2nthrt (problem 3.4.6)

Percentage Accurate: 54.4% → 85.3%

Time: 1.1min

Alternatives: 21

Speedup: 1.6×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{t\_0}{n} + t\_0 \cdot \frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-77)
     (/ (pow x (+ (/ 1.0 n) -1.0)) n)
     (if (<= (/ 1.0 n) 5e-80)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 2e-5)
         (/ (+ (/ t_0 n) (* t_0 (/ (+ (/ 0.5 (pow n 2.0)) (/ -0.5 n)) x))) x)
         (- (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-77) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = ((t_0 / n) + (t_0 * (((0.5 / pow(n, 2.0)) + (-0.5 / n)) / x))) / x;
	} 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-77) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = ((t_0 / n) + (t_0 * (((0.5 / Math.pow(n, 2.0)) + (-0.5 / n)) / x))) / x;
	} 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-77:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 5e-80:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e-5:
		tmp = ((t_0 / n) + (t_0 * (((0.5 / math.pow(n, 2.0)) + (-0.5 / n)) / x))) / x
	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-77)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 5e-80)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = Float64(Float64(Float64(t_0 / n) + Float64(t_0 * Float64(Float64(Float64(0.5 / (n ^ 2.0)) + Float64(-0.5 / n)) / x))) / x);
	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-77], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-80], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], N[(N[(N[(t$95$0 / n), $MachinePrecision] + N[(t$95$0 * N[(N[(N[(0.5 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-77

   1. Initial program 85.2%

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

   3. Taylor expanded in x around inf 91.8%

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

      1. mul-1-neg 91.8%

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

      2. log-rec 91.8%

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

      3. mul-1-neg 91.8%

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

      4. distribute-neg-frac 91.8%

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

      5. mul-1-neg 91.8%

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

      6. remove-double-neg 91.8%

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

      7. *-commutative 91.8%

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

   5. Simplified 91.8%

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

      1. *-un-lft-identity 91.8%

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

      2. associate-/r* 92.6%

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

      3. div-inv 92.6%

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

      4. pow-to-exp 92.6%

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

      5. pow1 92.6%

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

      6. pow-div 92.6%

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

   7. Applied egg-rr 92.6%

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

      1. *-lft-identity 92.6%

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

      2. sub-neg 92.6%

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

      3. metadata-eval 92.6%

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

   9. Simplified 92.6%

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

   if -1.9999999999999999e-77 < (/.f64 #s(literal 1 binary64) n) < 5e-80

   1. Initial program 33.4%

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

   3. Taylor expanded in n around inf 88.2%

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

      1. log1p-define 88.2%

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

   5. Simplified 88.2%

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

   if 5e-80 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 13.2%

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

      1. add-log-exp 12.9%

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

      2. pow-to-exp 12.9%

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

      3. un-div-inv 12.9%

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

      4. +-commutative 12.9%

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

      5. log1p-define 12.9%

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

   4. Applied egg-rr 12.9%

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

   5. Taylor expanded in x around inf 85.0%

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

   7. Simplified 85.0%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.8%

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

   3. Taylor expanded in n around 0 67.8%

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

      1. log1p-define 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n} + {x}^{\left(\frac{1}{n}\right)} \cdot \frac{\frac{0.5}{{n}^{2}} + \frac{-0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.5)
   (/
    (-
     (+
      (log1p x)
      (/
       (+
        (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0)))
        (*
         0.16666666666666666
         (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n)))
       n))
     (log x))
    n)
   (/ (pow x (+ (/ 1.0 n) -1.0)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.5) {
		tmp = ((log1p(x) + (((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) + (0.16666666666666666 * ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n))) / n)) - log(x)) / n;
	} else {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.5) {
		tmp = ((Math.log1p(x) + (((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) + (0.16666666666666666 * ((Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0)) / n))) / n)) - Math.log(x)) / n;
	} else {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.5:
		tmp = ((math.log1p(x) + (((0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0))) + (0.16666666666666666 * ((math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0)) / n))) / n)) - math.log(x)) / n
	else:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.5)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) + Float64(0.16666666666666666 * Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n))) / n)) - log(x)) / n);
	else
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.5], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(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]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 40.8%

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

   3. Taylor expanded in n around -inf 82.0%

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

   5. Simplified 82.0%

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

   if 3.5 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   5. Simplified 97.7%

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

      1. *-un-lft-identity 97.7%

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

      2. associate-/r* 99.2%

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

      3. div-inv 99.2%

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

      4. pow-to-exp 99.2%

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

      5. pow1 99.2%

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

      6. pow-div 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. *-lft-identity 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.52:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{2}} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.52)
   (/
    (-
     (* -0.16666666666666666 (/ (pow (log x) 3.0) (pow n 2.0)))
     (+ (log x) (* 0.5 (/ (pow (log x) 2.0) n))))
    n)
   (/ (pow x (+ (/ 1.0 n) -1.0)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.52) {
		tmp = ((-0.16666666666666666 * (pow(log(x), 3.0) / pow(n, 2.0))) - (log(x) + (0.5 * (pow(log(x), 2.0) / n)))) / n;
	} else {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.52d0) then
        tmp = (((-0.16666666666666666d0) * ((log(x) ** 3.0d0) / (n ** 2.0d0))) - (log(x) + (0.5d0 * ((log(x) ** 2.0d0) / n)))) / n
    else
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.52) {
		tmp = ((-0.16666666666666666 * (Math.pow(Math.log(x), 3.0) / Math.pow(n, 2.0))) - (Math.log(x) + (0.5 * (Math.pow(Math.log(x), 2.0) / n)))) / n;
	} else {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.52:
		tmp = ((-0.16666666666666666 * (math.pow(math.log(x), 3.0) / math.pow(n, 2.0))) - (math.log(x) + (0.5 * (math.pow(math.log(x), 2.0) / n)))) / n
	else:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.52)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * Float64((log(x) ^ 3.0) / (n ^ 2.0))) - Float64(log(x) + Float64(0.5 * Float64((log(x) ^ 2.0) / n)))) / n);
	else
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.52)
		tmp = ((-0.16666666666666666 * ((log(x) ^ 3.0) / (n ^ 2.0))) - (log(x) + (0.5 * ((log(x) ^ 2.0) / n)))) / n;
	else
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.52], N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] + N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.52000000000000002

   1. Initial program 40.8%

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

   3. Taylor expanded in x around 0 40.1%

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

   4. Taylor expanded in n around inf 81.4%

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

   if 0.52000000000000002 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   5. Simplified 97.7%

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

      1. *-un-lft-identity 97.7%

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

      2. associate-/r* 99.2%

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

      3. div-inv 99.2%

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

      4. pow-to-exp 99.2%

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

      5. pow1 99.2%

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

      6. pow-div 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. *-lft-identity 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.52:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{{n}^{2}} - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.24:\\ \;\;\;\;\frac{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + {\log x}^{2} \cdot -0.5}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.24)
   (/
    (-
     (/
      (+
       (* -0.16666666666666666 (/ (pow (log x) 3.0) n))
       (* (pow (log x) 2.0) -0.5))
      n)
     (log x))
    n)
   (/ (pow x (+ (/ 1.0 n) -1.0)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.24) {
		tmp = ((((-0.16666666666666666 * (pow(log(x), 3.0) / n)) + (pow(log(x), 2.0) * -0.5)) / n) - log(x)) / n;
	} else {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.24d0) then
        tmp = (((((-0.16666666666666666d0) * ((log(x) ** 3.0d0) / n)) + ((log(x) ** 2.0d0) * (-0.5d0))) / n) - log(x)) / n
    else
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.24) {
		tmp = ((((-0.16666666666666666 * (Math.pow(Math.log(x), 3.0) / n)) + (Math.pow(Math.log(x), 2.0) * -0.5)) / n) - Math.log(x)) / n;
	} else {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.24:
		tmp = ((((-0.16666666666666666 * (math.pow(math.log(x), 3.0) / n)) + (math.pow(math.log(x), 2.0) * -0.5)) / n) - math.log(x)) / n
	else:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.24)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.16666666666666666 * Float64((log(x) ^ 3.0) / n)) + Float64((log(x) ^ 2.0) * -0.5)) / n) - log(x)) / n);
	else
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.24)
		tmp = ((((-0.16666666666666666 * ((log(x) ^ 3.0) / n)) + ((log(x) ^ 2.0) * -0.5)) / n) - log(x)) / n;
	else
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.24], N[(N[(N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.23999999999999999

   1. Initial program 40.8%

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

   3. Taylor expanded in x around 0 40.1%

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

   4. Taylor expanded in n around -inf 81.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} - 0.5 \cdot {\log x}^{2}}{n} - -1 \cdot \log x}{n}} \]
   5. Step-by-step derivation

      1. mul-1-neg 81.4%

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

   6. Simplified 81.4%

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

   if 0.23999999999999999 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   5. Simplified 97.7%

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

      1. *-un-lft-identity 97.7%

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

      2. associate-/r* 99.2%

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

      3. div-inv 99.2%

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

      4. pow-to-exp 99.2%

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

      5. pow1 99.2%

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

      6. pow-div 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. *-lft-identity 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.24:\\ \;\;\;\;\frac{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + {\log x}^{2} \cdot -0.5}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\frac{0.3333333333333333}{{x}^{2}} - \frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) + 0.5 \cdot \frac{-1}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-77)
   (/ (pow x (+ (/ 1.0 n) -1.0)) n)
   (if (<= (/ 1.0 n) 5e-80)
     (/ (- (log1p x) (log x)) n)
     (if (<= (/ 1.0 n) 2e-5)
       (/
        (/
         (+
          (+ 1.0 (- (/ 0.3333333333333333 (pow x 2.0)) (/ (log (/ 1.0 x)) n)))
          (* 0.5 (/ -1.0 x)))
         x)
        n)
       (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = (((1.0 + ((0.3333333333333333 / pow(x, 2.0)) - (log((1.0 / x)) / n))) + (0.5 * (-1.0 / x))) / x) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = (((1.0 + ((0.3333333333333333 / Math.pow(x, 2.0)) - (Math.log((1.0 / x)) / n))) + (0.5 * (-1.0 / x))) / x) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-77:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 5e-80:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e-5:
		tmp = (((1.0 + ((0.3333333333333333 / math.pow(x, 2.0)) - (math.log((1.0 / x)) / n))) + (0.5 * (-1.0 / x))) / x) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-77)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 5e-80)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(0.3333333333333333 / (x ^ 2.0)) - Float64(log(Float64(1.0 / x)) / n))) + Float64(0.5 * Float64(-1.0 / x))) / x) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-77], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-80], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], N[(N[(N[(N[(1.0 + N[(N[(0.3333333333333333 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-77

   1. Initial program 85.2%

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

   3. Taylor expanded in x around inf 91.8%

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

      1. mul-1-neg 91.8%

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

      2. log-rec 91.8%

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

      3. mul-1-neg 91.8%

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

      4. distribute-neg-frac 91.8%

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

      5. mul-1-neg 91.8%

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

      6. remove-double-neg 91.8%

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

      7. *-commutative 91.8%

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

   5. Simplified 91.8%

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

      1. *-un-lft-identity 91.8%

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

      2. associate-/r* 92.6%

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

      3. div-inv 92.6%

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

      4. pow-to-exp 92.6%

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

      5. pow1 92.6%

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

      6. pow-div 92.6%

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

   7. Applied egg-rr 92.6%

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

      1. *-lft-identity 92.6%

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

      2. sub-neg 92.6%

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

      3. metadata-eval 92.6%

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

   9. Simplified 92.6%

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

   if -1.9999999999999999e-77 < (/.f64 #s(literal 1 binary64) n) < 5e-80

   1. Initial program 33.4%

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

   3. Taylor expanded in n around inf 88.2%

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

      1. log1p-define 88.2%

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

   5. Simplified 88.2%

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

   if 5e-80 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 13.2%

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

   3. Taylor expanded in n around inf 27.9%

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

   5. Simplified 27.9%

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

   6. Taylor expanded in x around inf 83.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 83.5%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.8%

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

   3. Taylor expanded in n around 0 67.8%

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

      1. log1p-define 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\frac{0.3333333333333333}{{x}^{2}} - \frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) + 0.5 \cdot \frac{-1}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\frac{0.3333333333333333}{{x}^{2}} - \frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) + 0.5 \cdot \frac{-1}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-77)
   (/ (pow x (+ (/ 1.0 n) -1.0)) n)
   (if (<= (/ 1.0 n) 5e-80)
     (/ (- (log1p x) (log x)) n)
     (if (<= (/ 1.0 n) 2e-5)
       (/
        (/
         (+
          (+ 1.0 (- (/ 0.3333333333333333 (pow x 2.0)) (/ (log (/ 1.0 x)) n)))
          (* 0.5 (/ -1.0 x)))
         x)
        n)
       (-
        (+
         1.0
         (*
          x
          (+
           (/ 1.0 n)
           (* x (+ (* 0.5 (/ 1.0 (pow n 2.0))) (* 0.5 (/ -1.0 n)))))))
        (pow x (/ 1.0 n)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = (((1.0 + ((0.3333333333333333 / pow(x, 2.0)) - (log((1.0 / x)) / n))) + (0.5 * (-1.0 / x))) / x) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + (x * ((0.5 * (1.0 / pow(n, 2.0))) + (0.5 * (-1.0 / n))))))) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = (((1.0 + ((0.3333333333333333 / Math.pow(x, 2.0)) - (Math.log((1.0 / x)) / n))) + (0.5 * (-1.0 / x))) / 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))))))) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-77:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 5e-80:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e-5:
		tmp = (((1.0 + ((0.3333333333333333 / math.pow(x, 2.0)) - (math.log((1.0 / x)) / n))) + (0.5 * (-1.0 / x))) / 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))))))) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-77)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 5e-80)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(0.3333333333333333 / (x ^ 2.0)) - Float64(log(Float64(1.0 / x)) / n))) + Float64(0.5 * Float64(-1.0 / x))) / 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))))))) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-77], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-80], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], N[(N[(N[(N[(1.0 + N[(N[(0.3333333333333333 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-77

   1. Initial program 85.2%

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

   3. Taylor expanded in x around inf 91.8%

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

      1. mul-1-neg 91.8%

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

      2. log-rec 91.8%

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

      3. mul-1-neg 91.8%

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

      4. distribute-neg-frac 91.8%

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

      5. mul-1-neg 91.8%

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

      6. remove-double-neg 91.8%

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

      7. *-commutative 91.8%

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

   5. Simplified 91.8%

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

      1. *-un-lft-identity 91.8%

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

      2. associate-/r* 92.6%

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

      3. div-inv 92.6%

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

      4. pow-to-exp 92.6%

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

      5. pow1 92.6%

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

      6. pow-div 92.6%

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

   7. Applied egg-rr 92.6%

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

      1. *-lft-identity 92.6%

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

      2. sub-neg 92.6%

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

      3. metadata-eval 92.6%

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

   9. Simplified 92.6%

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

   if -1.9999999999999999e-77 < (/.f64 #s(literal 1 binary64) n) < 5e-80

   1. Initial program 33.4%

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

   3. Taylor expanded in n around inf 88.2%

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

      1. log1p-define 88.2%

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

   5. Simplified 88.2%

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

   if 5e-80 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 13.2%

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

   3. Taylor expanded in n around inf 27.9%

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

   5. Simplified 27.9%

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

   6. Taylor expanded in x around inf 83.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 83.5%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.8%

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

   3. Taylor expanded in x around 0 70.3%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\frac{0.3333333333333333}{{x}^{2}} - \frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) + 0.5 \cdot \frac{-1}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\frac{0.3333333333333333}{{x}^{2}} - \frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) + 0.5 \cdot \frac{-1}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-77)
   (/ (pow x (+ (/ 1.0 n) -1.0)) n)
   (if (<= (/ 1.0 n) 5e-80)
     (/ (- (log1p x) (log x)) n)
     (if (<= (/ 1.0 n) 2e-5)
       (/
        (/
         (+
          (+ 1.0 (- (/ 0.3333333333333333 (pow x 2.0)) (/ (log (/ 1.0 x)) n)))
          (* 0.5 (/ -1.0 x)))
         x)
        n)
       (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = (((1.0 + ((0.3333333333333333 / pow(x, 2.0)) - (log((1.0 / x)) / n))) + (0.5 * (-1.0 / x))) / x) / n;
	} else {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = (((1.0 + ((0.3333333333333333 / Math.pow(x, 2.0)) - (Math.log((1.0 / x)) / n))) + (0.5 * (-1.0 / x))) / x) / n;
	} else {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-77:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 5e-80:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e-5:
		tmp = (((1.0 + ((0.3333333333333333 / math.pow(x, 2.0)) - (math.log((1.0 / x)) / n))) + (0.5 * (-1.0 / x))) / x) / n
	else:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-77)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 5e-80)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(0.3333333333333333 / (x ^ 2.0)) - Float64(log(Float64(1.0 / x)) / n))) + Float64(0.5 * Float64(-1.0 / x))) / x) / n);
	else
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-77], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-80], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], N[(N[(N[(N[(1.0 + N[(N[(0.3333333333333333 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-77

   1. Initial program 85.2%

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

   3. Taylor expanded in x around inf 91.8%

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

      1. mul-1-neg 91.8%

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

      2. log-rec 91.8%

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

      3. mul-1-neg 91.8%

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

      4. distribute-neg-frac 91.8%

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

      5. mul-1-neg 91.8%

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

      6. remove-double-neg 91.8%

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

      7. *-commutative 91.8%

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

   5. Simplified 91.8%

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

      1. *-un-lft-identity 91.8%

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

      2. associate-/r* 92.6%

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

      3. div-inv 92.6%

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

      4. pow-to-exp 92.6%

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

      5. pow1 92.6%

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

      6. pow-div 92.6%

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

   7. Applied egg-rr 92.6%

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

      1. *-lft-identity 92.6%

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

      2. sub-neg 92.6%

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

      3. metadata-eval 92.6%

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

   9. Simplified 92.6%

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

   if -1.9999999999999999e-77 < (/.f64 #s(literal 1 binary64) n) < 5e-80

   1. Initial program 33.4%

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

   3. Taylor expanded in n around inf 88.2%

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

      1. log1p-define 88.2%

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

   5. Simplified 88.2%

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

   if 5e-80 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 13.2%

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

   3. Taylor expanded in n around inf 27.9%

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

   5. Simplified 27.9%

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

   6. Taylor expanded in x around inf 83.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 83.5%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\left(1 + \left(\frac{0.3333333333333333}{{x}^{2}} - \frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) + 0.5 \cdot \frac{-1}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (pow x (+ (/ 1.0 n) -1.0)) n)))
   (if (<= (/ 1.0 n) -2e-77)
     t_0
     (if (<= (/ 1.0 n) 5e-80)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 2e-5)
         t_0
         (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, ((1.0 / n) + -1.0)) / n;
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_0;
	} else {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_0;
	} else {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, ((1.0 / n) + -1.0)) / n
	tmp = 0
	if (1.0 / n) <= -2e-77:
		tmp = t_0
	elif (1.0 / n) <= 5e-80:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e-5:
		tmp = t_0
	else:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-77)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-80)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = t_0;
	else
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-77], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-80], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], t$95$0, N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-77 or 5e-80 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 71.2%

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

   3. Taylor expanded in x around inf 90.0%

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

      1. mul-1-neg 90.0%

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

      2. log-rec 90.0%

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

      3. mul-1-neg 90.0%

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

      4. distribute-neg-frac 90.0%

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

      5. mul-1-neg 90.0%

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

      6. remove-double-neg 90.0%

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

      7. *-commutative 90.0%

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

   5. Simplified 90.0%

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

      1. *-un-lft-identity 90.0%

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

      2. associate-/r* 90.9%

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

      3. div-inv 90.9%

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

      4. pow-to-exp 90.9%

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

      5. pow1 90.9%

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

      6. pow-div 90.8%

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

   7. Applied egg-rr 90.8%

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

      1. *-lft-identity 90.8%

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

      2. sub-neg 90.8%

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

      3. metadata-eval 90.8%

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

   9. Simplified 90.8%

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

   if -1.9999999999999999e-77 < (/.f64 #s(literal 1 binary64) n) < 5e-80

   1. Initial program 33.4%

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

   3. Taylor expanded in n around inf 88.2%

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

      1. log1p-define 88.2%

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

   5. Simplified 88.2%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (pow x (+ (/ 1.0 n) -1.0)) n)))
   (if (<= (/ 1.0 n) -2e-77)
     t_0
     (if (<= (/ 1.0 n) 5e-80)
       (/ (- (log1p x) (log x)) n)
       (if (<= (/ 1.0 n) 2e-5) t_0 (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, ((1.0 / n) + -1.0)) / n;
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	double tmp;
	if ((1.0 / n) <= -2e-77) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-80) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e-5) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, ((1.0 / n) + -1.0)) / n
	tmp = 0
	if (1.0 / n) <= -2e-77:
		tmp = t_0
	elif (1.0 / n) <= 5e-80:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e-5:
		tmp = t_0
	else:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-77)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-80)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e-5)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-77], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-80], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-5], t$95$0, N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-77 or 5e-80 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000016e-5

   1. Initial program 71.2%

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

   3. Taylor expanded in x around inf 90.0%

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

      1. mul-1-neg 90.0%

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

      2. log-rec 90.0%

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

      3. mul-1-neg 90.0%

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

      4. distribute-neg-frac 90.0%

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

      5. mul-1-neg 90.0%

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

      6. remove-double-neg 90.0%

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

      7. *-commutative 90.0%

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

   5. Simplified 90.0%

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

      1. *-un-lft-identity 90.0%

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

      2. associate-/r* 90.9%

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

      3. div-inv 90.9%

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

      4. pow-to-exp 90.9%

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

      5. pow1 90.9%

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

      6. pow-div 90.8%

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

   7. Applied egg-rr 90.8%

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

      1. *-lft-identity 90.8%

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

      2. sub-neg 90.8%

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

      3. metadata-eval 90.8%

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

   9. Simplified 90.8%

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

   if -1.9999999999999999e-77 < (/.f64 #s(literal 1 binary64) n) < 5e-80

   1. Initial program 33.4%

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

   3. Taylor expanded in n around inf 88.2%

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

      1. log1p-define 88.2%

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

   5. Simplified 88.2%

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

   if 2.00000000000000016e-5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.8%

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

   3. Taylor expanded in x around 0 66.1%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{x}}{n}}{x}}{x}\\ \mathbf{if}\;n \leq -5.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;n \leq -2.9 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -330000:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{-202}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 58000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{+122}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0
         (/
          (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ (/ 0.3333333333333333 x) n)) x))
          x)))
   (if (<= n -5.8e+76)
     (/ (log x) (- n))
     (if (<= n -2.9e+17)
       t_0
       (if (<= n -330000.0)
         (/ -1.0 (/ n (log x)))
         (if (<= n 4.9e-202)
           (/ 0.3333333333333333 (* n (pow x 3.0)))
           (if (<= n 58000.0)
             (- 1.0 (pow x (/ 1.0 n)))
             (if (<= n 1.7e+79)
               t_0
               (if (<= n 7.2e+122) (/ (- x (log x)) n) (/ (/ 1.0 n) x))))))))))

C

double code(double x, double n) {
	double t_0 = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
	double tmp;
	if (n <= -5.8e+76) {
		tmp = log(x) / -n;
	} else if (n <= -2.9e+17) {
		tmp = t_0;
	} else if (n <= -330000.0) {
		tmp = -1.0 / (n / log(x));
	} else if (n <= 4.9e-202) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (n <= 58000.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (n <= 1.7e+79) {
		tmp = t_0;
	} else if (n <= 7.2e+122) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 / n) + ((((-0.5d0) / n) + ((0.3333333333333333d0 / x) / n)) / x)) / x
    if (n <= (-5.8d+76)) then
        tmp = log(x) / -n
    else if (n <= (-2.9d+17)) then
        tmp = t_0
    else if (n <= (-330000.0d0)) then
        tmp = (-1.0d0) / (n / log(x))
    else if (n <= 4.9d-202) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (n <= 58000.0d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (n <= 1.7d+79) then
        tmp = t_0
    else if (n <= 7.2d+122) then
        tmp = (x - log(x)) / n
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
	double tmp;
	if (n <= -5.8e+76) {
		tmp = Math.log(x) / -n;
	} else if (n <= -2.9e+17) {
		tmp = t_0;
	} else if (n <= -330000.0) {
		tmp = -1.0 / (n / Math.log(x));
	} else if (n <= 4.9e-202) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (n <= 58000.0) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (n <= 1.7e+79) {
		tmp = t_0;
	} else if (n <= 7.2e+122) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x
	tmp = 0
	if n <= -5.8e+76:
		tmp = math.log(x) / -n
	elif n <= -2.9e+17:
		tmp = t_0
	elif n <= -330000.0:
		tmp = -1.0 / (n / math.log(x))
	elif n <= 4.9e-202:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif n <= 58000.0:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif n <= 1.7e+79:
		tmp = t_0
	elif n <= 7.2e+122:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(Float64(0.3333333333333333 / x) / n)) / x)) / x)
	tmp = 0.0
	if (n <= -5.8e+76)
		tmp = Float64(log(x) / Float64(-n));
	elseif (n <= -2.9e+17)
		tmp = t_0;
	elseif (n <= -330000.0)
		tmp = Float64(-1.0 / Float64(n / log(x)));
	elseif (n <= 4.9e-202)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (n <= 58000.0)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (n <= 1.7e+79)
		tmp = t_0;
	elseif (n <= 7.2e+122)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
	tmp = 0.0;
	if (n <= -5.8e+76)
		tmp = log(x) / -n;
	elseif (n <= -2.9e+17)
		tmp = t_0;
	elseif (n <= -330000.0)
		tmp = -1.0 / (n / log(x));
	elseif (n <= 4.9e-202)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (n <= 58000.0)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (n <= 1.7e+79)
		tmp = t_0;
	elseif (n <= 7.2e+122)
		tmp = (x - log(x)) / n;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(N[(0.3333333333333333 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[n, -5.8e+76], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[n, -2.9e+17], t$95$0, If[LessEqual[n, -330000.0], N[(-1.0 / N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.9e-202], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 58000.0], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.7e+79], t$95$0, If[LessEqual[n, 7.2e+122], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if n < -5.8000000000000003e76

   1. Initial program 30.9%

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

   3. Taylor expanded in x around 0 30.9%

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

   4. Taylor expanded in n around inf 65.7%

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

      1. mul-1-neg 65.7%

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

      2. distribute-neg-frac2 65.7%

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

   6. Simplified 65.7%

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

   if -5.8000000000000003e76 < n < -2.9e17 or 58000 < n < 1.70000000000000016e79

   1. Initial program 12.6%

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

   3. Taylor expanded in n around inf 33.4%

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

   5. Simplified 33.4%

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

   6. Taylor expanded in x around inf 78.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 73.8%

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

      1. sub-neg 73.8%

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

      2. +-commutative 73.8%

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

      3. associate-+l+ 73.8%

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

      4. associate-*r/ 73.8%

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

      5. metadata-eval 73.8%

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

      6. associate-*r/ 73.8%

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

      7. metadata-eval 73.8%

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

      8. distribute-neg-frac 73.8%

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

      9. metadata-eval 73.8%

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

      10. *-commutative 73.8%

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

   9. Simplified 73.8%

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

   10. Taylor expanded in x around inf 75.5%

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

   12. Simplified 75.5%

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

   if -2.9e17 < n < -3.3e5

   1. Initial program 53.2%

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

   3. Taylor expanded in x around 0 53.2%

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

      1. add-cbrt-cube 53.2%

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

      2. pow3 53.2%

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

   5. Applied egg-rr 53.2%

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

   6. Taylor expanded in n around inf 69.9%

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

      1. mul-1-neg 69.9%

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

      2. distribute-neg-frac2 69.9%

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

   8. Simplified 69.9%

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

      1. rem-cbrt-cube 69.9%

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

      2. clear-num 70.0%

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

      3. frac-2neg 70.0%

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

      4. metadata-eval 70.0%

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

      5. add-sqr-sqrt 70.0%

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

      6. sqrt-unprod 70.0%

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

      7. sqr-neg 70.0%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 1.6%

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

      10. distribute-frac-neg 1.6%

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

      11. add-sqr-sqrt 1.6%

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

      12. sqrt-unprod 1.6%

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

      13. sqr-neg 1.6%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 70.0%

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

   10. Applied egg-rr 70.0%

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

   if -3.3e5 < n < 4.9000000000000004e-202

   1. Initial program 93.9%

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

   3. Taylor expanded in n around inf 76.4%

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

   5. Simplified 77.7%

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

   6. Taylor expanded in x around inf 22.4%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 47.5%

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

      1. sub-neg 47.5%

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

      2. +-commutative 47.5%

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

      3. associate-+l+ 47.5%

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

      4. associate-*r/ 47.5%

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

      5. metadata-eval 47.5%

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

      6. associate-*r/ 47.5%

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

      7. metadata-eval 47.5%

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

      8. distribute-neg-frac 47.5%

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

      9. metadata-eval 47.5%

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

      10. *-commutative 47.5%

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

   9. Simplified 47.5%

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

   10. Taylor expanded in x around 0 87.3%

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

   if 4.9000000000000004e-202 < n < 58000

   1. Initial program 80.8%

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

   3. Taylor expanded in x around 0 76.2%

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

   if 1.70000000000000016e79 < n < 7.2000000000000005e122

   1. Initial program 13.8%

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

   3. Taylor expanded in x around 0 3.8%

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

   4. Taylor expanded in n around inf 74.0%

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

   if 7.2000000000000005e122 < n

   1. Initial program 40.3%

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

   3. Taylor expanded in x around inf 54.5%

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

      1. mul-1-neg 54.5%

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

      2. log-rec 54.5%

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

      3. mul-1-neg 54.5%

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

      4. distribute-neg-frac 54.5%

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

      5. mul-1-neg 54.5%

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

      6. remove-double-neg 54.5%

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

      7. *-commutative 54.5%

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

   5. Simplified 54.5%

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

   6. Taylor expanded in n around inf 54.5%

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

      1. associate-/r* 55.0%

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

   8. Simplified 55.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;n \leq -2.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{x}}{n}}{x}}{x}\\ \mathbf{elif}\;n \leq -330000:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{-202}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 58000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 1.7 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{x}}{n}}{x}}{x}\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{+122}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{if}\;x \leq 2.7 \cdot 10^{-226}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-215}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-129}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-125}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.043:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* x n))))
   (if (<= x 2.7e-226)
     (* (log x) (/ -1.0 n))
     (if (<= x 3.5e-215)
       t_0
       (if (<= x 2.05e-129)
         (/ -1.0 (/ n (log x)))
         (if (<= x 7.8e-125)
           t_0
           (if (<= x 0.043)
             (/ (- x (log x)) n)
             (/ (pow x (+ (/ 1.0 n) -1.0)) n))))))))

C

double code(double x, double n) {
	double t_0 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	double tmp;
	if (x <= 2.7e-226) {
		tmp = log(x) * (-1.0 / n);
	} else if (x <= 3.5e-215) {
		tmp = t_0;
	} else if (x <= 2.05e-129) {
		tmp = -1.0 / (n / log(x));
	} else if (x <= 7.8e-125) {
		tmp = t_0;
	} else if (x <= 0.043) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (x * n)
    if (x <= 2.7d-226) then
        tmp = log(x) * ((-1.0d0) / n)
    else if (x <= 3.5d-215) then
        tmp = t_0
    else if (x <= 2.05d-129) then
        tmp = (-1.0d0) / (n / log(x))
    else if (x <= 7.8d-125) then
        tmp = t_0
    else if (x <= 0.043d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	double tmp;
	if (x <= 2.7e-226) {
		tmp = Math.log(x) * (-1.0 / n);
	} else if (x <= 3.5e-215) {
		tmp = t_0;
	} else if (x <= 2.05e-129) {
		tmp = -1.0 / (n / Math.log(x));
	} else if (x <= 7.8e-125) {
		tmp = t_0;
	} else if (x <= 0.043) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n)
	tmp = 0
	if x <= 2.7e-226:
		tmp = math.log(x) * (-1.0 / n)
	elif x <= 3.5e-215:
		tmp = t_0
	elif x <= 2.05e-129:
		tmp = -1.0 / (n / math.log(x))
	elif x <= 7.8e-125:
		tmp = t_0
	elif x <= 0.043:
		tmp = (x - math.log(x)) / n
	else:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(x * n))
	tmp = 0.0
	if (x <= 2.7e-226)
		tmp = Float64(log(x) * Float64(-1.0 / n));
	elseif (x <= 3.5e-215)
		tmp = t_0;
	elseif (x <= 2.05e-129)
		tmp = Float64(-1.0 / Float64(n / log(x)));
	elseif (x <= 7.8e-125)
		tmp = t_0;
	elseif (x <= 0.043)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	tmp = 0.0;
	if (x <= 2.7e-226)
		tmp = log(x) * (-1.0 / n);
	elseif (x <= 3.5e-215)
		tmp = t_0;
	elseif (x <= 2.05e-129)
		tmp = -1.0 / (n / log(x));
	elseif (x <= 7.8e-125)
		tmp = t_0;
	elseif (x <= 0.043)
		tmp = (x - log(x)) / n;
	else
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.7e-226], N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e-215], t$95$0, If[LessEqual[x, 2.05e-129], N[(-1.0 / N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.8e-125], t$95$0, If[LessEqual[x, 0.043], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 2.70000000000000014e-226

   1. Initial program 42.4%

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

   3. Taylor expanded in x around 0 42.4%

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

      1. add-cbrt-cube 42.4%

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

      2. pow3 42.5%

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

   5. Applied egg-rr 42.5%

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

   6. Taylor expanded in n around inf 43.7%

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

      1. mul-1-neg 57.9%

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

      2. distribute-neg-frac2 57.9%

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

   8. Simplified 43.7%

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

      1. rem-cbrt-cube 57.9%

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

      2. frac-2neg 57.9%

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

      3. div-inv 58.0%

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

      4. remove-double-neg 58.0%

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

   10. Applied egg-rr 58.0%

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

   if 2.70000000000000014e-226 < x < 3.5000000000000002e-215 or 2.05e-129 < x < 7.79999999999999965e-125

   1. Initial program 83.9%

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

   3. Taylor expanded in n around inf 35.8%

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

   5. Simplified 35.8%

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

   6. Taylor expanded in x around inf 25.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 91.9%

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

      1. sub-neg 91.9%

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

      2. +-commutative 91.9%

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

      3. associate-+l+ 91.9%

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

      4. associate-*r/ 91.9%

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

      5. metadata-eval 91.9%

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

      6. associate-*r/ 91.9%

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

      7. metadata-eval 91.9%

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

      8. distribute-neg-frac 91.9%

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

      9. metadata-eval 91.9%

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

      10. *-commutative 91.9%

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

   9. Simplified 91.9%

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

   10. Taylor expanded in x around -inf 91.9%

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

       1. mul-1-neg 91.9%

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

       2. unsub-neg 91.9%

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

       3. sub-neg 91.9%

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

       4. associate-*r/ 91.9%

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

       5. metadata-eval 91.9%

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

       6. distribute-neg-frac 91.9%

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

       7. metadata-eval 91.9%

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

   12. Simplified 91.9%

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

   if 3.5000000000000002e-215 < x < 2.05e-129

   1. Initial program 27.1%

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

   3. Taylor expanded in x around 0 27.1%

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

      1. add-cbrt-cube 27.1%

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

      2. pow3 27.1%

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

   5. Applied egg-rr 27.1%

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

   6. Taylor expanded in n around inf 18.0%

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

      1. mul-1-neg 78.4%

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

      2. distribute-neg-frac2 78.4%

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

   8. Simplified 18.0%

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

      1. rem-cbrt-cube 78.4%

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

      2. clear-num 78.5%

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

      3. frac-2neg 78.5%

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

      4. metadata-eval 78.5%

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

      5. add-sqr-sqrt 38.8%

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

      6. sqrt-unprod 14.1%

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

      7. sqr-neg 14.1%

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

      8. sqrt-unprod 1.9%

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

      9. add-sqr-sqrt 3.5%

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

      10. distribute-frac-neg 3.5%

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

      11. add-sqr-sqrt 1.6%

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

      12. sqrt-unprod 17.7%

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

      13. sqr-neg 17.7%

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

      14. sqrt-unprod 39.2%

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

      15. add-sqr-sqrt 78.5%

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

   10. Applied egg-rr 78.5%

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

   if 7.79999999999999965e-125 < x < 0.042999999999999997

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 0 36.1%

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

   4. Taylor expanded in n around inf 59.8%

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

   if 0.042999999999999997 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   5. Simplified 97.7%

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

      1. *-un-lft-identity 97.7%

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

      2. associate-/r* 99.2%

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

      3. div-inv 99.2%

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

      4. pow-to-exp 99.2%

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

      5. pow1 99.2%

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

      6. pow-div 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. *-lft-identity 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{-226}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-215}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-129}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.043:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-226}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-215}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.7e-226)
   (* (log x) (/ -1.0 n))
   (if (<= x 4.6e-215)
     (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
     (if (<= x 2.3e-129)
       (/ -1.0 (/ n (log x)))
       (if (<= x 4.5e-125)
         (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* x n))
         (if (<= x 0.046)
           (/ (- x (log x)) n)
           (/ (pow x (+ (/ 1.0 n) -1.0)) n)))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.7e-226) {
		tmp = log(x) * (-1.0 / n);
	} else if (x <= 4.6e-215) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else if (x <= 2.3e-129) {
		tmp = -1.0 / (n / log(x));
	} else if (x <= 4.5e-125) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else if (x <= 0.046) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.7d-226) then
        tmp = log(x) * ((-1.0d0) / n)
    else if (x <= 4.6d-215) then
        tmp = (1.0d0 + (x / n)) - (x ** (1.0d0 / n))
    else if (x <= 2.3d-129) then
        tmp = (-1.0d0) / (n / log(x))
    else if (x <= 4.5d-125) then
        tmp = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (x * n)
    else if (x <= 0.046d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.7e-226) {
		tmp = Math.log(x) * (-1.0 / n);
	} else if (x <= 4.6e-215) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else if (x <= 2.3e-129) {
		tmp = -1.0 / (n / Math.log(x));
	} else if (x <= 4.5e-125) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else if (x <= 0.046) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.7e-226:
		tmp = math.log(x) * (-1.0 / n)
	elif x <= 4.6e-215:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	elif x <= 2.3e-129:
		tmp = -1.0 / (n / math.log(x))
	elif x <= 4.5e-125:
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n)
	elif x <= 0.046:
		tmp = (x - math.log(x)) / n
	else:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.7e-226)
		tmp = Float64(log(x) * Float64(-1.0 / n));
	elseif (x <= 4.6e-215)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	elseif (x <= 2.3e-129)
		tmp = Float64(-1.0 / Float64(n / log(x)));
	elseif (x <= 4.5e-125)
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(x * n));
	elseif (x <= 0.046)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.7e-226)
		tmp = log(x) * (-1.0 / n);
	elseif (x <= 4.6e-215)
		tmp = (1.0 + (x / n)) - (x ^ (1.0 / n));
	elseif (x <= 2.3e-129)
		tmp = -1.0 / (n / log(x));
	elseif (x <= 4.5e-125)
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	elseif (x <= 0.046)
		tmp = (x - log(x)) / n;
	else
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.7e-226], N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.6e-215], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-129], N[(-1.0 / N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-125], N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.046], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < 1.70000000000000004e-226

   1. Initial program 42.4%

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

   3. Taylor expanded in x around 0 42.4%

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

      1. add-cbrt-cube 42.4%

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

      2. pow3 42.5%

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

   5. Applied egg-rr 42.5%

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

   6. Taylor expanded in n around inf 43.7%

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

      1. mul-1-neg 57.9%

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

      2. distribute-neg-frac2 57.9%

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

   8. Simplified 43.7%

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

      1. rem-cbrt-cube 57.9%

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

      2. frac-2neg 57.9%

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

      3. div-inv 58.0%

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

      4. remove-double-neg 58.0%

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

   10. Applied egg-rr 58.0%

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

   if 1.70000000000000004e-226 < x < 4.5999999999999998e-215

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0 83.9%

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

   if 4.5999999999999998e-215 < x < 2.3e-129

   1. Initial program 27.1%

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

   3. Taylor expanded in x around 0 27.1%

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

      1. add-cbrt-cube 27.1%

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

      2. pow3 27.1%

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

   5. Applied egg-rr 27.1%

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

   6. Taylor expanded in n around inf 18.0%

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

      1. mul-1-neg 78.4%

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

      2. distribute-neg-frac2 78.4%

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

   8. Simplified 18.0%

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

      1. rem-cbrt-cube 78.4%

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

      2. clear-num 78.5%

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

      3. frac-2neg 78.5%

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

      4. metadata-eval 78.5%

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

      5. add-sqr-sqrt 38.8%

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

      6. sqrt-unprod 14.1%

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

      7. sqr-neg 14.1%

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

      8. sqrt-unprod 1.9%

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

      9. add-sqr-sqrt 3.5%

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

      10. distribute-frac-neg 3.5%

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

      11. add-sqr-sqrt 1.6%

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

      12. sqrt-unprod 17.7%

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

      13. sqr-neg 17.7%

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

      14. sqrt-unprod 39.2%

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

      15. add-sqr-sqrt 78.5%

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

   10. Applied egg-rr 78.5%

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

   if 2.3e-129 < x < 4.50000000000000012e-125

   1. Initial program 83.9%

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

   3. Taylor expanded in n around inf 51.8%

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

   5. Simplified 51.8%

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

   6. Taylor expanded in x around inf 33.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*r/ 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. metadata-eval 100.0%

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

      10. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around -inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. unsub-neg 100.0%

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

       3. sub-neg 100.0%

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

       4. associate-*r/ 100.0%

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

       5. metadata-eval 100.0%

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

       6. distribute-neg-frac 100.0%

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

       7. metadata-eval 100.0%

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

   12. Simplified 100.0%

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

   if 4.50000000000000012e-125 < x < 0.045999999999999999

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 0 36.1%

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

   4. Taylor expanded in n around inf 59.8%

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

   if 0.045999999999999999 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. log-rec 97.7%

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

      3. mul-1-neg 97.7%

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

      4. distribute-neg-frac 97.7%

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

      5. mul-1-neg 97.7%

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

      6. remove-double-neg 97.7%

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

      7. *-commutative 97.7%

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

   5. Simplified 97.7%

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

      1. *-un-lft-identity 97.7%

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

      2. associate-/r* 99.2%

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

      3. div-inv 99.2%

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

      4. pow-to-exp 99.2%

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

      5. pow1 99.2%

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

      6. pow-div 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. *-lft-identity 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

   9. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-226}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-215}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\frac{n}{\log x}}\\ t_1 := \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{if}\;x \leq 6.1 \cdot 10^{-226}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-215}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{x}}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ -1.0 (/ n (log x))))
        (t_1 (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* x n))))
   (if (<= x 6.1e-226)
     t_0
     (if (<= x 3.5e-215)
       t_1
       (if (<= x 2.3e-129)
         t_0
         (if (<= x 4.5e-125)
           t_1
           (if (<= x 0.85)
             (/ (- x (log x)) n)
             (/
              (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ (/ 0.3333333333333333 x) n)) x))
              x))))))))

C

double code(double x, double n) {
	double t_0 = -1.0 / (n / log(x));
	double t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	double tmp;
	if (x <= 6.1e-226) {
		tmp = t_0;
	} else if (x <= 3.5e-215) {
		tmp = t_1;
	} else if (x <= 2.3e-129) {
		tmp = t_0;
	} else if (x <= 4.5e-125) {
		tmp = t_1;
	} else if (x <= 0.85) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) / (n / log(x))
    t_1 = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (x * n)
    if (x <= 6.1d-226) then
        tmp = t_0
    else if (x <= 3.5d-215) then
        tmp = t_1
    else if (x <= 2.3d-129) then
        tmp = t_0
    else if (x <= 4.5d-125) then
        tmp = t_1
    else if (x <= 0.85d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + ((0.3333333333333333d0 / x) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -1.0 / (n / Math.log(x));
	double t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	double tmp;
	if (x <= 6.1e-226) {
		tmp = t_0;
	} else if (x <= 3.5e-215) {
		tmp = t_1;
	} else if (x <= 2.3e-129) {
		tmp = t_0;
	} else if (x <= 4.5e-125) {
		tmp = t_1;
	} else if (x <= 0.85) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -1.0 / (n / math.log(x))
	t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n)
	tmp = 0
	if x <= 6.1e-226:
		tmp = t_0
	elif x <= 3.5e-215:
		tmp = t_1
	elif x <= 2.3e-129:
		tmp = t_0
	elif x <= 4.5e-125:
		tmp = t_1
	elif x <= 0.85:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(-1.0 / Float64(n / log(x)))
	t_1 = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(x * n))
	tmp = 0.0
	if (x <= 6.1e-226)
		tmp = t_0;
	elseif (x <= 3.5e-215)
		tmp = t_1;
	elseif (x <= 2.3e-129)
		tmp = t_0;
	elseif (x <= 4.5e-125)
		tmp = t_1;
	elseif (x <= 0.85)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(Float64(0.3333333333333333 / x) / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -1.0 / (n / log(x));
	t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	tmp = 0.0;
	if (x <= 6.1e-226)
		tmp = t_0;
	elseif (x <= 3.5e-215)
		tmp = t_1;
	elseif (x <= 2.3e-129)
		tmp = t_0;
	elseif (x <= 4.5e-125)
		tmp = t_1;
	elseif (x <= 0.85)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(-1.0 / N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 6.1e-226], t$95$0, If[LessEqual[x, 3.5e-215], t$95$1, If[LessEqual[x, 2.3e-129], t$95$0, If[LessEqual[x, 4.5e-125], t$95$1, If[LessEqual[x, 0.85], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(N[(0.3333333333333333 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 6.0999999999999998e-226 or 3.5000000000000002e-215 < x < 2.3e-129

   1. Initial program 35.8%

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

   3. Taylor expanded in x around 0 35.8%

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

      1. add-cbrt-cube 35.8%

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

      2. pow3 35.9%

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

   5. Applied egg-rr 35.9%

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

   6. Taylor expanded in n around inf 32.6%

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

      1. mul-1-neg 66.7%

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

      2. distribute-neg-frac2 66.7%

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

   8. Simplified 32.6%

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

      1. rem-cbrt-cube 66.7%

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

      2. clear-num 66.8%

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

      3. frac-2neg 66.8%

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

      4. metadata-eval 66.8%

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

      5. add-sqr-sqrt 33.1%

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

      6. sqrt-unprod 22.1%

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

      7. sqr-neg 22.1%

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

      8. sqrt-unprod 1.5%

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

      9. add-sqr-sqrt 2.8%

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

      10. distribute-frac-neg 2.8%

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

      11. add-sqr-sqrt 1.3%

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

      12. sqrt-unprod 18.2%

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

      13. sqr-neg 18.2%

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

      14. sqrt-unprod 33.4%

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

      15. add-sqr-sqrt 66.8%

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

   10. Applied egg-rr 66.8%

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

   if 6.0999999999999998e-226 < x < 3.5000000000000002e-215 or 2.3e-129 < x < 4.50000000000000012e-125

   1. Initial program 83.9%

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

   3. Taylor expanded in n around inf 35.8%

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

   5. Simplified 35.8%

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

   6. Taylor expanded in x around inf 25.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 91.9%

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

      1. sub-neg 91.9%

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

      2. +-commutative 91.9%

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

      3. associate-+l+ 91.9%

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

      4. associate-*r/ 91.9%

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

      5. metadata-eval 91.9%

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

      6. associate-*r/ 91.9%

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

      7. metadata-eval 91.9%

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

      8. distribute-neg-frac 91.9%

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

      9. metadata-eval 91.9%

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

      10. *-commutative 91.9%

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

   9. Simplified 91.9%

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

   10. Taylor expanded in x around -inf 91.9%

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

       1. mul-1-neg 91.9%

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

       2. unsub-neg 91.9%

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

       3. sub-neg 91.9%

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

       4. associate-*r/ 91.9%

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

       5. metadata-eval 91.9%

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

       6. distribute-neg-frac 91.9%

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

       7. metadata-eval 91.9%

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

   12. Simplified 91.9%

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

   if 4.50000000000000012e-125 < x < 0.849999999999999978

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 0 36.1%

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

   4. Taylor expanded in n around inf 59.8%

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

   if 0.849999999999999978 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in n around inf 66.5%

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

   5. Simplified 67.3%

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

   6. Taylor expanded in x around inf 65.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 63.1%

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

      1. sub-neg 63.1%

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

      2. +-commutative 63.1%

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

      3. associate-+l+ 63.1%

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

      4. associate-*r/ 63.1%

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

      5. metadata-eval 63.1%

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

      6. associate-*r/ 63.1%

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

      7. metadata-eval 63.1%

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

      8. distribute-neg-frac 63.1%

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

      9. metadata-eval 63.1%

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

      10. *-commutative 63.1%

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

   9. Simplified 63.1%

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

   10. Taylor expanded in x around inf 64.3%

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

   12. Simplified 64.3%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.1 \cdot 10^{-226}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-215}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{x}}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-226}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-215}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-129}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{x}}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* x n))))
   (if (<= x 5.5e-226)
     (* (log x) (/ -1.0 n))
     (if (<= x 3.5e-215)
       t_0
       (if (<= x 2.05e-129)
         (/ -1.0 (/ n (log x)))
         (if (<= x 4.5e-125)
           t_0
           (if (<= x 0.85)
             (/ (- x (log x)) n)
             (/
              (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ (/ 0.3333333333333333 x) n)) x))
              x))))))))

C

double code(double x, double n) {
	double t_0 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	double tmp;
	if (x <= 5.5e-226) {
		tmp = log(x) * (-1.0 / n);
	} else if (x <= 3.5e-215) {
		tmp = t_0;
	} else if (x <= 2.05e-129) {
		tmp = -1.0 / (n / log(x));
	} else if (x <= 4.5e-125) {
		tmp = t_0;
	} else if (x <= 0.85) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (x * n)
    if (x <= 5.5d-226) then
        tmp = log(x) * ((-1.0d0) / n)
    else if (x <= 3.5d-215) then
        tmp = t_0
    else if (x <= 2.05d-129) then
        tmp = (-1.0d0) / (n / log(x))
    else if (x <= 4.5d-125) then
        tmp = t_0
    else if (x <= 0.85d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + ((0.3333333333333333d0 / x) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	double tmp;
	if (x <= 5.5e-226) {
		tmp = Math.log(x) * (-1.0 / n);
	} else if (x <= 3.5e-215) {
		tmp = t_0;
	} else if (x <= 2.05e-129) {
		tmp = -1.0 / (n / Math.log(x));
	} else if (x <= 4.5e-125) {
		tmp = t_0;
	} else if (x <= 0.85) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n)
	tmp = 0
	if x <= 5.5e-226:
		tmp = math.log(x) * (-1.0 / n)
	elif x <= 3.5e-215:
		tmp = t_0
	elif x <= 2.05e-129:
		tmp = -1.0 / (n / math.log(x))
	elif x <= 4.5e-125:
		tmp = t_0
	elif x <= 0.85:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(x * n))
	tmp = 0.0
	if (x <= 5.5e-226)
		tmp = Float64(log(x) * Float64(-1.0 / n));
	elseif (x <= 3.5e-215)
		tmp = t_0;
	elseif (x <= 2.05e-129)
		tmp = Float64(-1.0 / Float64(n / log(x)));
	elseif (x <= 4.5e-125)
		tmp = t_0;
	elseif (x <= 0.85)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(Float64(0.3333333333333333 / x) / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	tmp = 0.0;
	if (x <= 5.5e-226)
		tmp = log(x) * (-1.0 / n);
	elseif (x <= 3.5e-215)
		tmp = t_0;
	elseif (x <= 2.05e-129)
		tmp = -1.0 / (n / log(x));
	elseif (x <= 4.5e-125)
		tmp = t_0;
	elseif (x <= 0.85)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.5e-226], N[(N[Log[x], $MachinePrecision] * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e-215], t$95$0, If[LessEqual[x, 2.05e-129], N[(-1.0 / N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-125], t$95$0, If[LessEqual[x, 0.85], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(N[(0.3333333333333333 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 5.5e-226

   1. Initial program 42.4%

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

   3. Taylor expanded in x around 0 42.4%

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

      1. add-cbrt-cube 42.4%

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

      2. pow3 42.5%

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

   5. Applied egg-rr 42.5%

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

   6. Taylor expanded in n around inf 43.7%

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

      1. mul-1-neg 57.9%

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

      2. distribute-neg-frac2 57.9%

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

   8. Simplified 43.7%

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

      1. rem-cbrt-cube 57.9%

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

      2. frac-2neg 57.9%

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

      3. div-inv 58.0%

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

      4. remove-double-neg 58.0%

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

   10. Applied egg-rr 58.0%

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

   if 5.5e-226 < x < 3.5000000000000002e-215 or 2.05e-129 < x < 4.50000000000000012e-125

   1. Initial program 83.9%

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

   3. Taylor expanded in n around inf 35.8%

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

   5. Simplified 35.8%

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

   6. Taylor expanded in x around inf 25.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 91.9%

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

      1. sub-neg 91.9%

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

      2. +-commutative 91.9%

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

      3. associate-+l+ 91.9%

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

      4. associate-*r/ 91.9%

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

      5. metadata-eval 91.9%

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

      6. associate-*r/ 91.9%

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

      7. metadata-eval 91.9%

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

      8. distribute-neg-frac 91.9%

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

      9. metadata-eval 91.9%

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

      10. *-commutative 91.9%

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

   9. Simplified 91.9%

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

   10. Taylor expanded in x around -inf 91.9%

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

       1. mul-1-neg 91.9%

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

       2. unsub-neg 91.9%

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

       3. sub-neg 91.9%

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

       4. associate-*r/ 91.9%

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

       5. metadata-eval 91.9%

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

       6. distribute-neg-frac 91.9%

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

       7. metadata-eval 91.9%

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

   12. Simplified 91.9%

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

   if 3.5000000000000002e-215 < x < 2.05e-129

   1. Initial program 27.1%

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

   3. Taylor expanded in x around 0 27.1%

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

      1. add-cbrt-cube 27.1%

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

      2. pow3 27.1%

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

   5. Applied egg-rr 27.1%

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

   6. Taylor expanded in n around inf 18.0%

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

      1. mul-1-neg 78.4%

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

      2. distribute-neg-frac2 78.4%

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

   8. Simplified 18.0%

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

      1. rem-cbrt-cube 78.4%

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

      2. clear-num 78.5%

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

      3. frac-2neg 78.5%

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

      4. metadata-eval 78.5%

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

      5. add-sqr-sqrt 38.8%

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

      6. sqrt-unprod 14.1%

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

      7. sqr-neg 14.1%

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

      8. sqrt-unprod 1.9%

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

      9. add-sqr-sqrt 3.5%

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

      10. distribute-frac-neg 3.5%

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

      11. add-sqr-sqrt 1.6%

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

      12. sqrt-unprod 17.7%

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

      13. sqr-neg 17.7%

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

      14. sqrt-unprod 39.2%

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

      15. add-sqr-sqrt 78.5%

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

   10. Applied egg-rr 78.5%

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

   if 4.50000000000000012e-125 < x < 0.849999999999999978

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 0 36.1%

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

   4. Taylor expanded in n around inf 59.8%

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

   if 0.849999999999999978 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in n around inf 66.5%

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

   5. Simplified 67.3%

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

   6. Taylor expanded in x around inf 65.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 63.1%

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

      1. sub-neg 63.1%

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

      2. +-commutative 63.1%

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

      3. associate-+l+ 63.1%

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

      4. associate-*r/ 63.1%

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

      5. metadata-eval 63.1%

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

      6. associate-*r/ 63.1%

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

      7. metadata-eval 63.1%

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

      8. distribute-neg-frac 63.1%

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

      9. metadata-eval 63.1%

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

      10. *-commutative 63.1%

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

   9. Simplified 63.1%

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

   10. Taylor expanded in x around inf 64.3%

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

   12. Simplified 64.3%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-226}:\\ \;\;\;\;\log x \cdot \frac{-1}{n}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-215}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-129}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{x}}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{if}\;x \leq 5.5 \cdot 10^{-226}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-215}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{x}}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n)))
        (t_1 (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* x n))))
   (if (<= x 5.5e-226)
     t_0
     (if (<= x 3.5e-215)
       t_1
       (if (<= x 2.3e-129)
         t_0
         (if (<= x 4.8e-125)
           t_1
           (if (<= x 0.6)
             t_0
             (/
              (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ (/ 0.3333333333333333 x) n)) x))
              x))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	double tmp;
	if (x <= 5.5e-226) {
		tmp = t_0;
	} else if (x <= 3.5e-215) {
		tmp = t_1;
	} else if (x <= 2.3e-129) {
		tmp = t_0;
	} else if (x <= 4.8e-125) {
		tmp = t_1;
	} else if (x <= 0.6) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (x * n)
    if (x <= 5.5d-226) then
        tmp = t_0
    else if (x <= 3.5d-215) then
        tmp = t_1
    else if (x <= 2.3d-129) then
        tmp = t_0
    else if (x <= 4.8d-125) then
        tmp = t_1
    else if (x <= 0.6d0) then
        tmp = t_0
    else
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + ((0.3333333333333333d0 / x) / n)) / x)) / 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 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	double tmp;
	if (x <= 5.5e-226) {
		tmp = t_0;
	} else if (x <= 3.5e-215) {
		tmp = t_1;
	} else if (x <= 2.3e-129) {
		tmp = t_0;
	} else if (x <= 4.8e-125) {
		tmp = t_1;
	} else if (x <= 0.6) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n)
	tmp = 0
	if x <= 5.5e-226:
		tmp = t_0
	elif x <= 3.5e-215:
		tmp = t_1
	elif x <= 2.3e-129:
		tmp = t_0
	elif x <= 4.8e-125:
		tmp = t_1
	elif x <= 0.6:
		tmp = t_0
	else:
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(x * n))
	tmp = 0.0
	if (x <= 5.5e-226)
		tmp = t_0;
	elseif (x <= 3.5e-215)
		tmp = t_1;
	elseif (x <= 2.3e-129)
		tmp = t_0;
	elseif (x <= 4.8e-125)
		tmp = t_1;
	elseif (x <= 0.6)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(Float64(0.3333333333333333 / x) / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	tmp = 0.0;
	if (x <= 5.5e-226)
		tmp = t_0;
	elseif (x <= 3.5e-215)
		tmp = t_1;
	elseif (x <= 2.3e-129)
		tmp = t_0;
	elseif (x <= 4.8e-125)
		tmp = t_1;
	elseif (x <= 0.6)
		tmp = t_0;
	else
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / 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[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.5e-226], t$95$0, If[LessEqual[x, 3.5e-215], t$95$1, If[LessEqual[x, 2.3e-129], t$95$0, If[LessEqual[x, 4.8e-125], t$95$1, If[LessEqual[x, 0.6], t$95$0, N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(N[(0.3333333333333333 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.5e-226 or 3.5000000000000002e-215 < x < 2.3e-129 or 4.8000000000000003e-125 < x < 0.599999999999999978

   1. Initial program 36.6%

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

   3. Taylor expanded in x around 0 35.8%

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

   4. Taylor expanded in n around inf 63.7%

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

      1. mul-1-neg 63.7%

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

      2. distribute-neg-frac2 63.7%

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

   6. Simplified 63.7%

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

   if 5.5e-226 < x < 3.5000000000000002e-215 or 2.3e-129 < x < 4.8000000000000003e-125

   1. Initial program 83.9%

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

   3. Taylor expanded in n around inf 35.8%

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

   5. Simplified 35.8%

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

   6. Taylor expanded in x around inf 25.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 91.9%

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

      1. sub-neg 91.9%

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

      2. +-commutative 91.9%

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

      3. associate-+l+ 91.9%

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

      4. associate-*r/ 91.9%

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

      5. metadata-eval 91.9%

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

      6. associate-*r/ 91.9%

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

      7. metadata-eval 91.9%

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

      8. distribute-neg-frac 91.9%

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

      9. metadata-eval 91.9%

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

      10. *-commutative 91.9%

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

   9. Simplified 91.9%

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

   10. Taylor expanded in x around -inf 91.9%

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

       1. mul-1-neg 91.9%

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

       2. unsub-neg 91.9%

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

       3. sub-neg 91.9%

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

       4. associate-*r/ 91.9%

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

       5. metadata-eval 91.9%

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

       6. distribute-neg-frac 91.9%

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

       7. metadata-eval 91.9%

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

   12. Simplified 91.9%

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

   if 0.599999999999999978 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in n around inf 66.5%

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

   5. Simplified 67.3%

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

   6. Taylor expanded in x around inf 65.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 63.1%

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

      1. sub-neg 63.1%

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

      2. +-commutative 63.1%

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

      3. associate-+l+ 63.1%

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

      4. associate-*r/ 63.1%

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

      5. metadata-eval 63.1%

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

      6. associate-*r/ 63.1%

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

      7. metadata-eval 63.1%

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

      8. distribute-neg-frac 63.1%

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

      9. metadata-eval 63.1%

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

      10. *-commutative 63.1%

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

   9. Simplified 63.1%

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

   10. Taylor expanded in x around inf 64.3%

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

   12. Simplified 64.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-226}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-215}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{x}}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\frac{n}{\log x}}\\ t_1 := \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{if}\;x \leq 6.1 \cdot 10^{-226}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-215}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{x}}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ -1.0 (/ n (log x))))
        (t_1 (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* x n))))
   (if (<= x 6.1e-226)
     t_0
     (if (<= x 3.5e-215)
       t_1
       (if (<= x 2.3e-129)
         t_0
         (if (<= x 1.35e-124)
           t_1
           (if (<= x 0.6)
             (/ (log x) (- n))
             (/
              (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ (/ 0.3333333333333333 x) n)) x))
              x))))))))

C

double code(double x, double n) {
	double t_0 = -1.0 / (n / log(x));
	double t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	double tmp;
	if (x <= 6.1e-226) {
		tmp = t_0;
	} else if (x <= 3.5e-215) {
		tmp = t_1;
	} else if (x <= 2.3e-129) {
		tmp = t_0;
	} else if (x <= 1.35e-124) {
		tmp = t_1;
	} else if (x <= 0.6) {
		tmp = log(x) / -n;
	} else {
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) / (n / log(x))
    t_1 = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (x * n)
    if (x <= 6.1d-226) then
        tmp = t_0
    else if (x <= 3.5d-215) then
        tmp = t_1
    else if (x <= 2.3d-129) then
        tmp = t_0
    else if (x <= 1.35d-124) then
        tmp = t_1
    else if (x <= 0.6d0) then
        tmp = log(x) / -n
    else
        tmp = ((1.0d0 / n) + ((((-0.5d0) / n) + ((0.3333333333333333d0 / x) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -1.0 / (n / Math.log(x));
	double t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	double tmp;
	if (x <= 6.1e-226) {
		tmp = t_0;
	} else if (x <= 3.5e-215) {
		tmp = t_1;
	} else if (x <= 2.3e-129) {
		tmp = t_0;
	} else if (x <= 1.35e-124) {
		tmp = t_1;
	} else if (x <= 0.6) {
		tmp = Math.log(x) / -n;
	} else {
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -1.0 / (n / math.log(x))
	t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n)
	tmp = 0
	if x <= 6.1e-226:
		tmp = t_0
	elif x <= 3.5e-215:
		tmp = t_1
	elif x <= 2.3e-129:
		tmp = t_0
	elif x <= 1.35e-124:
		tmp = t_1
	elif x <= 0.6:
		tmp = math.log(x) / -n
	else:
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(-1.0 / Float64(n / log(x)))
	t_1 = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(x * n))
	tmp = 0.0
	if (x <= 6.1e-226)
		tmp = t_0;
	elseif (x <= 3.5e-215)
		tmp = t_1;
	elseif (x <= 2.3e-129)
		tmp = t_0;
	elseif (x <= 1.35e-124)
		tmp = t_1;
	elseif (x <= 0.6)
		tmp = Float64(log(x) / Float64(-n));
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(Float64(0.3333333333333333 / x) / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -1.0 / (n / log(x));
	t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	tmp = 0.0;
	if (x <= 6.1e-226)
		tmp = t_0;
	elseif (x <= 3.5e-215)
		tmp = t_1;
	elseif (x <= 2.3e-129)
		tmp = t_0;
	elseif (x <= 1.35e-124)
		tmp = t_1;
	elseif (x <= 0.6)
		tmp = log(x) / -n;
	else
		tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(-1.0 / N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 6.1e-226], t$95$0, If[LessEqual[x, 3.5e-215], t$95$1, If[LessEqual[x, 2.3e-129], t$95$0, If[LessEqual[x, 1.35e-124], t$95$1, If[LessEqual[x, 0.6], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(N[(0.3333333333333333 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 6.0999999999999998e-226 or 3.5000000000000002e-215 < x < 2.3e-129

   1. Initial program 35.8%

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

   3. Taylor expanded in x around 0 35.8%

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

      1. add-cbrt-cube 35.8%

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

      2. pow3 35.9%

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

   5. Applied egg-rr 35.9%

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

   6. Taylor expanded in n around inf 32.6%

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

      1. mul-1-neg 66.7%

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

      2. distribute-neg-frac2 66.7%

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

   8. Simplified 32.6%

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

      1. rem-cbrt-cube 66.7%

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

      2. clear-num 66.8%

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

      3. frac-2neg 66.8%

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

      4. metadata-eval 66.8%

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

      5. add-sqr-sqrt 33.1%

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

      6. sqrt-unprod 22.1%

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

      7. sqr-neg 22.1%

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

      8. sqrt-unprod 1.5%

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

      9. add-sqr-sqrt 2.8%

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

      10. distribute-frac-neg 2.8%

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

      11. add-sqr-sqrt 1.3%

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

      12. sqrt-unprod 18.2%

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

      13. sqr-neg 18.2%

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

      14. sqrt-unprod 33.4%

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

      15. add-sqr-sqrt 66.8%

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

   10. Applied egg-rr 66.8%

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

   if 6.0999999999999998e-226 < x < 3.5000000000000002e-215 or 2.3e-129 < x < 1.35000000000000009e-124

   1. Initial program 83.9%

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

   3. Taylor expanded in n around inf 35.8%

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

   5. Simplified 35.8%

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

   6. Taylor expanded in x around inf 25.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 91.9%

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

      1. sub-neg 91.9%

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

      2. +-commutative 91.9%

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

      3. associate-+l+ 91.9%

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

      4. associate-*r/ 91.9%

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

      5. metadata-eval 91.9%

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

      6. associate-*r/ 91.9%

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

      7. metadata-eval 91.9%

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

      8. distribute-neg-frac 91.9%

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

      9. metadata-eval 91.9%

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

      10. *-commutative 91.9%

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

   9. Simplified 91.9%

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

   10. Taylor expanded in x around -inf 91.9%

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

       1. mul-1-neg 91.9%

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

       2. unsub-neg 91.9%

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

       3. sub-neg 91.9%

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

       4. associate-*r/ 91.9%

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

       5. metadata-eval 91.9%

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

       6. distribute-neg-frac 91.9%

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

       7. metadata-eval 91.9%

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

   12. Simplified 91.9%

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

   if 1.35000000000000009e-124 < x < 0.599999999999999978

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 0 35.8%

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

   4. Taylor expanded in n around inf 59.6%

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

      1. mul-1-neg 59.6%

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

      2. distribute-neg-frac2 59.6%

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

   6. Simplified 59.6%

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

   if 0.599999999999999978 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in n around inf 66.5%

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

   5. Simplified 67.3%

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

   6. Taylor expanded in x around inf 65.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

   7. Taylor expanded in n around inf 63.1%

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

      1. sub-neg 63.1%

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

      2. +-commutative 63.1%

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

      3. associate-+l+ 63.1%

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

      4. associate-*r/ 63.1%

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

      5. metadata-eval 63.1%

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

      6. associate-*r/ 63.1%

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

      7. metadata-eval 63.1%

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

      8. distribute-neg-frac 63.1%

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

      9. metadata-eval 63.1%

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

      10. *-commutative 63.1%

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

   9. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{{x}^{2}} + \left(1 + \frac{-0.5}{x}\right)}{x \cdot n}} \]

   10. Taylor expanded in x around inf 64.3%

       \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]
   11. Step-by-step derivation

   12. Simplified 64.3%

       \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{\frac{0.3333333333333333}{x}}{n} + \frac{-0.5}{n}}{x}}{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.1 \cdot 10^{-226}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-215}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-124}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 0.6:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{x}}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.2% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{x}}{n}}{x}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (+ (/ 1.0 n) (/ (+ (/ -0.5 n) (/ (/ 0.3333333333333333 x) n)) x)) x))

C

double code(double x, double n) {
	return ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 / n) + ((((-0.5d0) / n) + ((0.3333333333333333d0 / x) / n)) / x)) / x
end function

Java

public static double code(double x, double n) {
	return ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
}

Python

def code(x, n):
	return ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(-0.5 / n) + Float64(Float64(0.3333333333333333 / x) / n)) / x)) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 / n) + (((-0.5 / n) + ((0.3333333333333333 / x) / n)) / x)) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(-0.5 / n), $MachinePrecision] + N[(N[(0.3333333333333333 / x), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 53.6%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 70.1%

   \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
4. Step-by-step derivation

5. Simplified 70.5%

   \[\leadsto \color{blue}{\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right) - \log x}{n}} \]

6. Taylor expanded in x around inf 37.5%

   \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

7. Taylor expanded in n around inf 44.9%

   \[\leadsto \color{blue}{\frac{\left(1 + 0.3333333333333333 \cdot \frac{1}{{x}^{2}}\right) - 0.5 \cdot \frac{1}{x}}{n \cdot x}} \]
8. Step-by-step derivation

   1. sub-neg 44.9%

      \[\leadsto \frac{\color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{1}{{x}^{2}}\right) + \left(-0.5 \cdot \frac{1}{x}\right)}}{n \cdot x} \]

   2. +-commutative 44.9%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{2}} + 1\right)} + \left(-0.5 \cdot \frac{1}{x}\right)}{n \cdot x} \]

   3. associate-+l+ 44.9%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{{x}^{2}} + \left(1 + \left(-0.5 \cdot \frac{1}{x}\right)\right)}}{n \cdot x} \]

   4. associate-*r/ 44.9%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{2}}} + \left(1 + \left(-0.5 \cdot \frac{1}{x}\right)\right)}{n \cdot x} \]

   5. metadata-eval 44.9%

      \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333}}{{x}^{2}} + \left(1 + \left(-0.5 \cdot \frac{1}{x}\right)\right)}{n \cdot x} \]

   6. associate-*r/ 44.9%

      \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{2}} + \left(1 + \left(-\color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right)}{n \cdot x} \]

   7. metadata-eval 44.9%

      \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{2}} + \left(1 + \left(-\frac{\color{blue}{0.5}}{x}\right)\right)}{n \cdot x} \]

   8. distribute-neg-frac 44.9%

      \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{2}} + \left(1 + \color{blue}{\frac{-0.5}{x}}\right)}{n \cdot x} \]

   9. metadata-eval 44.9%

      \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{2}} + \left(1 + \frac{\color{blue}{-0.5}}{x}\right)}{n \cdot x} \]

   10. *-commutative 44.9%

       \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{2}} + \left(1 + \frac{-0.5}{x}\right)}{\color{blue}{x \cdot n}} \]

9. Simplified 44.9%

   \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{{x}^{2}} + \left(1 + \frac{-0.5}{x}\right)}{x \cdot n}} \]

10. Taylor expanded in x around inf 37.6%

    \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{n \cdot {x}^{2}} + \frac{1}{n}\right) - \frac{0.5}{n \cdot x}}{x}} \]
11. Step-by-step derivation

12. Simplified 45.5%

    \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\frac{\frac{0.3333333333333333}{x}}{n} + \frac{-0.5}{n}}{x}}{x}} \]

13. Final simplification 45.5%

    \[\leadsto \frac{\frac{1}{n} + \frac{\frac{-0.5}{n} + \frac{\frac{0.3333333333333333}{x}}{n}}{x}}{x} \]
14. Add Preprocessing

Alternative 18: 45.8% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* x n)))

C

double code(double x, double n) {
	return (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (x * n)
end function

Java

public static double code(double x, double n) {
	return (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
}

Python

def code(x, n):
	return (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n)

Julia

function code(x, n)
	return Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(x * n))
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
end

Wolfram

code[x_, n_] := N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 70.1%

   \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + 0.5 \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
4. Step-by-step derivation

5. Simplified 70.5%

   \[\leadsto \color{blue}{\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)\right) - \log x}{n}} \]

6. Taylor expanded in x around inf 37.5%

   \[\leadsto \frac{\color{blue}{\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(0.5 \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x} + \left(0.5 \cdot \frac{-0.6666666666666666 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} - \frac{1}{n}}{{x}^{2}} + \frac{0.3333333333333333}{{x}^{2}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{x}}{x}}}{n} \]

7. Taylor expanded in n around inf 44.9%

   \[\leadsto \color{blue}{\frac{\left(1 + 0.3333333333333333 \cdot \frac{1}{{x}^{2}}\right) - 0.5 \cdot \frac{1}{x}}{n \cdot x}} \]
8. Step-by-step derivation

   1. sub-neg 44.9%

      \[\leadsto \frac{\color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{1}{{x}^{2}}\right) + \left(-0.5 \cdot \frac{1}{x}\right)}}{n \cdot x} \]

   2. +-commutative 44.9%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{2}} + 1\right)} + \left(-0.5 \cdot \frac{1}{x}\right)}{n \cdot x} \]

   3. associate-+l+ 44.9%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{{x}^{2}} + \left(1 + \left(-0.5 \cdot \frac{1}{x}\right)\right)}}{n \cdot x} \]

   4. associate-*r/ 44.9%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{2}}} + \left(1 + \left(-0.5 \cdot \frac{1}{x}\right)\right)}{n \cdot x} \]

   5. metadata-eval 44.9%

      \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333}}{{x}^{2}} + \left(1 + \left(-0.5 \cdot \frac{1}{x}\right)\right)}{n \cdot x} \]

   6. associate-*r/ 44.9%

      \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{2}} + \left(1 + \left(-\color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right)}{n \cdot x} \]

   7. metadata-eval 44.9%

      \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{2}} + \left(1 + \left(-\frac{\color{blue}{0.5}}{x}\right)\right)}{n \cdot x} \]

   8. distribute-neg-frac 44.9%

      \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{2}} + \left(1 + \color{blue}{\frac{-0.5}{x}}\right)}{n \cdot x} \]

   9. metadata-eval 44.9%

      \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{2}} + \left(1 + \frac{\color{blue}{-0.5}}{x}\right)}{n \cdot x} \]

   10. *-commutative 44.9%

       \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{2}} + \left(1 + \frac{-0.5}{x}\right)}{\color{blue}{x \cdot n}} \]

9. Simplified 44.9%

   \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{{x}^{2}} + \left(1 + \frac{-0.5}{x}\right)}{x \cdot n}} \]

10. Taylor expanded in x around -inf 44.9%

    \[\leadsto \frac{\color{blue}{1 + -1 \cdot \frac{0.5 - 0.3333333333333333 \cdot \frac{1}{x}}{x}}}{x \cdot n} \]
11. Step-by-step derivation

    1. mul-1-neg 44.9%

       \[\leadsto \frac{1 + \color{blue}{\left(-\frac{0.5 - 0.3333333333333333 \cdot \frac{1}{x}}{x}\right)}}{x \cdot n} \]

    2. unsub-neg 44.9%

       \[\leadsto \frac{\color{blue}{1 - \frac{0.5 - 0.3333333333333333 \cdot \frac{1}{x}}{x}}}{x \cdot n} \]

    3. sub-neg 44.9%

       \[\leadsto \frac{1 - \frac{\color{blue}{0.5 + \left(-0.3333333333333333 \cdot \frac{1}{x}\right)}}{x}}{x \cdot n} \]

    4. associate-*r/ 44.9%

       \[\leadsto \frac{1 - \frac{0.5 + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{x}}\right)}{x}}{x \cdot n} \]

    5. metadata-eval 44.9%

       \[\leadsto \frac{1 - \frac{0.5 + \left(-\frac{\color{blue}{0.3333333333333333}}{x}\right)}{x}}{x \cdot n} \]

    6. distribute-neg-frac 44.9%

       \[\leadsto \frac{1 - \frac{0.5 + \color{blue}{\frac{-0.3333333333333333}{x}}}{x}}{x \cdot n} \]

    7. metadata-eval 44.9%

       \[\leadsto \frac{1 - \frac{0.5 + \frac{\color{blue}{-0.3333333333333333}}{x}}{x}}{x \cdot n} \]

12. Simplified 44.9%

    \[\leadsto \frac{\color{blue}{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}}{x \cdot n} \]

13. Final simplification 44.9%

    \[\leadsto \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n} \]
14. Add Preprocessing

Alternative 19: 40.2% 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 53.6%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 60.0%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
4. Step-by-step derivation

   1. mul-1-neg 60.0%

      \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

   2. log-rec 60.0%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 60.0%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   4. distribute-neg-frac 60.0%

      \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

   5. mul-1-neg 60.0%

      \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

   6. remove-double-neg 60.0%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

   7. *-commutative 60.0%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

5. Simplified 60.0%

   \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

6. Taylor expanded in n around inf 39.4%

   \[\leadsto \frac{\color{blue}{1}}{x \cdot n} \]

7. Final simplification 39.4%

   \[\leadsto \frac{1}{x \cdot n} \]
8. Add Preprocessing

Alternative 20: 40.6% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n}}{x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 n) x))

C

double code(double x, double n) {
	return (1.0 / n) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / n) / x
end function

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

function code(x, n)
	return Float64(Float64(1.0 / n) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / n) / x;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 53.6%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 60.0%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
4. Step-by-step derivation

   1. mul-1-neg 60.0%

      \[\leadsto \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]

   2. log-rec 60.0%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]

   3. mul-1-neg 60.0%

      \[\leadsto \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]

   4. distribute-neg-frac 60.0%

      \[\leadsto \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]

   5. mul-1-neg 60.0%

      \[\leadsto \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]

   6. remove-double-neg 60.0%

      \[\leadsto \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]

   7. *-commutative 60.0%

      \[\leadsto \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

5. Simplified 60.0%

   \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]

6. Taylor expanded in n around inf 39.4%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
7. Step-by-step derivation

   1. associate-/r* 40.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

8. Simplified 40.0%

   \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x}} \]

9. Final simplification 40.0%

   \[\leadsto \frac{\frac{1}{n}}{x} \]
10. Add Preprocessing

Alternative 21: 4.5% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ x n))

C

double code(double x, double n) {
	return x / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = x / n
end function

Java

public static double code(double x, double n) {
	return x / n;
}

Python

def code(x, n):
	return x / n

Julia

function code(x, n)
	return Float64(x / n)
end

MATLAB

function tmp = code(x, n)
	tmp = x / n;
end

Wolfram

code[x_, n_] := N[(x / n), $MachinePrecision]

Derivation

1. Initial program 53.6%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 28.5%

   \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]

4. Taylor expanded in x around inf 4.2%

   \[\leadsto \color{blue}{\frac{x}{n}} \]

5. Final simplification 4.2%

   \[\leadsto \frac{x}{n} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))