp = t_1;
	elseif (x <= 0.68)
		tmp = t_0;
	else
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.045e-286], t$95$0, If[LessEqual[x, 2.35e-209], t$95$1, If[LessEqual[x, 3.4e-44], t$95$0, If[LessEqual[x, 1.02e-30], t$95$1, If[LessEqual[x, 0.68], t$95$0, N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.04500000000000002e-286 or 2.35e-209 < x < 3.40000000000000016e-44 or 1.0199999999999999e-30 < x < 0.680000000000000049

   1. Initial program 31.4%

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

   3. Taylor expanded in x around 0 31.4%

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

      1. *-rgt-identity 31.4%

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

      2. associate-*l/ 31.4%

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

      3. associate-/l* 31.4%

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

      4. exp-to-pow 31.4%

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

   5. Simplified 31.4%

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

   6. Taylor expanded in n around inf 59.3%

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

      1. associate-*r/ 59.3%

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

      2. neg-mul-1 59.3%

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

   8. Simplified 59.3%

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

   if 1.04500000000000002e-286 < x < 2.35e-209 or 3.40000000000000016e-44 < x < 1.0199999999999999e-30

   1. Initial program 60.9%

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

   3. Taylor expanded in x around 0 60.9%

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

      1. *-rgt-identity 60.9%

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

      2. associate-*l/ 60.9%

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

      3. associate-/l* 60.9%

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

      4. exp-to-pow 60.9%

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

   5. Simplified 60.9%

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

   if 0.680000000000000049 < x

   1. Initial program 66.3%

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

   3. Taylor expanded in x around inf 75.7%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(0.16666666666666666 \cdot \frac{1}{{n}^{3}} + 0.3333333333333333 \cdot \frac{1}{n}\right) - 0.5 \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 75.7%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + {x}^{\left(\frac{1}{n}\right)} \cdot \left(\frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x} + \frac{\frac{0.16666666666666666}{{n}^{3}} + \left(\frac{0.3333333333333333}{n} - \frac{0.5}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   6. Taylor expanded in n around inf 77.7%

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

      1. associate-*r/ 77.7%

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

      2. metadata-eval 77.7%

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

      3. associate-*r/ 77.7%

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

      4. metadata-eval 77.7%

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

   8. Simplified 77.7%

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

   9. Taylor expanded in x around inf 77.6%

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

   10. Taylor expanded in n around inf 66.6%

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

       1. associate-*r/ 66.6%

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

       2. metadata-eval 66.6%

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

   12. Simplified 66.6%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.045 \cdot 10^{-286}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-209}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-30}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.8:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;n \leq 10^{-163}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{+15}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -5.8)
   (* (/ 1.0 n) (/ 1.0 x))
   (if (<= n 1e-163)
     (/ 0.3333333333333333 (* n (pow x 3.0)))
     (if (<= n 1.4e+15) (- 1.0 (pow x (/ 1.0 n))) (/ (/ 1.0 x) n)))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -5.8) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if (n <= 1e-163) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (n <= 1.4e+15) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} 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 <= (-5.8d0)) then
        tmp = (1.0d0 / n) * (1.0d0 / x)
    else if (n <= 1d-163) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (n <= 1.4d+15) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    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 <= -5.8) {
		tmp = (1.0 / n) * (1.0 / x);
	} else if (n <= 1e-163) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (n <= 1.4e+15) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -5.8:
		tmp = (1.0 / n) * (1.0 / x)
	elif n <= 1e-163:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif n <= 1.4e+15:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -5.8)
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	elseif (n <= 1e-163)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (n <= 1.4e+15)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -5.8)
		tmp = (1.0 / n) * (1.0 / x);
	elseif (n <= 1e-163)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (n <= 1.4e+15)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -5.8], N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1e-163], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.4e+15], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.79999999999999982

   1. Initial program 31.0%

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

   3. Taylor expanded in x around inf 55.5%

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

      1. mul-1-neg 55.5%

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

      2. log-rec 55.5%

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

      3. mul-1-neg 55.5%

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

      4. distribute-neg-frac 55.5%

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

      5. mul-1-neg 55.5%

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

      6. remove-double-neg 55.5%

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

      7. *-commutative 55.5%

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

   5. Simplified 55.5%

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

   6. Taylor expanded in n around inf 54.4%

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

      1. inv-pow 54.4%

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

      2. unpow-prod-down 56.1%

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

      3. inv-pow 56.1%

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

      4. inv-pow 56.1%

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

   8. Applied egg-rr 56.1%

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

   if -5.79999999999999982 < n < 9.99999999999999923e-164

   1. Initial program 80.5%

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

   3. Taylor expanded in x around inf 12.4%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(0.16666666666666666 \cdot \frac{1}{{n}^{3}} + 0.3333333333333333 \cdot \frac{1}{n}\right) - 0.5 \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 22.5%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + {x}^{\left(\frac{1}{n}\right)} \cdot \left(\frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x} + \frac{\frac{0.16666666666666666}{{n}^{3}} + \left(\frac{0.3333333333333333}{n} - \frac{0.5}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   6. Taylor expanded in n around inf 67.0%

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

      1. associate-*r/ 67.0%

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

      2. metadata-eval 67.0%

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

      3. associate-*r/ 67.0%

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

      4. metadata-eval 67.0%

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

   8. Simplified 67.0%

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

   9. Taylor expanded in x around 0 80.8%

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

   if 9.99999999999999923e-164 < n < 1.4e15

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0 88.3%

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

      1. *-rgt-identity 88.3%

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

      2. associate-*l/ 88.3%

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

      3. associate-/l* 88.3%

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

      4. exp-to-pow 88.3%

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

   5. Simplified 88.3%

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

   if 1.4e15 < n

   1. Initial program 28.8%

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

   3. Taylor expanded in x around inf 49.8%

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

      1. mul-1-neg 49.8%

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

      2. log-rec 49.8%

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

      3. mul-1-neg 49.8%

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

      4. distribute-neg-frac 49.8%

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

      5. mul-1-neg 49.8%

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

      6. remove-double-neg 49.8%

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

      7. *-commutative 49.8%

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

   5. Simplified 49.8%

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

   6. Taylor expanded in n around inf 49.8%

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

      1. *-commutative 49.8%

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

      2. associate-/r* 51.7%

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

   8. Simplified 51.7%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.8:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \mathbf{elif}\;n \leq 10^{-163}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{+15}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 1.045 \cdot 10^{-286}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 1.045e-286)
     t_0
     (if (<= x 7.2e-281)
       (/ 1.0 (* x n))
       (if (<= x 0.68) t_0 (/ (/ (- 1.0 (/ 0.5 x)) n) x))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 1.045e-286) {
		tmp = t_0;
	} else if (x <= 7.2e-281) {
		tmp = 1.0 / (x * n);
	} else if (x <= 0.68) {
		tmp = t_0;
	} else {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 1.045d-286) then
        tmp = t_0
    else if (x <= 7.2d-281) then
        tmp = 1.0d0 / (x * n)
    else if (x <= 0.68d0) then
        tmp = t_0
    else
        tmp = ((1.0d0 - (0.5d0 / x)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 1.045e-286) {
		tmp = t_0;
	} else if (x <= 7.2e-281) {
		tmp = 1.0 / (x * n);
	} else if (x <= 0.68) {
		tmp = t_0;
	} else {
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 1.045e-286:
		tmp = t_0
	elif x <= 7.2e-281:
		tmp = 1.0 / (x * n)
	elif x <= 0.68:
		tmp = t_0
	else:
		tmp = ((1.0 - (0.5 / x)) / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 1.045e-286)
		tmp = t_0;
	elseif (x <= 7.2e-281)
		tmp = Float64(1.0 / Float64(x * n));
	elseif (x <= 0.68)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 1.045e-286)
		tmp = t_0;
	elseif (x <= 7.2e-281)
		tmp = 1.0 / (x * n);
	elseif (x <= 0.68)
		tmp = t_0;
	else
		tmp = ((1.0 - (0.5 / x)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 1.045e-286], t$95$0, If[LessEqual[x, 7.2e-281], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.68], t$95$0, N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.04500000000000002e-286 or 7.20000000000000013e-281 < x < 0.680000000000000049

   1. Initial program 38.9%

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

   3. Taylor expanded in x around 0 38.9%

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

      1. *-rgt-identity 38.9%

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

      2. associate-*l/ 38.9%

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

      3. associate-/l* 38.9%

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

      4. exp-to-pow 38.9%

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

   5. Simplified 38.9%

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

   6. Taylor expanded in n around inf 51.3%

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

      1. associate-*r/ 51.3%

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

      2. neg-mul-1 51.3%

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

   8. Simplified 51.3%

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

   if 1.04500000000000002e-286 < x < 7.20000000000000013e-281

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 83.3%

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

      1. mul-1-neg 83.3%

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

      2. log-rec 83.3%

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

      3. mul-1-neg 83.3%

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

      4. distribute-neg-frac 83.3%

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

      5. mul-1-neg 83.3%

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

      6. remove-double-neg 83.3%

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

      7. *-commutative 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in n around inf 83.9%

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

   if 0.680000000000000049 < x

   1. Initial program 66.3%

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

   3. Taylor expanded in x around inf 75.7%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(0.16666666666666666 \cdot \frac{1}{{n}^{3}} + 0.3333333333333333 \cdot \frac{1}{n}\right) - 0.5 \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 75.7%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + {x}^{\left(\frac{1}{n}\right)} \cdot \left(\frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x} + \frac{\frac{0.16666666666666666}{{n}^{3}} + \left(\frac{0.3333333333333333}{n} - \frac{0.5}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   6. Taylor expanded in n around inf 77.7%

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

      1. associate-*r/ 77.7%

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

      2. metadata-eval 77.7%

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

      3. associate-*r/ 77.7%

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

      4. metadata-eval 77.7%

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

   8. Simplified 77.7%

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

   9. Taylor expanded in x around inf 77.6%

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

   10. Taylor expanded in n around inf 66.6%

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

       1. associate-*r/ 66.6%

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

       2. metadata-eval 66.6%

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

   12. Simplified 66.6%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.045 \cdot 10^{-286}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 40.6% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

3. Taylor expanded in x around inf 57.3%

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

   1. mul-1-neg 57.3%

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

   2. log-rec 57.3%

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

   3. mul-1-neg 57.3%

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

   4. distribute-neg-frac 57.3%

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

   5. mul-1-neg 57.3%

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

   6. remove-double-neg 57.3%

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

   7. *-commutative 57.3%

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

5. Simplified 57.3%

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

6. Taylor expanded in n around inf 41.0%

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

   1. *-commutative 41.0%

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

   2. associate-/r* 42.0%

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

8. Simplified 42.0%

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

Alternative 15: 40.0% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

3. Taylor expanded in x around inf 57.3%

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

   1. mul-1-neg 57.3%

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

   2. log-rec 57.3%

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

   3. mul-1-neg 57.3%

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

   4. distribute-neg-frac 57.3%

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

   5. mul-1-neg 57.3%

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

   6. remove-double-neg 57.3%

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

   7. *-commutative 57.3%

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

5. Simplified 57.3%

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

6. Taylor expanded in n around inf 41.0%

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

Reproduce

herbie shell --seed 2024086 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))