2nthrt (problem 3.4.6)

Percentage Accurate: 54.6% → 84.4%

Time: 43.6s

Alternatives: 20

Speedup: 1.7×

• 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: 54.6% 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: 84.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-173)
   (/ (pow x (- -1.0 (/ -1.0 n))) n)
   (if (<= (/ 1.0 n) 5e-15)
     (/ (log (/ (+ x 1.0) x)) n)
     (pow (cbrt (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n)))) 3.0))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-173) {
		tmp = pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-15) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = pow(cbrt((exp((log1p(x) / n)) - pow(x, (1.0 / n)))), 3.0);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-173) {
		tmp = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-15) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.pow(Math.cbrt((Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n)))), 3.0);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-173)
		tmp = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n);
	elseif (Float64(1.0 / n) <= 5e-15)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = cbrt(Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)))) ^ 3.0;
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-173], N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-15], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[Power[N[Power[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -1e-173

   1. Initial program 72.1%

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

   3. Taylor expanded in x around inf 85.8%

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

      1. mul-1-neg 85.8%

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

      2. log-rec 85.8%

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

      3. mul-1-neg 85.8%

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

      4. distribute-neg-frac 85.8%

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

      5. mul-1-neg 85.8%

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

      6. remove-double-neg 85.8%

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

      7. *-commutative 85.8%

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

   5. Simplified 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. associate-/r* 86.8%

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

      3. div-inv 86.8%

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

      4. pow-to-exp 86.8%

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

      5. pow1 86.8%

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

      6. pow-div 86.7%

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

   7. Applied egg-rr 86.7%

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

      1. *-lft-identity 86.7%

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

      2. sub-neg 86.7%

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

      3. metadata-eval 86.7%

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

   9. Simplified 86.7%

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

   if -1e-173 < (/.f64 1 n) < 4.99999999999999999e-15

   1. Initial program 39.5%

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

   3. Taylor expanded in n around inf 86.2%

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

      1. log1p-define 86.2%

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

   5. Simplified 86.2%

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

      1. log1p-undefine 86.2%

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

      2. diff-log 86.4%

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

   7. Applied egg-rr 86.4%

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

      1. +-commutative 86.4%

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

   9. Simplified 86.4%

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

   if 4.99999999999999999e-15 < (/.f64 1 n)

   1. Initial program 48.3%

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

      1. add-cube-cbrt 48.3%

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

      2. pow3 48.3%

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

      3. pow-to-exp 48.3%

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

      4. un-div-inv 48.3%

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

      5. +-commutative 48.3%

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

      6. log1p-define 96.3%

         \[\leadsto {\left(\sqrt[3]{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3} \]

   4. Applied egg-rr 96.3%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.052:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{0.16666666666666666 \cdot \left({\log x}^{3} - {\left(\mathsf{log1p}\left(x\right)\right)}^{3}\right)}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.052)
   (/
    (-
     (+
      (log1p x)
      (/
       (-
        (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0)))
        (/
         (* 0.16666666666666666 (- (pow (log x) 3.0) (pow (log1p x) 3.0)))
         n))
       n))
     (log x))
    n)
   (/ (pow x (- -1.0 (/ -1.0 n))) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.052) {
		tmp = ((log1p(x) + (((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) - ((0.16666666666666666 * (pow(log(x), 3.0) - pow(log1p(x), 3.0))) / n)) / n)) - log(x)) / n;
	} else {
		tmp = pow(x, (-1.0 - (-1.0 / n))) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.052) {
		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.log(x), 3.0) - Math.pow(Math.log1p(x), 3.0))) / n)) / n)) - Math.log(x)) / n;
	} else {
		tmp = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.052:
		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.log(x), 3.0) - math.pow(math.log1p(x), 3.0))) / n)) / n)) - math.log(x)) / n
	else:
		tmp = math.pow(x, (-1.0 - (-1.0 / n))) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.052)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) - Float64(Float64(0.16666666666666666 * Float64((log(x) ^ 3.0) - (log1p(x) ^ 3.0))) / n)) / n)) - log(x)) / n);
	else
		tmp = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.052], 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[(N[(0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0519999999999999976

   1. Initial program 45.2%

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

   3. Taylor expanded in n around -inf 76.7%

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

   5. Simplified 76.7%

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

   if 0.0519999999999999976 < x

   1. Initial program 70.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. log-rec 96.6%

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

      3. mul-1-neg 96.6%

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

      4. distribute-neg-frac 96.6%

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

      5. mul-1-neg 96.6%

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

      6. remove-double-neg 96.6%

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

      7. *-commutative 96.6%

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

   5. Simplified 96.6%

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

      1. *-un-lft-identity 96.6%

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

      2. associate-/r* 98.6%

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

      3. div-inv 98.6%

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

      4. pow-to-exp 98.6%

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

      5. pow1 98.6%

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

      6. pow-div 98.4%

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

   7. Applied egg-rr 98.4%

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

      1. *-lft-identity 98.4%

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

      2. sub-neg 98.4%

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

      3. metadata-eval 98.4%

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

   9. Simplified 98.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.052:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{0.16666666666666666 \cdot \left({\log x}^{3} - {\left(\mathsf{log1p}\left(x\right)\right)}^{3}\right)}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-173)
   (/ (pow x (- -1.0 (/ -1.0 n))) n)
   (if (<= (/ 1.0 n) 5e-15)
     (/ (log (/ (+ x 1.0) x)) n)
     (cbrt (pow (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))) 3.0)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-173) {
		tmp = pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-15) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = cbrt(pow((exp((log1p(x) / n)) - pow(x, (1.0 / n))), 3.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-173) {
		tmp = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-15) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.cbrt(Math.pow((Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n))), 3.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-173)
		tmp = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n);
	elseif (Float64(1.0 / n) <= 5e-15)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = cbrt((Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-173], N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-15], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[Power[N[Power[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -1e-173

   1. Initial program 72.1%

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

   3. Taylor expanded in x around inf 85.8%

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

      1. mul-1-neg 85.8%

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

      2. log-rec 85.8%

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

      3. mul-1-neg 85.8%

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

      4. distribute-neg-frac 85.8%

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

      5. mul-1-neg 85.8%

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

      6. remove-double-neg 85.8%

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

      7. *-commutative 85.8%

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

   5. Simplified 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. associate-/r* 86.8%

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

      3. div-inv 86.8%

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

      4. pow-to-exp 86.8%

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

      5. pow1 86.8%

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

      6. pow-div 86.7%

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

   7. Applied egg-rr 86.7%

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

      1. *-lft-identity 86.7%

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

      2. sub-neg 86.7%

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

      3. metadata-eval 86.7%

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

   9. Simplified 86.7%

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

   if -1e-173 < (/.f64 1 n) < 4.99999999999999999e-15

   1. Initial program 39.5%

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

   3. Taylor expanded in n around inf 86.2%

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

      1. log1p-define 86.2%

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

   5. Simplified 86.2%

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

      1. log1p-undefine 86.2%

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

      2. diff-log 86.4%

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

   7. Applied egg-rr 86.4%

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

      1. +-commutative 86.4%

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

   9. Simplified 86.4%

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

   if 4.99999999999999999e-15 < (/.f64 1 n)

   1. Initial program 48.3%

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

      1. add-cbrt-cube 48.3%

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

      2. pow3 48.3%

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

      3. pow-to-exp 48.3%

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

      4. un-div-inv 48.3%

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

      5. +-commutative 48.3%

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

      6. log1p-define 96.3%

         \[\leadsto \sqrt[3]{{\left(e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}} \]

   4. Applied egg-rr 96.3%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.025:\\ \;\;\;\;\frac{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + {\log x}^{2} \cdot -0.5}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.025)
   (/
    (-
     (/
      (+
       (* -0.16666666666666666 (/ (pow (log x) 3.0) n))
       (* (pow (log x) 2.0) -0.5))
      n)
     (log x))
    n)
   (/ (pow x (- -1.0 (/ -1.0 n))) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.025) {
		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 - (-1.0 / n))) / 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 <= 0.025d0) then
        tmp = (((((-0.16666666666666666d0) * ((log(x) ** 3.0d0) / n)) + ((log(x) ** 2.0d0) * (-0.5d0))) / n) - log(x)) / n
    else
        tmp = (x ** ((-1.0d0) - ((-1.0d0) / n))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.025) {
		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 - (-1.0 / n))) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.025:
		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 - (-1.0 / n))) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.025)
		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(-1.0 - Float64(-1.0 / n))) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.025)
		tmp = ((((-0.16666666666666666 * ((log(x) ^ 3.0) / n)) + ((log(x) ^ 2.0) * -0.5)) / n) - log(x)) / n;
	else
		tmp = (x ^ (-1.0 - (-1.0 / n))) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.025], 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[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.025000000000000001

   1. Initial program 45.2%

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

   3. Taylor expanded in x around 0 44.4%

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

   4. Taylor expanded in n around -inf 75.5%

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

      1. mul-1-neg 75.5%

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

   6. Simplified 75.5%

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

   if 0.025000000000000001 < x

   1. Initial program 70.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. log-rec 96.6%

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

      3. mul-1-neg 96.6%

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

      4. distribute-neg-frac 96.6%

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

      5. mul-1-neg 96.6%

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

      6. remove-double-neg 96.6%

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

      7. *-commutative 96.6%

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

   5. Simplified 96.6%

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

      1. *-un-lft-identity 96.6%

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

      2. associate-/r* 98.6%

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

      3. div-inv 98.6%

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

      4. pow-to-exp 98.6%

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

      5. pow1 98.6%

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

      6. pow-div 98.4%

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

   7. Applied egg-rr 98.4%

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

      1. *-lft-identity 98.4%

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

      2. sub-neg 98.4%

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

      3. metadata-eval 98.4%

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

   9. Simplified 98.4%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.025:\\ \;\;\;\;\frac{\frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + {\log x}^{2} \cdot -0.5}{n} - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-173)
   (/ (pow x (- -1.0 (/ -1.0 n))) n)
   (if (<= (/ 1.0 n) 5e-15)
     (/ (log (/ (+ x 1.0) x)) n)
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-173) {
		tmp = pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-15) {
		tmp = log(((x + 1.0) / 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) <= -1e-173) {
		tmp = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-15) {
		tmp = Math.log(((x + 1.0) / 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) <= -1e-173:
		tmp = math.pow(x, (-1.0 - (-1.0 / n))) / n
	elif (1.0 / n) <= 5e-15:
		tmp = math.log(((x + 1.0) / 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) <= -1e-173)
		tmp = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n);
	elseif (Float64(1.0 / n) <= 5e-15)
		tmp = Float64(log(Float64(Float64(x + 1.0) / 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], -1e-173], N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-15], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -1e-173

   1. Initial program 72.1%

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

   3. Taylor expanded in x around inf 85.8%

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

      1. mul-1-neg 85.8%

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

      2. log-rec 85.8%

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

      3. mul-1-neg 85.8%

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

      4. distribute-neg-frac 85.8%

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

      5. mul-1-neg 85.8%

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

      6. remove-double-neg 85.8%

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

      7. *-commutative 85.8%

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

   5. Simplified 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. associate-/r* 86.8%

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

      3. div-inv 86.8%

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

      4. pow-to-exp 86.8%

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

      5. pow1 86.8%

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

      6. pow-div 86.7%

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

   7. Applied egg-rr 86.7%

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

      1. *-lft-identity 86.7%

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

      2. sub-neg 86.7%

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

      3. metadata-eval 86.7%

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

   9. Simplified 86.7%

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

   if -1e-173 < (/.f64 1 n) < 4.99999999999999999e-15

   1. Initial program 39.5%

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

   3. Taylor expanded in n around inf 86.2%

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

      1. log1p-define 86.2%

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

   5. Simplified 86.2%

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

      1. log1p-undefine 86.2%

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

      2. diff-log 86.4%

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

   7. Applied egg-rr 86.4%

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

      1. +-commutative 86.4%

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

   9. Simplified 86.4%

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

   if 4.99999999999999999e-15 < (/.f64 1 n)

   1. Initial program 48.3%

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

   3. Taylor expanded in n around 0 48.3%

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

      1. log1p-define 96.3%

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

   5. Simplified 96.3%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+130}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-173)
   (/ (pow x (- -1.0 (/ -1.0 n))) n)
   (if (<= (/ 1.0 n) 5e-13)
     (/ (log (/ (+ x 1.0) x)) n)
     (if (<= (/ 1.0 n) 2e+130)
       (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
       (log1p (expm1 (/ 1.0 (* x n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-173) {
		tmp = pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+130) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = log1p(expm1((1.0 / (x * n))));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-173) {
		tmp = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+130) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log1p(Math.expm1((1.0 / (x * n))));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-173:
		tmp = math.pow(x, (-1.0 - (-1.0 / n))) / n
	elif (1.0 / n) <= 5e-13:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e+130:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.log1p(math.expm1((1.0 / (x * n))))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-173)
		tmp = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n);
	elseif (Float64(1.0 / n) <= 5e-13)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+130)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = log1p(expm1(Float64(1.0 / Float64(x * n))));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-173], N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-13], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+130], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1e-173

   1. Initial program 72.1%

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

   3. Taylor expanded in x around inf 85.8%

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

      1. mul-1-neg 85.8%

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

      2. log-rec 85.8%

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

      3. mul-1-neg 85.8%

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

      4. distribute-neg-frac 85.8%

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

      5. mul-1-neg 85.8%

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

      6. remove-double-neg 85.8%

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

      7. *-commutative 85.8%

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

   5. Simplified 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. associate-/r* 86.8%

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

      3. div-inv 86.8%

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

      4. pow-to-exp 86.8%

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

      5. pow1 86.8%

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

      6. pow-div 86.7%

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

   7. Applied egg-rr 86.7%

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

      1. *-lft-identity 86.7%

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

      2. sub-neg 86.7%

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

      3. metadata-eval 86.7%

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

   9. Simplified 86.7%

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

   if -1e-173 < (/.f64 1 n) < 4.9999999999999999e-13

   1. Initial program 39.2%

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

   3. Taylor expanded in n around inf 85.3%

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

      1. log1p-define 85.3%

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

   5. Simplified 85.3%

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

      1. log1p-undefine 85.3%

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

      2. diff-log 85.5%

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

   7. Applied egg-rr 85.5%

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

      1. +-commutative 85.5%

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

   9. Simplified 85.5%

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

   if 4.9999999999999999e-13 < (/.f64 1 n) < 2.0000000000000001e130

   1. Initial program 75.1%

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

   3. Taylor expanded in x around 0 75.2%

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

   if 2.0000000000000001e130 < (/.f64 1 n)

   1. Initial program 22.5%

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

   3. Taylor expanded in x around inf 0.8%

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

      1. mul-1-neg 0.8%

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

      2. log-rec 0.8%

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

      3. mul-1-neg 0.8%

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

      4. distribute-neg-frac 0.8%

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

      5. mul-1-neg 0.8%

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

      6. remove-double-neg 0.8%

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

      7. *-commutative 0.8%

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

   5. Simplified 0.8%

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

   6. Taylor expanded in n around inf 50.1%

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

      1. *-commutative 50.1%

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

   8. Simplified 50.1%

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

      1. log1p-expm1-u 87.1%

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

   10. Applied egg-rr 87.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+130}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-173)
   (/ (pow x (- -1.0 (/ -1.0 n))) n)
   (if (<= (/ 1.0 n) 5e-13)
     (/ (log (/ (+ x 1.0) x)) n)
     (- (exp (/ x n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-173) {
		tmp = pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d-173)) then
        tmp = (x ** ((-1.0d0) - ((-1.0d0) / n))) / n
    else if ((1.0d0 / n) <= 5d-13) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = exp((x / n)) - (x ** (1.0d0 / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-173) {
		tmp = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-173:
		tmp = math.pow(x, (-1.0 - (-1.0 / n))) / n
	elif (1.0 / n) <= 5e-13:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-173)
		tmp = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n);
	elseif (Float64(1.0 / n) <= 5e-13)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e-173)
		tmp = (x ^ (-1.0 - (-1.0 / n))) / n;
	elseif ((1.0 / n) <= 5e-13)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = exp((x / n)) - (x ^ (1.0 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-173], N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-13], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -1e-173

   1. Initial program 72.1%

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

   3. Taylor expanded in x around inf 85.8%

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

      1. mul-1-neg 85.8%

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

      2. log-rec 85.8%

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

      3. mul-1-neg 85.8%

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

      4. distribute-neg-frac 85.8%

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

      5. mul-1-neg 85.8%

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

      6. remove-double-neg 85.8%

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

      7. *-commutative 85.8%

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

   5. Simplified 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. associate-/r* 86.8%

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

      3. div-inv 86.8%

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

      4. pow-to-exp 86.8%

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

      5. pow1 86.8%

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

      6. pow-div 86.7%

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

   7. Applied egg-rr 86.7%

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

      1. *-lft-identity 86.7%

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

      2. sub-neg 86.7%

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

      3. metadata-eval 86.7%

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

   9. Simplified 86.7%

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

   if -1e-173 < (/.f64 1 n) < 4.9999999999999999e-13

   1. Initial program 39.2%

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

   3. Taylor expanded in n around inf 85.3%

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

      1. log1p-define 85.3%

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

   5. Simplified 85.3%

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

      1. log1p-undefine 85.3%

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

      2. diff-log 85.5%

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

   7. Applied egg-rr 85.5%

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

      1. +-commutative 85.5%

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

   9. Simplified 85.5%

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

   if 4.9999999999999999e-13 < (/.f64 1 n)

   1. Initial program 49.6%

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

   3. Taylor expanded in n around 0 49.6%

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

      1. log1p-define 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in x around 0 99.2%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - \log x}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5:\\ \;\;\;\;\frac{0.2}{n \cdot {x}^{5}}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-179}:\\ \;\;\;\;\frac{\frac{1 + \frac{\left(\frac{0.3333333333333333}{x} + \frac{\frac{\frac{0.2}{x} + -0.25}{x}}{x}\right) - 0.5}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-221}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-208}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+130}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333 - \frac{0.25 + \frac{-0.2}{x}}{x}}{x} - 0.5}{x}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- x (log x)) n)))
   (if (<= (/ 1.0 n) -5.0)
     (/ 0.2 (* n (pow x 5.0)))
     (if (<= (/ 1.0 n) -1e-179)
       (/
        (/
         (+
          1.0
          (/
           (- (+ (/ 0.3333333333333333 x) (/ (/ (+ (/ 0.2 x) -0.25) x) x)) 0.5)
           x))
         x)
        n)
       (if (<= (/ 1.0 n) -1e-221)
         t_0
         (if (<= (/ 1.0 n) 5e-208)
           (/ (/ 1.0 x) n)
           (if (<= (/ 1.0 n) 5e-15)
             t_0
             (if (<= (/ 1.0 n) 2e+130)
               (- 1.0 (pow x (/ 1.0 n)))
               (/
                (+
                 1.0
                 (/
                  (-
                   (/ (- 0.3333333333333333 (/ (+ 0.25 (/ -0.2 x)) x)) x)
                   0.5)
                  x))
                (* x n))))))))))

C

double code(double x, double n) {
	double t_0 = (x - log(x)) / n;
	double tmp;
	if ((1.0 / n) <= -5.0) {
		tmp = 0.2 / (n * pow(x, 5.0));
	} else if ((1.0 / n) <= -1e-179) {
		tmp = ((1.0 + ((((0.3333333333333333 / x) + ((((0.2 / x) + -0.25) / x) / x)) - 0.5) / x)) / x) / n;
	} else if ((1.0 / n) <= -1e-221) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-208) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-15) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+130) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - log(x)) / n
    if ((1.0d0 / n) <= (-5.0d0)) then
        tmp = 0.2d0 / (n * (x ** 5.0d0))
    else if ((1.0d0 / n) <= (-1d-179)) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 / x) + ((((0.2d0 / x) + (-0.25d0)) / x) / x)) - 0.5d0) / x)) / x) / n
    else if ((1.0d0 / n) <= (-1d-221)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-208) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 5d-15) then
        tmp = t_0
    else if ((1.0d0 / n) <= 2d+130) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (1.0d0 + ((((0.3333333333333333d0 - ((0.25d0 + ((-0.2d0) / x)) / x)) / x) - 0.5d0) / x)) / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (x - Math.log(x)) / n;
	double tmp;
	if ((1.0 / n) <= -5.0) {
		tmp = 0.2 / (n * Math.pow(x, 5.0));
	} else if ((1.0 / n) <= -1e-179) {
		tmp = ((1.0 + ((((0.3333333333333333 / x) + ((((0.2 / x) + -0.25) / x) / x)) - 0.5) / x)) / x) / n;
	} else if ((1.0 / n) <= -1e-221) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-208) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-15) {
		tmp = t_0;
	} else if ((1.0 / n) <= 2e+130) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (x - math.log(x)) / n
	tmp = 0
	if (1.0 / n) <= -5.0:
		tmp = 0.2 / (n * math.pow(x, 5.0))
	elif (1.0 / n) <= -1e-179:
		tmp = ((1.0 + ((((0.3333333333333333 / x) + ((((0.2 / x) + -0.25) / x) / x)) - 0.5) / x)) / x) / n
	elif (1.0 / n) <= -1e-221:
		tmp = t_0
	elif (1.0 / n) <= 5e-208:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e-15:
		tmp = t_0
	elif (1.0 / n) <= 2e+130:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(x - log(x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5.0)
		tmp = Float64(0.2 / Float64(n * (x ^ 5.0)));
	elseif (Float64(1.0 / n) <= -1e-179)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 / x) + Float64(Float64(Float64(Float64(0.2 / x) + -0.25) / x) / x)) - 0.5) / x)) / x) / n);
	elseif (Float64(1.0 / n) <= -1e-221)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-208)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-15)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 2e+130)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(Float64(0.25 + Float64(-0.2 / x)) / x)) / x) - 0.5) / x)) / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (x - log(x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5.0)
		tmp = 0.2 / (n * (x ^ 5.0));
	elseif ((1.0 / n) <= -1e-179)
		tmp = ((1.0 + ((((0.3333333333333333 / x) + ((((0.2 / x) + -0.25) / x) / x)) - 0.5) / x)) / x) / n;
	elseif ((1.0 / n) <= -1e-221)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-208)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 5e-15)
		tmp = t_0;
	elseif ((1.0 / n) <= 2e+130)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5.0], N[(0.2 / N[(n * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-179], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] + N[(N[(N[(N[(0.2 / x), $MachinePrecision] + -0.25), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-221], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-208], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-15], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+130], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 - N[(N[(0.25 + N[(-0.2 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 50.7%

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

      1. log1p-define 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in x around -inf 42.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in x around 0 79.5%

      \[\leadsto \color{blue}{\frac{0.2}{n \cdot {x}^{5}}} \]
   8. Step-by-step derivation

      1. *-commutative 79.5%

         \[\leadsto \frac{0.2}{\color{blue}{{x}^{5} \cdot n}} \]

   9. Simplified 79.5%

      \[\leadsto \color{blue}{\frac{0.2}{{x}^{5} \cdot n}} \]

   if -5 < (/.f64 1 n) < -1e-179

   1. Initial program 23.9%

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

   3. Taylor expanded in n around inf 53.6%

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

      1. log1p-define 53.6%

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

   5. Simplified 53.6%

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

   6. Taylor expanded in x around -inf 60.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
   7. Step-by-step derivation

      1. div-sub 60.4%

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \color{blue}{\left(\frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x}}{x} - \frac{0.3333333333333333}{x}\right)} - 0.5}{x} - 1}{x}}{n} \]

      2. mul-1-neg 60.4%

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \left(\frac{\color{blue}{-\frac{0.2 \cdot \frac{1}{x} - 0.25}{x}}}{x} - \frac{0.3333333333333333}{x}\right) - 0.5}{x} - 1}{x}}{n} \]

      3. sub-neg 60.4%

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \left(\frac{-\frac{\color{blue}{0.2 \cdot \frac{1}{x} + \left(-0.25\right)}}{x}}{x} - \frac{0.3333333333333333}{x}\right) - 0.5}{x} - 1}{x}}{n} \]

      4. un-div-inv 60.4%

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \left(\frac{-\frac{\color{blue}{\frac{0.2}{x}} + \left(-0.25\right)}{x}}{x} - \frac{0.3333333333333333}{x}\right) - 0.5}{x} - 1}{x}}{n} \]

      5. metadata-eval 60.4%

         \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \left(\frac{-\frac{\frac{0.2}{x} + \color{blue}{-0.25}}{x}}{x} - \frac{0.3333333333333333}{x}\right) - 0.5}{x} - 1}{x}}{n} \]

   8. Applied egg-rr 60.4%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \color{blue}{\left(\frac{-\frac{\frac{0.2}{x} + -0.25}{x}}{x} - \frac{0.3333333333333333}{x}\right)} - 0.5}{x} - 1}{x}}{n} \]

   if -1e-179 < (/.f64 1 n) < -1.00000000000000002e-221 or 4.99999999999999963e-208 < (/.f64 1 n) < 4.99999999999999999e-15

   1. Initial program 20.0%

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

   3. Taylor expanded in n around inf 79.6%

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

      1. log1p-define 79.6%

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

   5. Simplified 79.6%

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

   6. Taylor expanded in x around 0 63.3%

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

   if -1.00000000000000002e-221 < (/.f64 1 n) < 4.99999999999999963e-208

   1. Initial program 59.3%

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

   3. Taylor expanded in n around inf 92.5%

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

      1. log1p-define 92.5%

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

   5. Simplified 92.5%

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

   6. Taylor expanded in x around inf 65.9%

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

   if 4.99999999999999999e-15 < (/.f64 1 n) < 2.0000000000000001e130

   1. Initial program 71.0%

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

   3. Taylor expanded in x around 0 70.8%

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

   if 2.0000000000000001e130 < (/.f64 1 n)

   1. Initial program 22.5%

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

   3. Taylor expanded in n around inf 6.1%

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

      1. log1p-define 6.1%

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

   5. Simplified 6.1%

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

   6. Taylor expanded in x around -inf 80.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in n around 0 80.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{n \cdot x}} \]

   9. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25 + \frac{-0.2}{x}}{x}}{x}}{x}}{x \cdot n}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5:\\ \;\;\;\;\frac{0.2}{n \cdot {x}^{5}}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-179}:\\ \;\;\;\;\frac{\frac{1 + \frac{\left(\frac{0.3333333333333333}{x} + \frac{\frac{\frac{0.2}{x} + -0.25}{x}}{x}\right) - 0.5}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-221}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-208}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+130}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333 - \frac{0.25 + \frac{-0.2}{x}}{x}}{x} - 0.5}{x}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := -\frac{\log x}{n}\\ \mathbf{if}\;x \leq 4.5 \cdot 10^{-302}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-209}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{\frac{\frac{0.2}{x \cdot n} - \frac{0.25}{n}}{x} + \frac{0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + \frac{\frac{0.2 + 0.16666666666666666 \cdot \frac{-1}{x}}{x} - 0.25}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (- (/ (log x) n))))
   (if (<= x 4.5e-302)
     t_0
     (if (<= x 4.5e-209)
       t_1
       (if (<= x 8.5e-164)
         t_0
         (if (<= x 1.65e-107)
           t_1
           (if (<= x 2.6e-68)
             (/
              (+
               (/ 1.0 n)
               (/
                (-
                 (/
                  (+
                   (/ (- (/ 0.2 (* x n)) (/ 0.25 n)) x)
                   (/ 0.3333333333333333 n))
                  x)
                 (/ 0.5 n))
                x))
              x)
             (if (<= x 0.86)
               (/ (- x (log x)) n)
               (/
                (/
                 (+
                  1.0
                  (/
                   (-
                    (/
                     (+
                      0.3333333333333333
                      (/
                       (-
                        (/ (+ 0.2 (* 0.16666666666666666 (/ -1.0 x))) x)
                        0.25)
                       x))
                     x)
                    0.5)
                   x))
                 x)
                n)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = -(log(x) / n);
	double tmp;
	if (x <= 4.5e-302) {
		tmp = t_0;
	} else if (x <= 4.5e-209) {
		tmp = t_1;
	} else if (x <= 8.5e-164) {
		tmp = t_0;
	} else if (x <= 1.65e-107) {
		tmp = t_1;
	} else if (x <= 2.6e-68) {
		tmp = ((1.0 / n) + (((((((0.2 / (x * n)) - (0.25 / n)) / x) + (0.3333333333333333 / n)) / x) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.86) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + ((((0.2 + (0.16666666666666666 * (-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = -(log(x) / n)
    if (x <= 4.5d-302) then
        tmp = t_0
    else if (x <= 4.5d-209) then
        tmp = t_1
    else if (x <= 8.5d-164) then
        tmp = t_0
    else if (x <= 1.65d-107) then
        tmp = t_1
    else if (x <= 2.6d-68) then
        tmp = ((1.0d0 / n) + (((((((0.2d0 / (x * n)) - (0.25d0 / n)) / x) + (0.3333333333333333d0 / n)) / x) - (0.5d0 / n)) / x)) / x
    else if (x <= 0.86d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 + ((((0.3333333333333333d0 + ((((0.2d0 + (0.16666666666666666d0 * ((-1.0d0) / x))) / x) - 0.25d0) / x)) / x) - 0.5d0) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = -(Math.log(x) / n);
	double tmp;
	if (x <= 4.5e-302) {
		tmp = t_0;
	} else if (x <= 4.5e-209) {
		tmp = t_1;
	} else if (x <= 8.5e-164) {
		tmp = t_0;
	} else if (x <= 1.65e-107) {
		tmp = t_1;
	} else if (x <= 2.6e-68) {
		tmp = ((1.0 / n) + (((((((0.2 / (x * n)) - (0.25 / n)) / x) + (0.3333333333333333 / n)) / x) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.86) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + ((((0.2 + (0.16666666666666666 * (-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = -(math.log(x) / n)
	tmp = 0
	if x <= 4.5e-302:
		tmp = t_0
	elif x <= 4.5e-209:
		tmp = t_1
	elif x <= 8.5e-164:
		tmp = t_0
	elif x <= 1.65e-107:
		tmp = t_1
	elif x <= 2.6e-68:
		tmp = ((1.0 / n) + (((((((0.2 / (x * n)) - (0.25 / n)) / x) + (0.3333333333333333 / n)) / x) - (0.5 / n)) / x)) / x
	elif x <= 0.86:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 + ((((0.3333333333333333 + ((((0.2 + (0.16666666666666666 * (-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(-Float64(log(x) / n))
	tmp = 0.0
	if (x <= 4.5e-302)
		tmp = t_0;
	elseif (x <= 4.5e-209)
		tmp = t_1;
	elseif (x <= 8.5e-164)
		tmp = t_0;
	elseif (x <= 1.65e-107)
		tmp = t_1;
	elseif (x <= 2.6e-68)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.2 / Float64(x * n)) - Float64(0.25 / n)) / x) + Float64(0.3333333333333333 / n)) / x) - Float64(0.5 / n)) / x)) / x);
	elseif (x <= 0.86)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(Float64(Float64(Float64(0.2 + Float64(0.16666666666666666 * Float64(-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = -(log(x) / n);
	tmp = 0.0;
	if (x <= 4.5e-302)
		tmp = t_0;
	elseif (x <= 4.5e-209)
		tmp = t_1;
	elseif (x <= 8.5e-164)
		tmp = t_0;
	elseif (x <= 1.65e-107)
		tmp = t_1;
	elseif (x <= 2.6e-68)
		tmp = ((1.0 / n) + (((((((0.2 / (x * n)) - (0.25 / n)) / x) + (0.3333333333333333 / n)) / x) - (0.5 / n)) / x)) / x;
	elseif (x <= 0.86)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 + ((((0.3333333333333333 + ((((0.2 + (0.16666666666666666 * (-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision])}, If[LessEqual[x, 4.5e-302], t$95$0, If[LessEqual[x, 4.5e-209], t$95$1, If[LessEqual[x, 8.5e-164], t$95$0, If[LessEqual[x, 1.65e-107], t$95$1, If[LessEqual[x, 2.6e-68], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(N[(N[(N[(0.2 / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[(0.25 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.3333333333333333 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.86], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(N[(N[(N[(0.2 + N[(0.16666666666666666 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.25), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 4.50000000000000009e-302 or 4.4999999999999998e-209 < x < 8.50000000000000035e-164

   1. Initial program 76.2%

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

   3. Taylor expanded in x around 0 76.2%

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

   if 4.50000000000000009e-302 < x < 4.4999999999999998e-209 or 8.50000000000000035e-164 < x < 1.65000000000000002e-107

   1. Initial program 38.9%

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

   3. Taylor expanded in x around 0 38.9%

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

   4. Taylor expanded in n around inf 64.0%

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

      1. neg-mul-1 64.0%

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

      2. distribute-neg-frac2 64.0%

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

   6. Simplified 64.0%

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

   if 1.65000000000000002e-107 < x < 2.5999999999999998e-68

   1. Initial program 38.4%

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

   3. Taylor expanded in n around inf 33.2%

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

      1. log1p-define 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in x around -inf 65.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{n \cdot x} - 0.25 \cdot \frac{1}{n}}{x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 65.7%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\left(-\frac{\frac{0.2}{x \cdot n} - \frac{0.25}{n}}{x}\right) - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]

   if 2.5999999999999998e-68 < x < 0.859999999999999987

   1. Initial program 40.7%

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

   3. Taylor expanded in n around inf 44.0%

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

      1. log1p-define 44.0%

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

   5. Simplified 44.0%

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

   6. Taylor expanded in x around 0 42.2%

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

   if 0.859999999999999987 < x

   1. Initial program 70.0%

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

   3. Taylor expanded in n around inf 69.9%

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

      1. log1p-define 69.9%

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

   5. Simplified 69.9%

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

   6. Taylor expanded in x around -inf 64.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.16666666666666666 \cdot \frac{1}{x} - 0.2}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-302}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-209}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-164}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-107}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{\frac{\frac{0.2}{x \cdot n} - \frac{0.25}{n}}{x} + \frac{0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + \frac{\frac{0.2 + 0.16666666666666666 \cdot \frac{-1}{x}}{x} - 0.25}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+130}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333 - \frac{0.25 + \frac{-0.2}{x}}{x}}{x} - 0.5}{x}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-173)
   (/ (pow x (- -1.0 (/ -1.0 n))) n)
   (if (<= (/ 1.0 n) 5e-13)
     (/ (log (/ (+ x 1.0) x)) n)
     (if (<= (/ 1.0 n) 2e+130)
       (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
       (/
        (+
         1.0
         (/ (- (/ (- 0.3333333333333333 (/ (+ 0.25 (/ -0.2 x)) x)) x) 0.5) x))
        (* x n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-173) {
		tmp = pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+130) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d-173)) then
        tmp = (x ** ((-1.0d0) - ((-1.0d0) / n))) / n
    else if ((1.0d0 / n) <= 5d-13) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d+130) then
        tmp = (1.0d0 + (x / n)) - (x ** (1.0d0 / n))
    else
        tmp = (1.0d0 + ((((0.3333333333333333d0 - ((0.25d0 + ((-0.2d0) / x)) / x)) / x) - 0.5d0) / x)) / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-173) {
		tmp = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+130) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-173:
		tmp = math.pow(x, (-1.0 - (-1.0 / n))) / n
	elif (1.0 / n) <= 5e-13:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e+130:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-173)
		tmp = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n);
	elseif (Float64(1.0 / n) <= 5e-13)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+130)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(Float64(0.25 + Float64(-0.2 / x)) / x)) / x) - 0.5) / x)) / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e-173)
		tmp = (x ^ (-1.0 - (-1.0 / n))) / n;
	elseif ((1.0 / n) <= 5e-13)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e+130)
		tmp = (1.0 + (x / n)) - (x ^ (1.0 / n));
	else
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-173], N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-13], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+130], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 - N[(N[(0.25 + N[(-0.2 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1e-173

   1. Initial program 72.1%

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

   3. Taylor expanded in x around inf 85.8%

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

      1. mul-1-neg 85.8%

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

      2. log-rec 85.8%

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

      3. mul-1-neg 85.8%

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

      4. distribute-neg-frac 85.8%

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

      5. mul-1-neg 85.8%

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

      6. remove-double-neg 85.8%

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

      7. *-commutative 85.8%

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

   5. Simplified 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. associate-/r* 86.8%

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

      3. div-inv 86.8%

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

      4. pow-to-exp 86.8%

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

      5. pow1 86.8%

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

      6. pow-div 86.7%

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

   7. Applied egg-rr 86.7%

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

      1. *-lft-identity 86.7%

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

      2. sub-neg 86.7%

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

      3. metadata-eval 86.7%

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

   9. Simplified 86.7%

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

   if -1e-173 < (/.f64 1 n) < 4.9999999999999999e-13

   1. Initial program 39.2%

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

   3. Taylor expanded in n around inf 85.3%

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

      1. log1p-define 85.3%

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

   5. Simplified 85.3%

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

      1. log1p-undefine 85.3%

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

      2. diff-log 85.5%

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

   7. Applied egg-rr 85.5%

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

      1. +-commutative 85.5%

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

   9. Simplified 85.5%

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

   if 4.9999999999999999e-13 < (/.f64 1 n) < 2.0000000000000001e130

   1. Initial program 75.1%

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

   3. Taylor expanded in x around 0 75.2%

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

   if 2.0000000000000001e130 < (/.f64 1 n)

   1. Initial program 22.5%

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

   3. Taylor expanded in n around inf 6.1%

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

      1. log1p-define 6.1%

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

   5. Simplified 6.1%

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

   6. Taylor expanded in x around -inf 80.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in n around 0 80.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{n \cdot x}} \]

   9. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25 + \frac{-0.2}{x}}{x}}{x}}{x}}{x \cdot n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+130}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333 - \frac{0.25 + \frac{-0.2}{x}}{x}}{x} - 0.5}{x}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{\log x}{n}\\ \mathbf{if}\;x \leq 9 \cdot 10^{-180}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{0.5}{n} - \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10^{-69}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{\frac{\frac{0.2}{x \cdot n} - \frac{0.25}{n}}{x} + \frac{0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + \frac{\frac{0.2 + 0.16666666666666666 \cdot \frac{-1}{x}}{x} - 0.25}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (/ (log x) n))))
   (if (<= x 9e-180)
     t_0
     (if (<= x 2.8e-164)
       (/ (- (/ 1.0 n) (/ (- (/ 0.5 n) (/ 0.3333333333333333 (* x n))) x)) x)
       (if (<= x 1.2e-109)
         t_0
         (if (<= x 1e-69)
           (/
            (+
             (/ 1.0 n)
             (/
              (-
               (/
                (+
                 (/ (- (/ 0.2 (* x n)) (/ 0.25 n)) x)
                 (/ 0.3333333333333333 n))
                x)
               (/ 0.5 n))
              x))
            x)
           (if (<= x 0.86)
             (/ (- x (log x)) n)
             (/
              (/
               (+
                1.0
                (/
                 (-
                  (/
                   (+
                    0.3333333333333333
                    (/
                     (- (/ (+ 0.2 (* 0.16666666666666666 (/ -1.0 x))) x) 0.25)
                     x))
                   x)
                  0.5)
                 x))
               x)
              n))))))))

C

double code(double x, double n) {
	double t_0 = -(log(x) / n);
	double tmp;
	if (x <= 9e-180) {
		tmp = t_0;
	} else if (x <= 2.8e-164) {
		tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (x * n))) / x)) / x;
	} else if (x <= 1.2e-109) {
		tmp = t_0;
	} else if (x <= 1e-69) {
		tmp = ((1.0 / n) + (((((((0.2 / (x * n)) - (0.25 / n)) / x) + (0.3333333333333333 / n)) / x) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.86) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + ((((0.2 + (0.16666666666666666 * (-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(log(x) / n)
    if (x <= 9d-180) then
        tmp = t_0
    else if (x <= 2.8d-164) then
        tmp = ((1.0d0 / n) - (((0.5d0 / n) - (0.3333333333333333d0 / (x * n))) / x)) / x
    else if (x <= 1.2d-109) then
        tmp = t_0
    else if (x <= 1d-69) then
        tmp = ((1.0d0 / n) + (((((((0.2d0 / (x * n)) - (0.25d0 / n)) / x) + (0.3333333333333333d0 / n)) / x) - (0.5d0 / n)) / x)) / x
    else if (x <= 0.86d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 + ((((0.3333333333333333d0 + ((((0.2d0 + (0.16666666666666666d0 * ((-1.0d0) / x))) / x) - 0.25d0) / x)) / x) - 0.5d0) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -(Math.log(x) / n);
	double tmp;
	if (x <= 9e-180) {
		tmp = t_0;
	} else if (x <= 2.8e-164) {
		tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (x * n))) / x)) / x;
	} else if (x <= 1.2e-109) {
		tmp = t_0;
	} else if (x <= 1e-69) {
		tmp = ((1.0 / n) + (((((((0.2 / (x * n)) - (0.25 / n)) / x) + (0.3333333333333333 / n)) / x) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.86) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + ((((0.2 + (0.16666666666666666 * (-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -(math.log(x) / n)
	tmp = 0
	if x <= 9e-180:
		tmp = t_0
	elif x <= 2.8e-164:
		tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (x * n))) / x)) / x
	elif x <= 1.2e-109:
		tmp = t_0
	elif x <= 1e-69:
		tmp = ((1.0 / n) + (((((((0.2 / (x * n)) - (0.25 / n)) / x) + (0.3333333333333333 / n)) / x) - (0.5 / n)) / x)) / x
	elif x <= 0.86:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 + ((((0.3333333333333333 + ((((0.2 + (0.16666666666666666 * (-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(-Float64(log(x) / n))
	tmp = 0.0
	if (x <= 9e-180)
		tmp = t_0;
	elseif (x <= 2.8e-164)
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(0.5 / n) - Float64(0.3333333333333333 / Float64(x * n))) / x)) / x);
	elseif (x <= 1.2e-109)
		tmp = t_0;
	elseif (x <= 1e-69)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.2 / Float64(x * n)) - Float64(0.25 / n)) / x) + Float64(0.3333333333333333 / n)) / x) - Float64(0.5 / n)) / x)) / x);
	elseif (x <= 0.86)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(Float64(Float64(Float64(0.2 + Float64(0.16666666666666666 * Float64(-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -(log(x) / n);
	tmp = 0.0;
	if (x <= 9e-180)
		tmp = t_0;
	elseif (x <= 2.8e-164)
		tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (x * n))) / x)) / x;
	elseif (x <= 1.2e-109)
		tmp = t_0;
	elseif (x <= 1e-69)
		tmp = ((1.0 / n) + (((((((0.2 / (x * n)) - (0.25 / n)) / x) + (0.3333333333333333 / n)) / x) - (0.5 / n)) / x)) / x;
	elseif (x <= 0.86)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 + ((((0.3333333333333333 + ((((0.2 + (0.16666666666666666 * (-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision])}, If[LessEqual[x, 9e-180], t$95$0, If[LessEqual[x, 2.8e-164], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(0.5 / n), $MachinePrecision] - N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.2e-109], t$95$0, If[LessEqual[x, 1e-69], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(N[(N[(N[(0.2 / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[(0.25 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.3333333333333333 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.86], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(N[(N[(N[(0.2 + N[(0.16666666666666666 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.25), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 9.00000000000000019e-180 or 2.8000000000000001e-164 < x < 1.19999999999999994e-109

   1. Initial program 47.0%

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

   3. Taylor expanded in x around 0 47.0%

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

   4. Taylor expanded in n around inf 56.7%

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

      1. neg-mul-1 56.7%

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

      2. distribute-neg-frac2 56.7%

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

   6. Simplified 56.7%

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

   if 9.00000000000000019e-180 < x < 2.8000000000000001e-164

   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 inf 16.0%

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

      1. log1p-define 16.0%

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

   5. Simplified 16.0%

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

   6. Taylor expanded in x around -inf 78.4%

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

      1. mul-1-neg 78.4%

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

      2. mul-1-neg 78.4%

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

      3. associate-*r/ 78.4%

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

      4. metadata-eval 78.4%

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

      5. *-commutative 78.4%

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

      6. associate-*r/ 78.4%

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

      7. metadata-eval 78.4%

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

   8. Simplified 78.4%

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

   if 1.19999999999999994e-109 < x < 9.9999999999999996e-70

   1. Initial program 38.4%

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

   3. Taylor expanded in n around inf 33.2%

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

      1. log1p-define 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in x around -inf 65.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{n \cdot x} - 0.25 \cdot \frac{1}{n}}{x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 65.7%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\left(-\frac{\frac{0.2}{x \cdot n} - \frac{0.25}{n}}{x}\right) - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]

   if 9.9999999999999996e-70 < x < 0.859999999999999987

   1. Initial program 40.7%

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

   3. Taylor expanded in n around inf 44.0%

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

      1. log1p-define 44.0%

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

   5. Simplified 44.0%

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

   6. Taylor expanded in x around 0 42.2%

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

   if 0.859999999999999987 < x

   1. Initial program 70.0%

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

   3. Taylor expanded in n around inf 69.9%

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

      1. log1p-define 69.9%

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

   5. Simplified 69.9%

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

   6. Taylor expanded in x around -inf 64.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.16666666666666666 \cdot \frac{1}{x} - 0.2}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-180}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{0.5}{n} - \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-109}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 10^{-69}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{\frac{\frac{0.2}{x \cdot n} - \frac{0.25}{n}}{x} + \frac{0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + \frac{\frac{0.2 + 0.16666666666666666 \cdot \frac{-1}{x}}{x} - 0.25}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{\log x}{n}\\ \mathbf{if}\;x \leq 1.4 \cdot 10^{-180}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10^{-162}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{0.5}{n} - \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{\frac{\frac{0.2}{x \cdot n} - \frac{0.25}{n}}{x} + \frac{0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + \frac{\frac{0.2 + 0.16666666666666666 \cdot \frac{-1}{x}}{x} - 0.25}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- (/ (log x) n))))
   (if (<= x 1.4e-180)
     t_0
     (if (<= x 1e-162)
       (/ (- (/ 1.0 n) (/ (- (/ 0.5 n) (/ 0.3333333333333333 (* x n))) x)) x)
       (if (<= x 9.5e-109)
         t_0
         (if (<= x 2.4e-68)
           (/
            (+
             (/ 1.0 n)
             (/
              (-
               (/
                (+
                 (/ (- (/ 0.2 (* x n)) (/ 0.25 n)) x)
                 (/ 0.3333333333333333 n))
                x)
               (/ 0.5 n))
              x))
            x)
           (if (<= x 0.75)
             t_0
             (/
              (/
               (+
                1.0
                (/
                 (-
                  (/
                   (+
                    0.3333333333333333
                    (/
                     (- (/ (+ 0.2 (* 0.16666666666666666 (/ -1.0 x))) x) 0.25)
                     x))
                   x)
                  0.5)
                 x))
               x)
              n))))))))

C

double code(double x, double n) {
	double t_0 = -(log(x) / n);
	double tmp;
	if (x <= 1.4e-180) {
		tmp = t_0;
	} else if (x <= 1e-162) {
		tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (x * n))) / x)) / x;
	} else if (x <= 9.5e-109) {
		tmp = t_0;
	} else if (x <= 2.4e-68) {
		tmp = ((1.0 / n) + (((((((0.2 / (x * n)) - (0.25 / n)) / x) + (0.3333333333333333 / n)) / x) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.75) {
		tmp = t_0;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + ((((0.2 + (0.16666666666666666 * (-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(log(x) / n)
    if (x <= 1.4d-180) then
        tmp = t_0
    else if (x <= 1d-162) then
        tmp = ((1.0d0 / n) - (((0.5d0 / n) - (0.3333333333333333d0 / (x * n))) / x)) / x
    else if (x <= 9.5d-109) then
        tmp = t_0
    else if (x <= 2.4d-68) then
        tmp = ((1.0d0 / n) + (((((((0.2d0 / (x * n)) - (0.25d0 / n)) / x) + (0.3333333333333333d0 / n)) / x) - (0.5d0 / n)) / x)) / x
    else if (x <= 0.75d0) then
        tmp = t_0
    else
        tmp = ((1.0d0 + ((((0.3333333333333333d0 + ((((0.2d0 + (0.16666666666666666d0 * ((-1.0d0) / x))) / x) - 0.25d0) / x)) / x) - 0.5d0) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -(Math.log(x) / n);
	double tmp;
	if (x <= 1.4e-180) {
		tmp = t_0;
	} else if (x <= 1e-162) {
		tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (x * n))) / x)) / x;
	} else if (x <= 9.5e-109) {
		tmp = t_0;
	} else if (x <= 2.4e-68) {
		tmp = ((1.0 / n) + (((((((0.2 / (x * n)) - (0.25 / n)) / x) + (0.3333333333333333 / n)) / x) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.75) {
		tmp = t_0;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + ((((0.2 + (0.16666666666666666 * (-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -(math.log(x) / n)
	tmp = 0
	if x <= 1.4e-180:
		tmp = t_0
	elif x <= 1e-162:
		tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (x * n))) / x)) / x
	elif x <= 9.5e-109:
		tmp = t_0
	elif x <= 2.4e-68:
		tmp = ((1.0 / n) + (((((((0.2 / (x * n)) - (0.25 / n)) / x) + (0.3333333333333333 / n)) / x) - (0.5 / n)) / x)) / x
	elif x <= 0.75:
		tmp = t_0
	else:
		tmp = ((1.0 + ((((0.3333333333333333 + ((((0.2 + (0.16666666666666666 * (-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(-Float64(log(x) / n))
	tmp = 0.0
	if (x <= 1.4e-180)
		tmp = t_0;
	elseif (x <= 1e-162)
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(0.5 / n) - Float64(0.3333333333333333 / Float64(x * n))) / x)) / x);
	elseif (x <= 9.5e-109)
		tmp = t_0;
	elseif (x <= 2.4e-68)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.2 / Float64(x * n)) - Float64(0.25 / n)) / x) + Float64(0.3333333333333333 / n)) / x) - Float64(0.5 / n)) / x)) / x);
	elseif (x <= 0.75)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(Float64(Float64(Float64(0.2 + Float64(0.16666666666666666 * Float64(-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -(log(x) / n);
	tmp = 0.0;
	if (x <= 1.4e-180)
		tmp = t_0;
	elseif (x <= 1e-162)
		tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (x * n))) / x)) / x;
	elseif (x <= 9.5e-109)
		tmp = t_0;
	elseif (x <= 2.4e-68)
		tmp = ((1.0 / n) + (((((((0.2 / (x * n)) - (0.25 / n)) / x) + (0.3333333333333333 / n)) / x) - (0.5 / n)) / x)) / x;
	elseif (x <= 0.75)
		tmp = t_0;
	else
		tmp = ((1.0 + ((((0.3333333333333333 + ((((0.2 + (0.16666666666666666 * (-1.0 / x))) / x) - 0.25) / x)) / x) - 0.5) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = (-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision])}, If[LessEqual[x, 1.4e-180], t$95$0, If[LessEqual[x, 1e-162], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(0.5 / n), $MachinePrecision] - N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 9.5e-109], t$95$0, If[LessEqual[x, 2.4e-68], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(N[(N[(N[(0.2 / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[(0.25 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.3333333333333333 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.75], t$95$0, N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(N[(N[(N[(0.2 + N[(0.16666666666666666 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.25), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.39999999999999999e-180 or 9.99999999999999954e-163 < x < 9.49999999999999933e-109 or 2.39999999999999991e-68 < x < 0.75

   1. Initial program 45.4%

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

   3. Taylor expanded in x around 0 44.4%

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

   4. Taylor expanded in n around inf 52.4%

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

      1. neg-mul-1 52.4%

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

      2. distribute-neg-frac2 52.4%

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

   6. Simplified 52.4%

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

   if 1.39999999999999999e-180 < x < 9.99999999999999954e-163

   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 inf 16.0%

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

      1. log1p-define 16.0%

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

   5. Simplified 16.0%

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

   6. Taylor expanded in x around -inf 78.4%

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

      1. mul-1-neg 78.4%

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

      2. mul-1-neg 78.4%

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

      3. associate-*r/ 78.4%

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

      4. metadata-eval 78.4%

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

      5. *-commutative 78.4%

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

      6. associate-*r/ 78.4%

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

      7. metadata-eval 78.4%

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

   8. Simplified 78.4%

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

   if 9.49999999999999933e-109 < x < 2.39999999999999991e-68

   1. Initial program 38.4%

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

   3. Taylor expanded in n around inf 33.2%

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

      1. log1p-define 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in x around -inf 65.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{n \cdot x} - 0.25 \cdot \frac{1}{n}}{x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 65.7%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\left(-\frac{\frac{0.2}{x \cdot n} - \frac{0.25}{n}}{x}\right) - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]

   if 0.75 < x

   1. Initial program 70.0%

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

   3. Taylor expanded in n around inf 69.9%

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

      1. log1p-define 69.9%

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

   5. Simplified 69.9%

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

   6. Taylor expanded in x around -inf 64.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.16666666666666666 \cdot \frac{1}{x} - 0.2}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-180}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 10^{-162}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{0.5}{n} - \frac{0.3333333333333333}{x \cdot n}}{x}}{x}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-109}:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{\frac{\frac{0.2}{x \cdot n} - \frac{0.25}{n}}{x} + \frac{0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;-\frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + \frac{\frac{0.2 + 0.16666666666666666 \cdot \frac{-1}{x}}{x} - 0.25}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10000:\\ \;\;\;\;\frac{0.2}{n \cdot {x}^{5}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+130}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333 - \frac{0.25 + \frac{-0.2}{x}}{x}}{x} - 0.5}{x}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -10000.0)
   (/ 0.2 (* n (pow x 5.0)))
   (if (<= (/ 1.0 n) 5e-13)
     (/ (log (/ (+ x 1.0) x)) n)
     (if (<= (/ 1.0 n) 2e+130)
       (- 1.0 (pow x (/ 1.0 n)))
       (/
        (+
         1.0
         (/ (- (/ (- 0.3333333333333333 (/ (+ 0.25 (/ -0.2 x)) x)) x) 0.5) x))
        (* x n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -10000.0) {
		tmp = 0.2 / (n * pow(x, 5.0));
	} else if ((1.0 / n) <= 5e-13) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+130) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-10000.0d0)) then
        tmp = 0.2d0 / (n * (x ** 5.0d0))
    else if ((1.0d0 / n) <= 5d-13) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d+130) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (1.0d0 + ((((0.3333333333333333d0 - ((0.25d0 + ((-0.2d0) / x)) / x)) / x) - 0.5d0) / x)) / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -10000.0) {
		tmp = 0.2 / (n * Math.pow(x, 5.0));
	} else if ((1.0 / n) <= 5e-13) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+130) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -10000.0:
		tmp = 0.2 / (n * math.pow(x, 5.0))
	elif (1.0 / n) <= 5e-13:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e+130:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -10000.0)
		tmp = Float64(0.2 / Float64(n * (x ^ 5.0)));
	elseif (Float64(1.0 / n) <= 5e-13)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+130)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(Float64(0.25 + Float64(-0.2 / x)) / x)) / x) - 0.5) / x)) / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -10000.0)
		tmp = 0.2 / (n * (x ^ 5.0));
	elseif ((1.0 / n) <= 5e-13)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e+130)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -10000.0], N[(0.2 / N[(n * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-13], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+130], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 - N[(N[(0.25 + N[(-0.2 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1e4

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 49.4%

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

      1. log1p-define 49.4%

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

   5. Simplified 49.4%

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

   6. Taylor expanded in x around -inf 42.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in x around 0 79.0%

      \[\leadsto \color{blue}{\frac{0.2}{n \cdot {x}^{5}}} \]
   8. Step-by-step derivation

      1. *-commutative 79.0%

         \[\leadsto \frac{0.2}{\color{blue}{{x}^{5} \cdot n}} \]

   9. Simplified 79.0%

      \[\leadsto \color{blue}{\frac{0.2}{{x}^{5} \cdot n}} \]

   if -1e4 < (/.f64 1 n) < 4.9999999999999999e-13

   1. Initial program 34.6%

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

   3. Taylor expanded in n around inf 74.6%

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

      1. log1p-define 74.5%

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

   5. Simplified 74.5%

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

      1. log1p-undefine 74.6%

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

      2. diff-log 74.9%

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

   7. Applied egg-rr 74.9%

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

      1. +-commutative 74.9%

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

   9. Simplified 74.9%

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

   if 4.9999999999999999e-13 < (/.f64 1 n) < 2.0000000000000001e130

   1. Initial program 75.1%

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

   3. Taylor expanded in x around 0 75.1%

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

   if 2.0000000000000001e130 < (/.f64 1 n)

   1. Initial program 22.5%

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

   3. Taylor expanded in n around inf 6.1%

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

      1. log1p-define 6.1%

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

   5. Simplified 6.1%

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

   6. Taylor expanded in x around -inf 80.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in n around 0 80.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{n \cdot x}} \]

   9. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25 + \frac{-0.2}{x}}{x}}{x}}{x}}{x \cdot n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10000:\\ \;\;\;\;\frac{0.2}{n \cdot {x}^{5}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+130}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333 - \frac{0.25 + \frac{-0.2}{x}}{x}}{x} - 0.5}{x}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+130}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333 - \frac{0.25 + \frac{-0.2}{x}}{x}}{x} - 0.5}{x}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e-173)
   (/ (pow x (- -1.0 (/ -1.0 n))) n)
   (if (<= (/ 1.0 n) 5e-13)
     (/ (log (/ (+ x 1.0) x)) n)
     (if (<= (/ 1.0 n) 2e+130)
       (- 1.0 (pow x (/ 1.0 n)))
       (/
        (+
         1.0
         (/ (- (/ (- 0.3333333333333333 (/ (+ 0.25 (/ -0.2 x)) x)) x) 0.5) x))
        (* x n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-173) {
		tmp = pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+130) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d-173)) then
        tmp = (x ** ((-1.0d0) - ((-1.0d0) / n))) / n
    else if ((1.0d0 / n) <= 5d-13) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d+130) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (1.0d0 + ((((0.3333333333333333d0 - ((0.25d0 + ((-0.2d0) / x)) / x)) / x) - 0.5d0) / x)) / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e-173) {
		tmp = Math.pow(x, (-1.0 - (-1.0 / n))) / n;
	} else if ((1.0 / n) <= 5e-13) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+130) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e-173:
		tmp = math.pow(x, (-1.0 - (-1.0 / n))) / n
	elif (1.0 / n) <= 5e-13:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e+130:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-173)
		tmp = Float64((x ^ Float64(-1.0 - Float64(-1.0 / n))) / n);
	elseif (Float64(1.0 / n) <= 5e-13)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+130)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 - Float64(Float64(0.25 + Float64(-0.2 / x)) / x)) / x) - 0.5) / x)) / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e-173)
		tmp = (x ^ (-1.0 - (-1.0 / n))) / n;
	elseif ((1.0 / n) <= 5e-13)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e+130)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (1.0 + ((((0.3333333333333333 - ((0.25 + (-0.2 / x)) / x)) / x) - 0.5) / x)) / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-173], N[(N[Power[x, N[(-1.0 - N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-13], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+130], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 - N[(N[(0.25 + N[(-0.2 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1e-173

   1. Initial program 72.1%

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

   3. Taylor expanded in x around inf 85.8%

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

      1. mul-1-neg 85.8%

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

      2. log-rec 85.8%

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

      3. mul-1-neg 85.8%

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

      4. distribute-neg-frac 85.8%

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

      5. mul-1-neg 85.8%

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

      6. remove-double-neg 85.8%

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

      7. *-commutative 85.8%

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

   5. Simplified 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. associate-/r* 86.8%

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

      3. div-inv 86.8%

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

      4. pow-to-exp 86.8%

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

      5. pow1 86.8%

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

      6. pow-div 86.7%

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

   7. Applied egg-rr 86.7%

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

      1. *-lft-identity 86.7%

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

      2. sub-neg 86.7%

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

      3. metadata-eval 86.7%

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

   9. Simplified 86.7%

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

   if -1e-173 < (/.f64 1 n) < 4.9999999999999999e-13

   1. Initial program 39.2%

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

   3. Taylor expanded in n around inf 85.3%

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

      1. log1p-define 85.3%

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

   5. Simplified 85.3%

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

      1. log1p-undefine 85.3%

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

      2. diff-log 85.5%

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

   7. Applied egg-rr 85.5%

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

      1. +-commutative 85.5%

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

   9. Simplified 85.5%

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

   if 4.9999999999999999e-13 < (/.f64 1 n) < 2.0000000000000001e130

   1. Initial program 75.1%

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

   3. Taylor expanded in x around 0 75.1%

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

   if 2.0000000000000001e130 < (/.f64 1 n)

   1. Initial program 22.5%

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

   3. Taylor expanded in n around inf 6.1%

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

      1. log1p-define 6.1%

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

   5. Simplified 6.1%

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

   6. Taylor expanded in x around -inf 80.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in n around 0 80.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{n \cdot x}} \]

   9. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25 + \frac{-0.2}{x}}{x}}{x}}{x}}{x \cdot n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\frac{{x}^{\left(-1 - \frac{-1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+130}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333 - \frac{0.25 + \frac{-0.2}{x}}{x}}{x} - 0.5}{x}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.8% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 + \frac{\left(\frac{0.3333333333333333}{x} + \frac{\frac{\frac{0.2}{x} + -0.25}{x}}{x}\right) - 0.5}{x}}{x}}{n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/
  (/
   (+
    1.0
    (/ (- (+ (/ 0.3333333333333333 x) (/ (/ (+ (/ 0.2 x) -0.25) x) x)) 0.5) x))
   x)
  n))

C

double code(double x, double n) {
	return ((1.0 + ((((0.3333333333333333 / x) + ((((0.2 / x) + -0.25) / x) / x)) - 0.5) / x)) / x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 + ((((0.3333333333333333d0 / x) + ((((0.2d0 / x) + (-0.25d0)) / x) / x)) - 0.5d0) / x)) / x) / n
end function

Java

public static double code(double x, double n) {
	return ((1.0 + ((((0.3333333333333333 / x) + ((((0.2 / x) + -0.25) / x) / x)) - 0.5) / x)) / x) / n;
}

Python

def code(x, n):
	return ((1.0 + ((((0.3333333333333333 / x) + ((((0.2 / x) + -0.25) / x) / x)) - 0.5) / x)) / x) / n

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 / x) + Float64(Float64(Float64(Float64(0.2 / x) + -0.25) / x) / x)) - 0.5) / x)) / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 + ((((0.3333333333333333 / x) + ((((0.2 / x) + -0.25) / x) / x)) - 0.5) / x)) / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] + N[(N[(N[(N[(0.2 / x), $MachinePrecision] + -0.25), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in n around inf 58.5%

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

   1. log1p-define 58.5%

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

5. Simplified 58.5%

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

6. Taylor expanded in x around -inf 49.2%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
7. Step-by-step derivation

   1. div-sub 49.2%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \color{blue}{\left(\frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x}}{x} - \frac{0.3333333333333333}{x}\right)} - 0.5}{x} - 1}{x}}{n} \]

   2. mul-1-neg 49.2%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \left(\frac{\color{blue}{-\frac{0.2 \cdot \frac{1}{x} - 0.25}{x}}}{x} - \frac{0.3333333333333333}{x}\right) - 0.5}{x} - 1}{x}}{n} \]

   3. sub-neg 49.2%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \left(\frac{-\frac{\color{blue}{0.2 \cdot \frac{1}{x} + \left(-0.25\right)}}{x}}{x} - \frac{0.3333333333333333}{x}\right) - 0.5}{x} - 1}{x}}{n} \]

   4. un-div-inv 49.2%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \left(\frac{-\frac{\color{blue}{\frac{0.2}{x}} + \left(-0.25\right)}{x}}{x} - \frac{0.3333333333333333}{x}\right) - 0.5}{x} - 1}{x}}{n} \]

   5. metadata-eval 49.2%

      \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \left(\frac{-\frac{\frac{0.2}{x} + \color{blue}{-0.25}}{x}}{x} - \frac{0.3333333333333333}{x}\right) - 0.5}{x} - 1}{x}}{n} \]

8. Applied egg-rr 49.2%

   \[\leadsto \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \color{blue}{\left(\frac{-\frac{\frac{0.2}{x} + -0.25}{x}}{x} - \frac{0.3333333333333333}{x}\right)} - 0.5}{x} - 1}{x}}{n} \]

9. Final simplification 49.2%

   \[\leadsto \frac{\frac{1 + \frac{\left(\frac{0.3333333333333333}{x} + \frac{\frac{\frac{0.2}{x} + -0.25}{x}}{x}\right) - 0.5}{x}}{x}}{n} \]
10. Add Preprocessing

Alternative 16: 46.7% accurate, 12.4× speedup

Math

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

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(0.3333333333333333 / Float64(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[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in n around inf 58.5%

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

   1. log1p-define 58.5%

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

5. Simplified 58.5%

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

6. Taylor expanded in x around -inf 48.9%

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

   1. mul-1-neg 48.9%

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

   2. mul-1-neg 48.9%

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

   3. associate-*r/ 48.9%

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

   4. metadata-eval 48.9%

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

   5. *-commutative 48.9%

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

   6. associate-*r/ 48.9%

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

   7. metadata-eval 48.9%

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

8. Simplified 48.9%

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

9. Final simplification 48.9%

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

Alternative 17: 46.6% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in n around inf 58.5%

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

   1. log1p-define 58.5%

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

5. Simplified 58.5%

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

6. Taylor expanded in x around -inf 49.2%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{x} - 0.25}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

7. Taylor expanded in x around inf 48.9%

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

8. Final simplification 48.9%

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

Alternative 18: 40.3% 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 57.1%

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

3. Taylor expanded in x around inf 62.8%

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

   1. mul-1-neg 62.8%

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

   2. log-rec 62.8%

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

   3. mul-1-neg 62.8%

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

   4. distribute-neg-frac 62.8%

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

   5. mul-1-neg 62.8%

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

   6. remove-double-neg 62.8%

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

   7. *-commutative 62.8%

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

5. Simplified 62.8%

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

6. Taylor expanded in n around inf 41.1%

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

   1. *-commutative 41.1%

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

8. Simplified 41.1%

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

9. Final simplification 41.1%

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

Alternative 19: 40.8% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in n around inf 58.5%

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

   1. log1p-define 58.5%

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

5. Simplified 58.5%

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

6. Taylor expanded in x around inf 42.1%

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

7. Final simplification 42.1%

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

Alternative 20: 4.5% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in n around inf 58.5%

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

   1. log1p-define 58.5%

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

5. Simplified 58.5%

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

6. Taylor expanded in x around 0 27.4%

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

7. Taylor expanded in x around inf 4.3%

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

8. Final simplification 4.3%

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

Reproduce

herbie shell --seed 2024055 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))