2nthrt (problem 3.4.6)

Percentage Accurate: 53.4% → 83.4%

Time: 59.2s

Alternatives: 21

Speedup: 1.8×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 2.4 \cdot 10^{-280}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \mathbf{elif}\;x \leq 1400:\\ \;\;\;\;\frac{\log \left(\frac{e^{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}, \mathsf{log1p}\left(x\right)\right)}}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 2.4e-280)
     (- (exp (/ x n)) t_0)
     (if (<= x 1400.0)
       (/
        (log
         (/
          (exp
           (fma 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n) (log1p x)))
          x))
        n)
       (/ (/ t_0 n) x)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 2.4e-280) {
		tmp = exp((x / n)) - t_0;
	} else if (x <= 1400.0) {
		tmp = log((exp(fma(0.5, ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n), log1p(x))) / x)) / n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 2.4e-280)
		tmp = Float64(exp(Float64(x / n)) - t_0);
	elseif (x <= 1400.0)
		tmp = Float64(log(Float64(exp(fma(0.5, Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n), log1p(x))) / x)) / n);
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.4e-280], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 1400.0], N[(N[Log[N[(N[Exp[N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.3999999999999998e-280

   1. Initial program 78.1%

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

   3. Taylor expanded in n around 0 78.1%

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

      1. log1p-define 85.5%

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

      2. *-rgt-identity 85.5%

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

      3. associate-/l* 85.5%

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

      4. exp-to-pow 85.5%

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

   5. Simplified 85.5%

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

   6. Taylor expanded in x around 0 85.5%

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

   if 2.3999999999999998e-280 < x < 1400

   1. Initial program 42.6%

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

   3. Taylor expanded in n around inf 66.3%

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

   5. Simplified 66.3%

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

      1. add-log-exp 77.4%

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

      2. diff-log 77.4%

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

      3. +-commutative 77.4%

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

      4. fma-define 77.4%

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

   7. Applied egg-rr 77.4%

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

   if 1400 < x

   1. Initial program 63.4%

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

   3. Taylor expanded in x around inf 98.6%

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

      1. associate-/r* 99.3%

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

      2. mul-1-neg 99.3%

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

      3. log-rec 99.3%

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

      4. mul-1-neg 99.3%

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

      5. distribute-neg-frac 99.3%

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

      6. mul-1-neg 99.3%

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

      7. remove-double-neg 99.3%

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

      8. *-rgt-identity 99.3%

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

      9. associate-/l* 99.3%

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

      10. exp-to-pow 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-280}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1400:\\ \;\;\;\;\frac{\log \left(\frac{e^{\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}, \mathsf{log1p}\left(x\right)\right)}}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1.1e-78)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-23)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1.1e-78) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-23) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1.1e-78) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-23) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1.1e-78:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-23:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.1e-78)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-23)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.1e-78], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-23], 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] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.0999999999999999e-78

   1. Initial program 83.2%

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

   3. Taylor expanded in x around inf 94.4%

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

      1. associate-/r* 94.5%

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

      2. mul-1-neg 94.5%

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

      3. log-rec 94.5%

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

      4. mul-1-neg 94.5%

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

      5. distribute-neg-frac 94.5%

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

      6. mul-1-neg 94.5%

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

      7. remove-double-neg 94.5%

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

      8. *-rgt-identity 94.5%

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

      9. associate-/l* 94.5%

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

      10. exp-to-pow 94.5%

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

   5. Simplified 94.5%

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

   if -1.0999999999999999e-78 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000002e-23

   1. Initial program 29.3%

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

   3. Taylor expanded in n around inf 77.5%

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

      1. log1p-define 77.5%

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

   5. Simplified 77.5%

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

      1. log1p-undefine 77.5%

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

      2. diff-log 77.6%

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

   7. Applied egg-rr 77.6%

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

      1. +-commutative 77.6%

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

   9. Simplified 77.6%

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

   if 5.0000000000000002e-23 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 48.7%

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

   3. Taylor expanded in n around 0 48.7%

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

      1. log1p-define 94.9%

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

      2. *-rgt-identity 94.9%

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

      3. associate-/l* 94.9%

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

      4. exp-to-pow 94.9%

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

   5. Simplified 94.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ t_1 := x \cdot \left(\frac{1}{n} - \frac{\log x}{x \cdot n}\right)\\ \mathbf{if}\;x \leq 1.8 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log1p (expm1 (/ 1.0 (* x n)))))
        (t_1 (* x (- (/ 1.0 n) (/ (log x) (* x n))))))
   (if (<= x 1.8e-48)
     t_1
     (if (<= x 1.55e-33)
       t_0
       (if (<= x 1.25e-7)
         t_1
         (if (<= x 1.0) t_0 (/ (/ (pow x (/ 1.0 n)) n) x)))))))

C

double code(double x, double n) {
	double t_0 = log1p(expm1((1.0 / (x * n))));
	double t_1 = x * ((1.0 / n) - (log(x) / (x * n)));
	double tmp;
	if (x <= 1.8e-48) {
		tmp = t_1;
	} else if (x <= 1.55e-33) {
		tmp = t_0;
	} else if (x <= 1.25e-7) {
		tmp = t_1;
	} else if (x <= 1.0) {
		tmp = t_0;
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log1p(Math.expm1((1.0 / (x * n))));
	double t_1 = x * ((1.0 / n) - (Math.log(x) / (x * n)));
	double tmp;
	if (x <= 1.8e-48) {
		tmp = t_1;
	} else if (x <= 1.55e-33) {
		tmp = t_0;
	} else if (x <= 1.25e-7) {
		tmp = t_1;
	} else if (x <= 1.0) {
		tmp = t_0;
	} else {
		tmp = (Math.pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log1p(math.expm1((1.0 / (x * n))))
	t_1 = x * ((1.0 / n) - (math.log(x) / (x * n)))
	tmp = 0
	if x <= 1.8e-48:
		tmp = t_1
	elif x <= 1.55e-33:
		tmp = t_0
	elif x <= 1.25e-7:
		tmp = t_1
	elif x <= 1.0:
		tmp = t_0
	else:
		tmp = (math.pow(x, (1.0 / n)) / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = log1p(expm1(Float64(1.0 / Float64(x * n))))
	t_1 = Float64(x * Float64(Float64(1.0 / n) - Float64(log(x) / Float64(x * n))))
	tmp = 0.0
	if (x <= 1.8e-48)
		tmp = t_1;
	elseif (x <= 1.55e-33)
		tmp = t_0;
	elseif (x <= 1.25e-7)
		tmp = t_1;
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[1 + N[(Exp[N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(N[(1.0 / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.8e-48], t$95$1, If[LessEqual[x, 1.55e-33], t$95$0, If[LessEqual[x, 1.25e-7], t$95$1, If[LessEqual[x, 1.0], t$95$0, N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.8000000000000001e-48 or 1.54999999999999998e-33 < x < 1.24999999999999994e-7

   1. Initial program 44.0%

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

   3. Taylor expanded in n around inf 52.4%

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

      1. log1p-define 52.4%

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

   5. Simplified 52.4%

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

      1. clear-num 52.4%

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

      2. inv-pow 52.4%

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

   7. Applied egg-rr 52.4%

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

      1. unpow-1 52.4%

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

   9. Simplified 52.4%

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

   10. Taylor expanded in x around 0 52.4%

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

   11. Taylor expanded in x around inf 74.4%

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

       1. log-rec 74.4%

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

       2. distribute-frac-neg 74.4%

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

       3. unsub-neg 74.4%

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

   13. Simplified 74.4%

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

   if 1.8000000000000001e-48 < x < 1.54999999999999998e-33 or 1.24999999999999994e-7 < x < 1

   1. Initial program 70.2%

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

   3. Taylor expanded in n around inf 12.6%

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

      1. log1p-define 12.6%

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

   5. Simplified 12.6%

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

   6. Taylor expanded in x around inf 8.2%

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

      1. *-commutative 8.2%

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

   8. Simplified 8.2%

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

      1. log1p-expm1-u 92.9%

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

   10. Applied egg-rr 92.9%

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

   if 1 < x

   1. Initial program 62.4%

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

   3. Taylor expanded in x around inf 97.4%

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

      1. associate-/r* 98.1%

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

      2. mul-1-neg 98.1%

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

      3. log-rec 98.1%

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

      4. mul-1-neg 98.1%

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

      5. distribute-neg-frac 98.1%

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

      6. mul-1-neg 98.1%

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

      7. remove-double-neg 98.1%

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

      8. *-rgt-identity 98.1%

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

      9. associate-/l* 98.1%

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

      10. exp-to-pow 98.1%

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

   5. Simplified 98.1%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-48}:\\ \;\;\;\;x \cdot \left(\frac{1}{n} - \frac{\log x}{x \cdot n}\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(\frac{1}{n} - \frac{\log x}{x \cdot n}\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0001:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1.1e-78)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 0.0001)
       (/ (log (/ (+ x 1.0) x)) n)
       (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1.1e-78) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 0.0001) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1.1d-78)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 0.0001d0) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1.1e-78) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 0.0001) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1.1e-78:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 0.0001:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.1e-78)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 0.0001)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1.1e-78)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 0.0001)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.1e-78], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0001], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.0999999999999999e-78

   1. Initial program 83.2%

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

   3. Taylor expanded in x around inf 94.4%

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

      1. associate-/r* 94.5%

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

      2. mul-1-neg 94.5%

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

      3. log-rec 94.5%

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

      4. mul-1-neg 94.5%

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

      5. distribute-neg-frac 94.5%

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

      6. mul-1-neg 94.5%

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

      7. remove-double-neg 94.5%

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

      8. *-rgt-identity 94.5%

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

      9. associate-/l* 94.5%

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

      10. exp-to-pow 94.5%

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

   5. Simplified 94.5%

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

   if -1.0999999999999999e-78 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e-4

   1. Initial program 28.9%

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

   3. Taylor expanded in n around inf 76.3%

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

      1. log1p-define 76.3%

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

   5. Simplified 76.3%

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

      1. log1p-undefine 76.3%

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

      2. diff-log 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. +-commutative 76.4%

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

   9. Simplified 76.4%

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

   if 1.00000000000000005e-4 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 51.2%

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

   3. Taylor expanded in n around 0 51.2%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-/l* 100.0%

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

      4. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0001:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq -1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + 0.5 \cdot \frac{n}{x}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0001:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{\frac{-0.25}{x} + 0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5.0)
   (/ 0.3333333333333333 (* n (pow x 3.0)))
   (if (<= (/ 1.0 n) -1.1e-78)
     (/ 1.0 (* x (+ n (* 0.5 (/ n x)))))
     (if (<= (/ 1.0 n) 0.0001)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+163)
         (- 1.0 (pow x (/ 1.0 n)))
         (/
          (-
           (/ 1.0 n)
           (/ (+ (/ (+ (/ -0.25 x) 0.3333333333333333) (* x n)) (/ -0.5 n)) x))
          x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5.0) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if ((1.0 / n) <= -1.1e-78) {
		tmp = 1.0 / (x * (n + (0.5 * (n / x))));
	} else if ((1.0 / n) <= 0.0001) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+163) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 / n) - (((((-0.25 / x) + 0.3333333333333333) / (x * n)) + (-0.5 / n)) / x)) / x;
	}
	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) <= (-5.0d0)) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if ((1.0d0 / n) <= (-1.1d-78)) then
        tmp = 1.0d0 / (x * (n + (0.5d0 * (n / x))))
    else if ((1.0d0 / n) <= 0.0001d0) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+163) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = ((1.0d0 / n) - ((((((-0.25d0) / x) + 0.3333333333333333d0) / (x * n)) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5.0) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if ((1.0 / n) <= -1.1e-78) {
		tmp = 1.0 / (x * (n + (0.5 * (n / x))));
	} else if ((1.0 / n) <= 0.0001) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+163) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 / n) - (((((-0.25 / x) + 0.3333333333333333) / (x * n)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5.0:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif (1.0 / n) <= -1.1e-78:
		tmp = 1.0 / (x * (n + (0.5 * (n / x))))
	elif (1.0 / n) <= 0.0001:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+163:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = ((1.0 / n) - (((((-0.25 / x) + 0.3333333333333333) / (x * n)) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5.0)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (Float64(1.0 / n) <= -1.1e-78)
		tmp = Float64(1.0 / Float64(x * Float64(n + Float64(0.5 * Float64(n / x)))));
	elseif (Float64(1.0 / n) <= 0.0001)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+163)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(Float64(Float64(-0.25 / x) + 0.3333333333333333) / Float64(x * n)) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -5.0)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif ((1.0 / n) <= -1.1e-78)
		tmp = 1.0 / (x * (n + (0.5 * (n / x))));
	elseif ((1.0 / n) <= 0.0001)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+163)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = ((1.0 / n) - (((((-0.25 / x) + 0.3333333333333333) / (x * n)) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5.0], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.1e-78], N[(1.0 / N[(x * N[(n + N[(0.5 * N[(n / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0001], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+163], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(N[(N[(-0.25 / x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) 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 51.7%

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

      1. log1p-define 51.7%

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

   5. Simplified 51.7%

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

      1. clear-num 51.7%

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

      2. inv-pow 51.7%

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

   7. Applied egg-rr 51.7%

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

      1. unpow-1 51.7%

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

   9. Simplified 51.7%

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

   10. Taylor expanded in x around -inf 42.6%

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

   11. Taylor expanded in x around 0 74.3%

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

   if -5 < (/.f64 #s(literal 1 binary64) n) < -1.0999999999999999e-78

   1. Initial program 17.5%

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

   3. Taylor expanded in n around inf 23.7%

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

      1. log1p-define 23.7%

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

   5. Simplified 23.7%

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

      1. clear-num 23.7%

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

      2. inv-pow 23.7%

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

   7. Applied egg-rr 23.7%

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

      1. unpow-1 23.7%

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

   9. Simplified 23.7%

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

   10. Taylor expanded in x around inf 63.4%

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

   if -1.0999999999999999e-78 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e-4

   1. Initial program 28.9%

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

   3. Taylor expanded in n around inf 76.3%

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

      1. log1p-define 76.3%

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

   5. Simplified 76.3%

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

      1. log1p-undefine 76.3%

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

      2. diff-log 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. +-commutative 76.4%

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

   9. Simplified 76.4%

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

   if 1.00000000000000005e-4 < (/.f64 #s(literal 1 binary64) n) < 5e163

   1. Initial program 74.9%

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

   3. Taylor expanded in x around 0 68.0%

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

      1. *-rgt-identity 68.0%

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

      2. associate-/l* 68.0%

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

      3. exp-to-pow 68.0%

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

   5. Simplified 68.0%

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

   if 5e163 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 10.7%

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

      1. log1p-define 10.7%

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

   5. Simplified 10.7%

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

   6. Taylor expanded in x around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

   8. Simplified 0.2%

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

      1. distribute-frac-neg 0.2%

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

      2. neg-sub0 0.2%

         \[\leadsto \frac{-\left(\color{blue}{\left(0 - \frac{\frac{-\left(\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}\right)}{x} - \frac{0.5}{n}}{x}\right)} - \frac{1}{n}\right)}{x} \]

   10. Applied egg-rr 72.5%

       \[\leadsto \frac{-\left(\color{blue}{\left(0 - \frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} + \frac{0.5}{n}}{x}\right)} - \frac{1}{n}\right)}{x} \]

   12. Simplified 72.5%

       \[\leadsto \frac{-\left(\color{blue}{\frac{\frac{\frac{-0.25}{x} + 0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}} - \frac{1}{n}\right)}{x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq -1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{1}{x \cdot \left(n + 0.5 \cdot \frac{n}{x}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0001:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{\frac{-0.25}{x} + 0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{-232}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.05e-232)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 1.95e-144)
     (/ (log x) (- n))
     (if (<= x 4.2e-122)
       (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* x n))
       (if (<= x 1.4e-7)
         (/ (- x (log x)) n)
         (if (<= x 8.6e+172)
           (/
            (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* x n)) (/ 0.5 n)) x))
            x)
           (/ 0.3333333333333333 (* n (pow x 3.0)))))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.05e-232) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 1.95e-144) {
		tmp = log(x) / -n;
	} else if (x <= 4.2e-122) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else if (x <= 1.4e-7) {
		tmp = (x - log(x)) / n;
	} else if (x <= 8.6e+172) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.05d-232) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 1.95d-144) then
        tmp = log(x) / -n
    else if (x <= 4.2d-122) then
        tmp = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (x * n)
    else if (x <= 1.4d-7) then
        tmp = (x - log(x)) / n
    else if (x <= 8.6d+172) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) - (0.5d0 / n)) / x)) / x
    else
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.05e-232) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 1.95e-144) {
		tmp = Math.log(x) / -n;
	} else if (x <= 4.2e-122) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else if (x <= 1.4e-7) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 8.6e+172) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.05e-232:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 1.95e-144:
		tmp = math.log(x) / -n
	elif x <= 4.2e-122:
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n)
	elif x <= 1.4e-7:
		tmp = (x - math.log(x)) / n
	elif x <= 8.6e+172:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x
	else:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.05e-232)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.95e-144)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 4.2e-122)
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(x * n));
	elseif (x <= 1.4e-7)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 8.6e+172)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) - Float64(0.5 / n)) / x)) / x);
	else
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.05e-232)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 1.95e-144)
		tmp = log(x) / -n;
	elseif (x <= 4.2e-122)
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	elseif (x <= 1.4e-7)
		tmp = (x - log(x)) / n;
	elseif (x <= 8.6e+172)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	else
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.05e-232], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.95e-144], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 4.2e-122], N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-7], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 8.6e+172], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < 1.05e-232

   1. Initial program 61.9%

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

   3. Taylor expanded in x around 0 61.9%

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

      1. *-rgt-identity 61.9%

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

      2. associate-/l* 61.9%

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

      3. exp-to-pow 61.9%

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

   5. Simplified 61.9%

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

   if 1.05e-232 < x < 1.95000000000000007e-144

   1. Initial program 34.7%

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

   3. Taylor expanded in n around inf 61.9%

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

      1. log1p-define 61.9%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in x around 0 61.9%

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

      1. neg-mul-1 61.9%

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

   8. Simplified 61.9%

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

   if 1.95000000000000007e-144 < x < 4.19999999999999985e-122

   1. Initial program 55.5%

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

   3. Taylor expanded in n around inf 34.2%

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

      1. log1p-define 34.2%

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

   5. Simplified 34.2%

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

      1. clear-num 34.2%

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

      2. inv-pow 34.2%

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

   7. Applied egg-rr 34.2%

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

      1. unpow-1 34.2%

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

   9. Simplified 34.2%

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

   10. Taylor expanded in x around -inf 77.5%

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

   11. Taylor expanded in n around 0 77.5%

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

       1. mul-1-neg 77.5%

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

       2. distribute-neg-frac2 77.5%

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

       3. sub-neg 77.5%

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

       4. associate-*r/ 77.5%

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

       5. sub-neg 77.5%

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

       6. metadata-eval 77.5%

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

       7. distribute-lft-in 77.5%

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

       8. neg-mul-1 77.5%

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

       9. associate-*r/ 77.5%

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

       10. metadata-eval 77.5%

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

       11. distribute-neg-frac 77.5%

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

       12. metadata-eval 77.5%

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

       13. metadata-eval 77.5%

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

       14. metadata-eval 77.5%

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

       15. distribute-lft-neg-out 77.5%

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

       16. neg-mul-1 77.5%

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

       17. *-commutative 77.5%

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

       18. associate-*l* 77.5%

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

       19. neg-mul-1 77.5%

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

   13. Simplified 77.5%

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

   if 4.19999999999999985e-122 < x < 1.4000000000000001e-7

   1. Initial program 33.1%

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

   3. Taylor expanded in n around inf 56.2%

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

      1. log1p-define 56.2%

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

   5. Simplified 56.2%

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

   6. Taylor expanded in x around 0 56.2%

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

   if 1.4000000000000001e-7 < x < 8.6000000000000005e172

   1. Initial program 50.5%

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

   3. Taylor expanded in n around inf 44.7%

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

      1. log1p-define 44.7%

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

   5. Simplified 44.7%

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

   6. Taylor expanded in x around -inf 64.6%

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

      1. associate-*r/ 64.6%

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

   8. Simplified 64.6%

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

   9. Taylor expanded in x around inf 65.8%

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

       1. associate-*r/ 65.8%

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

       2. metadata-eval 65.8%

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

       3. associate-*r/ 65.8%

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

       4. metadata-eval 65.8%

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

       5. *-commutative 65.8%

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

   11. Simplified 65.8%

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

   if 8.6000000000000005e172 < x

   1. Initial program 81.9%

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

   3. Taylor expanded in n around inf 81.9%

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

      1. log1p-define 81.9%

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

   5. Simplified 81.9%

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

      1. clear-num 81.9%

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

      2. inv-pow 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. unpow-1 81.9%

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

   9. Simplified 81.9%

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

   10. Taylor expanded in x around -inf 53.6%

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

   11. Taylor expanded in x around 0 81.9%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{-232}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-144}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-232}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-144}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 1.48 \cdot 10^{-120}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 10^{+173}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 6e-232)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 2.4e-144)
     (/ -1.0 (/ n (log x)))
     (if (<= x 1.48e-120)
       (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* x n))
       (if (<= x 1.4e-7)
         (/ (- x (log x)) n)
         (if (<= x 1e+173)
           (/
            (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* x n)) (/ 0.5 n)) x))
            x)
           (/ 0.3333333333333333 (* n (pow x 3.0)))))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 6e-232) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 2.4e-144) {
		tmp = -1.0 / (n / log(x));
	} else if (x <= 1.48e-120) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else if (x <= 1.4e-7) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1e+173) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 6d-232) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 2.4d-144) then
        tmp = (-1.0d0) / (n / log(x))
    else if (x <= 1.48d-120) then
        tmp = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (x * n)
    else if (x <= 1.4d-7) then
        tmp = (x - log(x)) / n
    else if (x <= 1d+173) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) - (0.5d0 / n)) / x)) / x
    else
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 6e-232) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 2.4e-144) {
		tmp = -1.0 / (n / Math.log(x));
	} else if (x <= 1.48e-120) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else if (x <= 1.4e-7) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1e+173) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 6e-232:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 2.4e-144:
		tmp = -1.0 / (n / math.log(x))
	elif x <= 1.48e-120:
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n)
	elif x <= 1.4e-7:
		tmp = (x - math.log(x)) / n
	elif x <= 1e+173:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x
	else:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 6e-232)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 2.4e-144)
		tmp = Float64(-1.0 / Float64(n / log(x)));
	elseif (x <= 1.48e-120)
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(x * n));
	elseif (x <= 1.4e-7)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1e+173)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) - Float64(0.5 / n)) / x)) / x);
	else
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 6e-232)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 2.4e-144)
		tmp = -1.0 / (n / log(x));
	elseif (x <= 1.48e-120)
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	elseif (x <= 1.4e-7)
		tmp = (x - log(x)) / n;
	elseif (x <= 1e+173)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	else
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 6e-232], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-144], N[(-1.0 / N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.48e-120], N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-7], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1e+173], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < 5.99999999999999979e-232

   1. Initial program 61.9%

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

   3. Taylor expanded in x around 0 61.9%

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

      1. *-rgt-identity 61.9%

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

      2. associate-/l* 61.9%

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

      3. exp-to-pow 61.9%

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

   5. Simplified 61.9%

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

   if 5.99999999999999979e-232 < x < 2.39999999999999994e-144

   1. Initial program 34.7%

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

   3. Taylor expanded in n around inf 61.9%

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

      1. log1p-define 61.9%

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

   5. Simplified 61.9%

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

      1. clear-num 62.0%

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

      2. inv-pow 62.0%

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

   7. Applied egg-rr 62.0%

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

      1. unpow-1 62.0%

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

   9. Simplified 62.0%

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

   10. Taylor expanded in x around 0 62.0%

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

       1. associate-*r/ 62.0%

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

       2. neg-mul-1 62.0%

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

   12. Simplified 62.0%

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

   if 2.39999999999999994e-144 < x < 1.4800000000000001e-120

   1. Initial program 55.5%

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

   3. Taylor expanded in n around inf 34.2%

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

      1. log1p-define 34.2%

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

   5. Simplified 34.2%

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

      1. clear-num 34.2%

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

      2. inv-pow 34.2%

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

   7. Applied egg-rr 34.2%

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

      1. unpow-1 34.2%

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

   9. Simplified 34.2%

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

   10. Taylor expanded in x around -inf 77.5%

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

   11. Taylor expanded in n around 0 77.5%

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

       1. mul-1-neg 77.5%

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

       2. distribute-neg-frac2 77.5%

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

       3. sub-neg 77.5%

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

       4. associate-*r/ 77.5%

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

       5. sub-neg 77.5%

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

       6. metadata-eval 77.5%

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

       7. distribute-lft-in 77.5%

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

       8. neg-mul-1 77.5%

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

       9. associate-*r/ 77.5%

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

       10. metadata-eval 77.5%

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

       11. distribute-neg-frac 77.5%

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

       12. metadata-eval 77.5%

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

       13. metadata-eval 77.5%

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

       14. metadata-eval 77.5%

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

       15. distribute-lft-neg-out 77.5%

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

       16. neg-mul-1 77.5%

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

       17. *-commutative 77.5%

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

       18. associate-*l* 77.5%

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

       19. neg-mul-1 77.5%

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

   13. Simplified 77.5%

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

   if 1.4800000000000001e-120 < x < 1.4000000000000001e-7

   1. Initial program 33.1%

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

   3. Taylor expanded in n around inf 56.2%

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

      1. log1p-define 56.2%

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

   5. Simplified 56.2%

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

   6. Taylor expanded in x around 0 56.2%

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

   if 1.4000000000000001e-7 < x < 1e173

   1. Initial program 50.5%

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

   3. Taylor expanded in n around inf 44.7%

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

      1. log1p-define 44.7%

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

   5. Simplified 44.7%

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

   6. Taylor expanded in x around -inf 64.6%

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

      1. associate-*r/ 64.6%

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

   8. Simplified 64.6%

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

   9. Taylor expanded in x around inf 65.8%

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

       1. associate-*r/ 65.8%

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

       2. metadata-eval 65.8%

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

       3. associate-*r/ 65.8%

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

       4. metadata-eval 65.8%

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

       5. *-commutative 65.8%

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

   11. Simplified 65.8%

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

   if 1e173 < x

   1. Initial program 81.9%

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

   3. Taylor expanded in n around inf 81.9%

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

      1. log1p-define 81.9%

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

   5. Simplified 81.9%

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

      1. clear-num 81.9%

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

      2. inv-pow 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. unpow-1 81.9%

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

   9. Simplified 81.9%

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

   10. Taylor expanded in x around -inf 53.6%

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

   11. Taylor expanded in x around 0 81.9%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-232}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-144}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 1.48 \cdot 10^{-120}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 10^{+173}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-144}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-122}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.1e-233)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 1.05e-144)
     (/ -1.0 (/ n (log x)))
     (if (<= x 4.7e-122)
       (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* x n))
       (if (<= x 1.4e-7)
         (- (/ x n) (/ (log x) n))
         (if (<= x 9.5e+172)
           (/
            (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* x n)) (/ 0.5 n)) x))
            x)
           (/ 0.3333333333333333 (* n (pow x 3.0)))))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.1e-233) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 1.05e-144) {
		tmp = -1.0 / (n / log(x));
	} else if (x <= 4.7e-122) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else if (x <= 1.4e-7) {
		tmp = (x / n) - (log(x) / n);
	} else if (x <= 9.5e+172) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.1d-233) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 1.05d-144) then
        tmp = (-1.0d0) / (n / log(x))
    else if (x <= 4.7d-122) then
        tmp = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (x * n)
    else if (x <= 1.4d-7) then
        tmp = (x / n) - (log(x) / n)
    else if (x <= 9.5d+172) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) - (0.5d0 / n)) / x)) / x
    else
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.1e-233) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 1.05e-144) {
		tmp = -1.0 / (n / Math.log(x));
	} else if (x <= 4.7e-122) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else if (x <= 1.4e-7) {
		tmp = (x / n) - (Math.log(x) / n);
	} else if (x <= 9.5e+172) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.1e-233:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 1.05e-144:
		tmp = -1.0 / (n / math.log(x))
	elif x <= 4.7e-122:
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n)
	elif x <= 1.4e-7:
		tmp = (x / n) - (math.log(x) / n)
	elif x <= 9.5e+172:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x
	else:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.1e-233)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.05e-144)
		tmp = Float64(-1.0 / Float64(n / log(x)));
	elseif (x <= 4.7e-122)
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(x * n));
	elseif (x <= 1.4e-7)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	elseif (x <= 9.5e+172)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) - Float64(0.5 / n)) / x)) / x);
	else
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.1e-233)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 1.05e-144)
		tmp = -1.0 / (n / log(x));
	elseif (x <= 4.7e-122)
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	elseif (x <= 1.4e-7)
		tmp = (x / n) - (log(x) / n);
	elseif (x <= 9.5e+172)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	else
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.1e-233], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e-144], N[(-1.0 / N[(n / N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.7e-122], N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-7], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e+172], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < 4.1000000000000004e-233

   1. Initial program 61.9%

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

   3. Taylor expanded in x around 0 61.9%

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

      1. *-rgt-identity 61.9%

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

      2. associate-/l* 61.9%

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

      3. exp-to-pow 61.9%

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

   5. Simplified 61.9%

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

   if 4.1000000000000004e-233 < x < 1.0500000000000001e-144

   1. Initial program 34.7%

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

   3. Taylor expanded in n around inf 61.9%

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

      1. log1p-define 61.9%

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

   5. Simplified 61.9%

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

      1. clear-num 62.0%

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

      2. inv-pow 62.0%

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

   7. Applied egg-rr 62.0%

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

      1. unpow-1 62.0%

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

   9. Simplified 62.0%

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

   10. Taylor expanded in x around 0 62.0%

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

       1. associate-*r/ 62.0%

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

       2. neg-mul-1 62.0%

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

   12. Simplified 62.0%

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

   if 1.0500000000000001e-144 < x < 4.6999999999999999e-122

   1. Initial program 55.5%

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

   3. Taylor expanded in n around inf 34.2%

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

      1. log1p-define 34.2%

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

   5. Simplified 34.2%

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

      1. clear-num 34.2%

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

      2. inv-pow 34.2%

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

   7. Applied egg-rr 34.2%

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

      1. unpow-1 34.2%

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

   9. Simplified 34.2%

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

   10. Taylor expanded in x around -inf 77.5%

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

   11. Taylor expanded in n around 0 77.5%

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

       1. mul-1-neg 77.5%

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

       2. distribute-neg-frac2 77.5%

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

       3. sub-neg 77.5%

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

       4. associate-*r/ 77.5%

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

       5. sub-neg 77.5%

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

       6. metadata-eval 77.5%

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

       7. distribute-lft-in 77.5%

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

       8. neg-mul-1 77.5%

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

       9. associate-*r/ 77.5%

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

       10. metadata-eval 77.5%

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

       11. distribute-neg-frac 77.5%

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

       12. metadata-eval 77.5%

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

       13. metadata-eval 77.5%

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

       14. metadata-eval 77.5%

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

       15. distribute-lft-neg-out 77.5%

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

       16. neg-mul-1 77.5%

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

       17. *-commutative 77.5%

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

       18. associate-*l* 77.5%

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

       19. neg-mul-1 77.5%

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

   13. Simplified 77.5%

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

   if 4.6999999999999999e-122 < x < 1.4000000000000001e-7

   1. Initial program 33.1%

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

   3. Taylor expanded in n around inf 56.2%

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

      1. log1p-define 56.2%

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

   5. Simplified 56.2%

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

   6. Taylor expanded in x around 0 56.2%

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

      1. neg-mul-1 56.2%

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

      2. +-commutative 56.2%

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

      3. unsub-neg 56.2%

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

   8. Simplified 56.2%

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

   if 1.4000000000000001e-7 < x < 9.50000000000000027e172

   1. Initial program 50.5%

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

   3. Taylor expanded in n around inf 44.7%

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

      1. log1p-define 44.7%

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

   5. Simplified 44.7%

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

   6. Taylor expanded in x around -inf 64.6%

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

      1. associate-*r/ 64.6%

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

   8. Simplified 64.6%

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

   9. Taylor expanded in x around inf 65.8%

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

       1. associate-*r/ 65.8%

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

       2. metadata-eval 65.8%

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

       3. associate-*r/ 65.8%

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

       4. metadata-eval 65.8%

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

       5. *-commutative 65.8%

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

   11. Simplified 65.8%

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

   if 9.50000000000000027e172 < x

   1. Initial program 81.9%

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

   3. Taylor expanded in n around inf 81.9%

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

      1. log1p-define 81.9%

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

   5. Simplified 81.9%

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

      1. clear-num 81.9%

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

      2. inv-pow 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. unpow-1 81.9%

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

   9. Simplified 81.9%

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

   10. Taylor expanded in x around -inf 53.6%

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

   11. Taylor expanded in x around 0 81.9%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-144}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x}}\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-122}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0001:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{\frac{-0.25}{x} + 0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1.1e-78)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 0.0001)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+163)
         (- (+ (/ x n) 1.0) t_0)
         (/
          (-
           (/ 1.0 n)
           (/ (+ (/ (+ (/ -0.25 x) 0.3333333333333333) (* x n)) (/ -0.5 n)) x))
          x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1.1e-78) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 0.0001) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+163) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((1.0 / n) - (((((-0.25 / x) + 0.3333333333333333) / (x * n)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1.1d-78)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 0.0001d0) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+163) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = ((1.0d0 / n) - ((((((-0.25d0) / x) + 0.3333333333333333d0) / (x * n)) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1.1e-78) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 0.0001) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+163) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((1.0 / n) - (((((-0.25 / x) + 0.3333333333333333) / (x * n)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1.1e-78:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 0.0001:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+163:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = ((1.0 / n) - (((((-0.25 / x) + 0.3333333333333333) / (x * n)) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.1e-78)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 0.0001)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+163)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(Float64(Float64(-0.25 / x) + 0.3333333333333333) / Float64(x * n)) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1.1e-78)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 0.0001)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+163)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = ((1.0 / n) - (((((-0.25 / x) + 0.3333333333333333) / (x * n)) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.1e-78], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0001], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+163], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(N[(N[(-0.25 / x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.0999999999999999e-78

   1. Initial program 83.2%

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

   3. Taylor expanded in x around inf 94.4%

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

      1. associate-/r* 94.5%

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

      2. mul-1-neg 94.5%

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

      3. log-rec 94.5%

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

      4. mul-1-neg 94.5%

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

      5. distribute-neg-frac 94.5%

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

      6. mul-1-neg 94.5%

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

      7. remove-double-neg 94.5%

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

      8. *-rgt-identity 94.5%

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

      9. associate-/l* 94.5%

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

      10. exp-to-pow 94.5%

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

   5. Simplified 94.5%

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

   if -1.0999999999999999e-78 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e-4

   1. Initial program 28.9%

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

   3. Taylor expanded in n around inf 76.3%

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

      1. log1p-define 76.3%

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

   5. Simplified 76.3%

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

      1. log1p-undefine 76.3%

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

      2. diff-log 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. +-commutative 76.4%

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

   9. Simplified 76.4%

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

   if 1.00000000000000005e-4 < (/.f64 #s(literal 1 binary64) n) < 5e163

   1. Initial program 74.9%

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

   3. Taylor expanded in x around 0 69.6%

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

      1. *-rgt-identity 69.6%

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

      2. associate-/l* 69.6%

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

      3. exp-to-pow 69.6%

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

   5. Simplified 69.6%

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

   if 5e163 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 10.7%

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

      1. log1p-define 10.7%

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

   5. Simplified 10.7%

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

   6. Taylor expanded in x around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

   8. Simplified 0.2%

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

      1. distribute-frac-neg 0.2%

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

      2. neg-sub0 0.2%

         \[\leadsto \frac{-\left(\color{blue}{\left(0 - \frac{\frac{-\left(\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}\right)}{x} - \frac{0.5}{n}}{x}\right)} - \frac{1}{n}\right)}{x} \]

   10. Applied egg-rr 72.5%

       \[\leadsto \frac{-\left(\color{blue}{\left(0 - \frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} + \frac{0.5}{n}}{x}\right)} - \frac{1}{n}\right)}{x} \]

   12. Simplified 72.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0001:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{\frac{-0.25}{x} + 0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0001:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{\frac{-0.25}{x} + 0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1.1e-78)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 0.0001)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+163)
         (- 1.0 t_0)
         (/
          (-
           (/ 1.0 n)
           (/ (+ (/ (+ (/ -0.25 x) 0.3333333333333333) (* x n)) (/ -0.5 n)) x))
          x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1.1e-78) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 0.0001) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+163) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((1.0 / n) - (((((-0.25 / x) + 0.3333333333333333) / (x * n)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-1.1d-78)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 0.0001d0) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+163) then
        tmp = 1.0d0 - t_0
    else
        tmp = ((1.0d0 / n) - ((((((-0.25d0) / x) + 0.3333333333333333d0) / (x * n)) + ((-0.5d0) / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1.1e-78) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 0.0001) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+163) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((1.0 / n) - (((((-0.25 / x) + 0.3333333333333333) / (x * n)) + (-0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1.1e-78:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 0.0001:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+163:
		tmp = 1.0 - t_0
	else:
		tmp = ((1.0 / n) - (((((-0.25 / x) + 0.3333333333333333) / (x * n)) + (-0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.1e-78)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 0.0001)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+163)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(Float64(Float64(-0.25 / x) + 0.3333333333333333) / Float64(x * n)) + Float64(-0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -1.1e-78)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 0.0001)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+163)
		tmp = 1.0 - t_0;
	else
		tmp = ((1.0 / n) - (((((-0.25 / x) + 0.3333333333333333) / (x * n)) + (-0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.1e-78], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0001], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+163], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(N[(N[(-0.25 / x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.0999999999999999e-78

   1. Initial program 83.2%

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

   3. Taylor expanded in x around inf 94.4%

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

      1. associate-/r* 94.5%

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

      2. mul-1-neg 94.5%

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

      3. log-rec 94.5%

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

      4. mul-1-neg 94.5%

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

      5. distribute-neg-frac 94.5%

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

      6. mul-1-neg 94.5%

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

      7. remove-double-neg 94.5%

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

      8. *-rgt-identity 94.5%

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

      9. associate-/l* 94.5%

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

      10. exp-to-pow 94.5%

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

   5. Simplified 94.5%

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

   if -1.0999999999999999e-78 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000005e-4

   1. Initial program 28.9%

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

   3. Taylor expanded in n around inf 76.3%

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

      1. log1p-define 76.3%

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

   5. Simplified 76.3%

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

      1. log1p-undefine 76.3%

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

      2. diff-log 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. +-commutative 76.4%

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

   9. Simplified 76.4%

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

   if 1.00000000000000005e-4 < (/.f64 #s(literal 1 binary64) n) < 5e163

   1. Initial program 74.9%

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

   3. Taylor expanded in x around 0 68.0%

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

      1. *-rgt-identity 68.0%

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

      2. associate-/l* 68.0%

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

      3. exp-to-pow 68.0%

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

   5. Simplified 68.0%

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

   if 5e163 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf 10.7%

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

      1. log1p-define 10.7%

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

   5. Simplified 10.7%

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

   6. Taylor expanded in x around -inf 0.2%

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

      1. associate-*r/ 0.2%

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

   8. Simplified 0.2%

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

      1. distribute-frac-neg 0.2%

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

      2. neg-sub0 0.2%

         \[\leadsto \frac{-\left(\color{blue}{\left(0 - \frac{\frac{-\left(\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}\right)}{x} - \frac{0.5}{n}}{x}\right)} - \frac{1}{n}\right)}{x} \]

   10. Applied egg-rr 72.5%

       \[\leadsto \frac{-\left(\color{blue}{\left(0 - \frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} + \frac{0.5}{n}}{x}\right)} - \frac{1}{n}\right)}{x} \]

   12. Simplified 72.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1.1 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0001:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+163}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{\frac{\frac{-0.25}{x} + 0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-122}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 7.5e-233)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 1.65e-144)
     (/ (log x) (- n))
     (if (<= x 5.8e-122)
       (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* x n))
       (if (<= x 1.4e-7)
         (/ (- x (log x)) n)
         (/
          (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* x n)) (/ 0.5 n)) x))
          x))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 7.5e-233) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 1.65e-144) {
		tmp = log(x) / -n;
	} else if (x <= 5.8e-122) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else if (x <= 1.4e-7) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 7.5d-233) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 1.65d-144) then
        tmp = log(x) / -n
    else if (x <= 5.8d-122) then
        tmp = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (x * n)
    else if (x <= 1.4d-7) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) - (0.5d0 / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 7.5e-233) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 1.65e-144) {
		tmp = Math.log(x) / -n;
	} else if (x <= 5.8e-122) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else if (x <= 1.4e-7) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 7.5e-233:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 1.65e-144:
		tmp = math.log(x) / -n
	elif x <= 5.8e-122:
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n)
	elif x <= 1.4e-7:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 7.5e-233)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.65e-144)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 5.8e-122)
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(x * n));
	elseif (x <= 1.4e-7)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) - Float64(0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 7.5e-233)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 1.65e-144)
		tmp = log(x) / -n;
	elseif (x <= 5.8e-122)
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	elseif (x <= 1.4e-7)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 7.5e-233], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e-144], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 5.8e-122], N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-7], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 7.49999999999999974e-233

   1. Initial program 61.9%

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

   3. Taylor expanded in x around 0 61.9%

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

      1. *-rgt-identity 61.9%

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

      2. associate-/l* 61.9%

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

      3. exp-to-pow 61.9%

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

   5. Simplified 61.9%

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

   if 7.49999999999999974e-233 < x < 1.64999999999999998e-144

   1. Initial program 34.7%

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

   3. Taylor expanded in n around inf 61.9%

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

      1. log1p-define 61.9%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in x around 0 61.9%

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

      1. neg-mul-1 61.9%

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

   8. Simplified 61.9%

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

   if 1.64999999999999998e-144 < x < 5.8000000000000005e-122

   1. Initial program 55.5%

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

   3. Taylor expanded in n around inf 34.2%

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

      1. log1p-define 34.2%

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

   5. Simplified 34.2%

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

      1. clear-num 34.2%

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

      2. inv-pow 34.2%

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

   7. Applied egg-rr 34.2%

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

      1. unpow-1 34.2%

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

   9. Simplified 34.2%

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

   10. Taylor expanded in x around -inf 77.5%

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

   11. Taylor expanded in n around 0 77.5%

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

       1. mul-1-neg 77.5%

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

       2. distribute-neg-frac2 77.5%

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

       3. sub-neg 77.5%

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

       4. associate-*r/ 77.5%

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

       5. sub-neg 77.5%

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

       6. metadata-eval 77.5%

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

       7. distribute-lft-in 77.5%

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

       8. neg-mul-1 77.5%

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

       9. associate-*r/ 77.5%

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

       10. metadata-eval 77.5%

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

       11. distribute-neg-frac 77.5%

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

       12. metadata-eval 77.5%

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

       13. metadata-eval 77.5%

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

       14. metadata-eval 77.5%

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

       15. distribute-lft-neg-out 77.5%

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

       16. neg-mul-1 77.5%

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

       17. *-commutative 77.5%

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

       18. associate-*l* 77.5%

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

       19. neg-mul-1 77.5%

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

   13. Simplified 77.5%

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

   if 5.8000000000000005e-122 < x < 1.4000000000000001e-7

   1. Initial program 33.1%

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

   3. Taylor expanded in n around inf 56.2%

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

      1. log1p-define 56.2%

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

   5. Simplified 56.2%

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

   6. Taylor expanded in x around 0 56.2%

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

   if 1.4000000000000001e-7 < x

   1. Initial program 63.6%

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

   3. Taylor expanded in n around inf 60.2%

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

      1. log1p-define 60.2%

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

   5. Simplified 60.2%

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

   6. Taylor expanded in x around -inf 60.1%

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

      1. associate-*r/ 60.1%

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

   8. Simplified 60.1%

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

   9. Taylor expanded in x around inf 60.9%

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

       1. associate-*r/ 60.9%

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

       2. metadata-eval 60.9%

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

       3. associate-*r/ 60.9%

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

       4. metadata-eval 60.9%

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

       5. *-commutative 60.9%

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

   11. Simplified 60.9%

       \[\leadsto \frac{-\left(\color{blue}{\frac{\frac{0.5}{n} - \frac{0.3333333333333333}{x \cdot n}}{x}} - \frac{1}{n}\right)}{x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-122}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-121}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2.5e-144)
   (/ (log x) (- n))
   (if (<= x 2.1e-121)
     (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* x n))
     (if (<= x 1.4e-7)
       (/ (- x (log x)) n)
       (/
        (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* x n)) (/ 0.5 n)) x))
        x)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2.5e-144) {
		tmp = log(x) / -n;
	} else if (x <= 2.1e-121) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else if (x <= 1.4e-7) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 2.5d-144) then
        tmp = log(x) / -n
    else if (x <= 2.1d-121) then
        tmp = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (x * n)
    else if (x <= 1.4d-7) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) - (0.5d0 / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2.5e-144) {
		tmp = Math.log(x) / -n;
	} else if (x <= 2.1e-121) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else if (x <= 1.4e-7) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2.5e-144:
		tmp = math.log(x) / -n
	elif x <= 2.1e-121:
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n)
	elif x <= 1.4e-7:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2.5e-144)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 2.1e-121)
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(x * n));
	elseif (x <= 1.4e-7)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) - Float64(0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 2.5e-144)
		tmp = log(x) / -n;
	elseif (x <= 2.1e-121)
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	elseif (x <= 1.4e-7)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2.5e-144], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 2.1e-121], N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-7], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.4999999999999999e-144

   1. Initial program 51.2%

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

   3. Taylor expanded in n around inf 48.4%

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

      1. log1p-define 48.4%

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

   5. Simplified 48.4%

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

   6. Taylor expanded in x around 0 48.4%

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

      1. neg-mul-1 48.4%

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

   8. Simplified 48.4%

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

   if 2.4999999999999999e-144 < x < 2.0999999999999999e-121

   1. Initial program 55.5%

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

   3. Taylor expanded in n around inf 34.2%

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

      1. log1p-define 34.2%

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

   5. Simplified 34.2%

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

      1. clear-num 34.2%

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

      2. inv-pow 34.2%

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

   7. Applied egg-rr 34.2%

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

      1. unpow-1 34.2%

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

   9. Simplified 34.2%

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

   10. Taylor expanded in x around -inf 77.5%

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

   11. Taylor expanded in n around 0 77.5%

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

       1. mul-1-neg 77.5%

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

       2. distribute-neg-frac2 77.5%

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

       3. sub-neg 77.5%

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

       4. associate-*r/ 77.5%

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

       5. sub-neg 77.5%

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

       6. metadata-eval 77.5%

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

       7. distribute-lft-in 77.5%

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

       8. neg-mul-1 77.5%

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

       9. associate-*r/ 77.5%

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

       10. metadata-eval 77.5%

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

       11. distribute-neg-frac 77.5%

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

       12. metadata-eval 77.5%

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

       13. metadata-eval 77.5%

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

       14. metadata-eval 77.5%

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

       15. distribute-lft-neg-out 77.5%

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

       16. neg-mul-1 77.5%

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

       17. *-commutative 77.5%

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

       18. associate-*l* 77.5%

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

       19. neg-mul-1 77.5%

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

   13. Simplified 77.5%

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

   if 2.0999999999999999e-121 < x < 1.4000000000000001e-7

   1. Initial program 33.1%

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

   3. Taylor expanded in n around inf 56.2%

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

      1. log1p-define 56.2%

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

   5. Simplified 56.2%

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

   6. Taylor expanded in x around 0 56.2%

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

   if 1.4000000000000001e-7 < x

   1. Initial program 63.6%

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

   3. Taylor expanded in n around inf 60.2%

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

      1. log1p-define 60.2%

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

   5. Simplified 60.2%

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

   6. Taylor expanded in x around -inf 60.1%

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

      1. associate-*r/ 60.1%

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

   8. Simplified 60.1%

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

   9. Taylor expanded in x around inf 60.9%

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

       1. associate-*r/ 60.9%

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

       2. metadata-eval 60.9%

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

       3. associate-*r/ 60.9%

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

       4. metadata-eval 60.9%

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

       5. *-commutative 60.9%

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

   11. Simplified 60.9%

       \[\leadsto \frac{-\left(\color{blue}{\frac{\frac{0.5}{n} - \frac{0.3333333333333333}{x \cdot n}}{x}} - \frac{1}{n}\right)}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-121}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 2 \cdot 10^{-144}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 2e-144)
     t_0
     (if (<= x 4.5e-122)
       (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* x n))
       (if (<= x 1.4e-7)
         t_0
         (/
          (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* x n)) (/ 0.5 n)) x))
          x))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 2e-144) {
		tmp = t_0;
	} else if (x <= 4.5e-122) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else if (x <= 1.4e-7) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 2d-144) then
        tmp = t_0
    else if (x <= 4.5d-122) then
        tmp = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (x * n)
    else if (x <= 1.4d-7) then
        tmp = t_0
    else
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) - (0.5d0 / n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 2e-144) {
		tmp = t_0;
	} else if (x <= 4.5e-122) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	} else if (x <= 1.4e-7) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 2e-144:
		tmp = t_0
	elif x <= 4.5e-122:
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n)
	elif x <= 1.4e-7:
		tmp = t_0
	else:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 2e-144)
		tmp = t_0;
	elseif (x <= 4.5e-122)
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(x * n));
	elseif (x <= 1.4e-7)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) - Float64(0.5 / n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 2e-144)
		tmp = t_0;
	elseif (x <= 4.5e-122)
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (x * n);
	elseif (x <= 1.4e-7)
		tmp = t_0;
	else
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 2e-144], t$95$0, If[LessEqual[x, 4.5e-122], N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-7], t$95$0, N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.9999999999999999e-144 or 4.4999999999999998e-122 < x < 1.4000000000000001e-7

   1. Initial program 43.7%

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

   3. Taylor expanded in n around inf 51.6%

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

      1. log1p-define 51.6%

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

   5. Simplified 51.6%

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

   6. Taylor expanded in x around 0 51.3%

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

      1. neg-mul-1 51.3%

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

   8. Simplified 51.3%

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

   if 1.9999999999999999e-144 < x < 4.4999999999999998e-122

   1. Initial program 55.5%

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

   3. Taylor expanded in n around inf 34.2%

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

      1. log1p-define 34.2%

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

   5. Simplified 34.2%

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

      1. clear-num 34.2%

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

      2. inv-pow 34.2%

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

   7. Applied egg-rr 34.2%

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

      1. unpow-1 34.2%

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

   9. Simplified 34.2%

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

   10. Taylor expanded in x around -inf 77.5%

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

   11. Taylor expanded in n around 0 77.5%

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

       1. mul-1-neg 77.5%

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

       2. distribute-neg-frac2 77.5%

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

       3. sub-neg 77.5%

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

       4. associate-*r/ 77.5%

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

       5. sub-neg 77.5%

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

       6. metadata-eval 77.5%

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

       7. distribute-lft-in 77.5%

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

       8. neg-mul-1 77.5%

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

       9. associate-*r/ 77.5%

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

       10. metadata-eval 77.5%

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

       11. distribute-neg-frac 77.5%

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

       12. metadata-eval 77.5%

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

       13. metadata-eval 77.5%

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

       14. metadata-eval 77.5%

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

       15. distribute-lft-neg-out 77.5%

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

       16. neg-mul-1 77.5%

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

       17. *-commutative 77.5%

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

       18. associate-*l* 77.5%

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

       19. neg-mul-1 77.5%

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

   13. Simplified 77.5%

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

   if 1.4000000000000001e-7 < x

   1. Initial program 63.6%

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

   3. Taylor expanded in n around inf 60.2%

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

      1. log1p-define 60.2%

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

   5. Simplified 60.2%

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

   6. Taylor expanded in x around -inf 60.1%

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

      1. associate-*r/ 60.1%

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

   8. Simplified 60.1%

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

   9. Taylor expanded in x around inf 60.9%

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

       1. associate-*r/ 60.9%

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

       2. metadata-eval 60.9%

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

       3. associate-*r/ 60.9%

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

       4. metadata-eval 60.9%

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

       5. *-commutative 60.9%

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

   11. Simplified 60.9%

       \[\leadsto \frac{-\left(\color{blue}{\frac{\frac{0.5}{n} - \frac{0.3333333333333333}{x \cdot n}}{x}} - \frac{1}{n}\right)}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-144}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{1}{n} - \frac{\log x}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 7.2e-6)
   (* x (- (/ 1.0 n) (/ (log x) (* x n))))
   (/ (/ (pow x (/ 1.0 n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 7.2e-6) {
		tmp = x * ((1.0 / n) - (log(x) / (x * n)));
	} else {
		tmp = (pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 7.2d-6) then
        tmp = x * ((1.0d0 / n) - (log(x) / (x * n)))
    else
        tmp = ((x ** (1.0d0 / n)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 7.2e-6) {
		tmp = x * ((1.0 / n) - (Math.log(x) / (x * n)));
	} else {
		tmp = (Math.pow(x, (1.0 / n)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 7.2e-6:
		tmp = x * ((1.0 / n) - (math.log(x) / (x * n)))
	else:
		tmp = (math.pow(x, (1.0 / n)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 7.2e-6)
		tmp = Float64(x * Float64(Float64(1.0 / n) - Float64(log(x) / Float64(x * n))));
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 7.2e-6)
		tmp = x * ((1.0 / n) - (log(x) / (x * n)));
	else
		tmp = ((x ^ (1.0 / n)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 7.2e-6], N[(x * N[(N[(1.0 / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.19999999999999967e-6

   1. Initial program 45.7%

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

   3. Taylor expanded in n around inf 49.3%

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

      1. log1p-define 49.3%

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

   5. Simplified 49.3%

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

      1. clear-num 49.3%

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

      2. inv-pow 49.3%

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

   7. Applied egg-rr 49.3%

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

      1. unpow-1 49.3%

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

   9. Simplified 49.3%

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

   10. Taylor expanded in x around 0 49.3%

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

   11. Taylor expanded in x around inf 70.2%

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

       1. log-rec 70.2%

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

       2. distribute-frac-neg 70.2%

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

       3. unsub-neg 70.2%

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

   13. Simplified 70.2%

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

   if 7.19999999999999967e-6 < x

   1. Initial program 63.0%

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

   3. Taylor expanded in x around inf 96.7%

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

      1. associate-/r* 97.3%

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

      2. mul-1-neg 97.3%

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

      3. log-rec 97.3%

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

      4. mul-1-neg 97.3%

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

      5. distribute-neg-frac 97.3%

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

      6. mul-1-neg 97.3%

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

      7. remove-double-neg 97.3%

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

      8. *-rgt-identity 97.3%

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

      9. associate-/l* 97.3%

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

      10. exp-to-pow 97.3%

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

   5. Simplified 97.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{1}{n} - \frac{\log x}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.6% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in n around inf 54.9%

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

   1. log1p-define 54.9%

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

5. Simplified 54.9%

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

6. Taylor expanded in x around -inf 29.1%

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

   1. associate-*r/ 29.1%

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

8. Simplified 29.1%

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

   1. add-sqr-sqrt 13.3%

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

   2. sqrt-unprod 35.8%

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

   3. sqr-neg 35.8%

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

   4. sqrt-unprod 22.3%

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

   5. add-sqr-sqrt 49.1%

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

   6. div-sub 36.4%

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

10. Applied egg-rr 36.4%

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

    1. div-sub 49.1%

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

    2. associate-/r* 49.1%

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

    3. div-sub 49.1%

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

    4. associate-/l/ 49.1%

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

    5. sub-neg 49.1%

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

    6. metadata-eval 49.1%

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

12. Simplified 49.1%

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

13. Final simplification 49.1%

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

Alternative 16: 46.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in n around inf 54.9%

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

   1. log1p-define 54.9%

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

5. Simplified 54.9%

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

6. Taylor expanded in x around -inf 29.1%

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

   1. associate-*r/ 29.1%

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

8. Simplified 29.1%

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

9. Taylor expanded in x around inf 49.0%

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

    1. associate-*r/ 49.0%

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

    2. metadata-eval 49.0%

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

    3. associate-*r/ 49.0%

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

    4. metadata-eval 49.0%

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

    5. *-commutative 49.0%

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

11. Simplified 49.0%

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

12. Final simplification 49.0%

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

Alternative 17: 46.4% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in n around inf 54.9%

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

   1. log1p-define 54.9%

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

5. Simplified 54.9%

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

   1. clear-num 54.8%

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

   2. inv-pow 54.8%

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

7. Applied egg-rr 54.8%

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

   1. unpow-1 54.8%

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

9. Simplified 54.8%

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

10. Taylor expanded in x around -inf 48.6%

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

11. Taylor expanded in n around 0 48.7%

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

    1. mul-1-neg 48.7%

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

    2. distribute-neg-frac2 48.7%

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

    3. sub-neg 48.7%

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

    4. associate-*r/ 48.7%

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

    5. sub-neg 48.7%

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

    6. metadata-eval 48.7%

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

    7. distribute-lft-in 48.7%

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

    8. neg-mul-1 48.7%

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

    9. associate-*r/ 48.7%

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

    10. metadata-eval 48.7%

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

    11. distribute-neg-frac 48.7%

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

    12. metadata-eval 48.7%

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

    13. metadata-eval 48.7%

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

    14. metadata-eval 48.7%

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

    15. distribute-lft-neg-out 48.7%

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

    16. neg-mul-1 48.7%

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

    17. *-commutative 48.7%

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

    18. associate-*l* 48.7%

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

    19. neg-mul-1 48.7%

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

13. Simplified 48.7%

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

14. Final simplification 48.7%

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

Alternative 18: 46.9% 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 53.8%

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

3. Taylor expanded in n around inf 54.9%

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

   1. log1p-define 54.9%

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

5. Simplified 54.9%

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

   1. add-sqr-sqrt 54.7%

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

   2. pow2 54.7%

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

7. Applied egg-rr 54.7%

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

8. Taylor expanded in x around -inf 48.9%

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

   1. mul-1-neg 48.9%

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

   2. distribute-neg-frac2 48.9%

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

   3. sub-neg 48.9%

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

   4. associate-*r/ 48.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{-1 \cdot \left(0.3333333333333333 \cdot \frac{1}{x} - 0.5\right)}{x}} + \left(-1\right)}{-x}}{n} \]

   5. sub-neg 48.9%

      \[\leadsto \frac{\frac{\frac{-1 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{x} + \left(-0.5\right)\right)}}{x} + \left(-1\right)}{-x}}{n} \]

   6. metadata-eval 48.9%

      \[\leadsto \frac{\frac{\frac{-1 \cdot \left(0.3333333333333333 \cdot \frac{1}{x} + \color{blue}{-0.5}\right)}{x} + \left(-1\right)}{-x}}{n} \]

   7. distribute-lft-in 48.9%

      \[\leadsto \frac{\frac{\frac{\color{blue}{-1 \cdot \left(0.3333333333333333 \cdot \frac{1}{x}\right) + -1 \cdot -0.5}}{x} + \left(-1\right)}{-x}}{n} \]

   8. neg-mul-1 48.9%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{x}\right)} + -1 \cdot -0.5}{x} + \left(-1\right)}{-x}}{n} \]

   9. associate-*r/ 48.9%

      \[\leadsto \frac{\frac{\frac{\left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{x}}\right) + -1 \cdot -0.5}{x} + \left(-1\right)}{-x}}{n} \]

   10. metadata-eval 48.9%

       \[\leadsto \frac{\frac{\frac{\left(-\frac{\color{blue}{0.3333333333333333}}{x}\right) + -1 \cdot -0.5}{x} + \left(-1\right)}{-x}}{n} \]

   11. distribute-neg-frac 48.9%

       \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{-0.3333333333333333}{x}} + -1 \cdot -0.5}{x} + \left(-1\right)}{-x}}{n} \]

   12. metadata-eval 48.9%

       \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{-0.3333333333333333}}{x} + -1 \cdot -0.5}{x} + \left(-1\right)}{-x}}{n} \]

   13. metadata-eval 48.9%

       \[\leadsto \frac{\frac{\frac{\frac{-0.3333333333333333}{x} + \color{blue}{0.5}}{x} + \left(-1\right)}{-x}}{n} \]

   14. metadata-eval 48.9%

       \[\leadsto \frac{\frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + \color{blue}{-1}}{-x}}{n} \]

10. Simplified 48.9%

    \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + -1}{-x}}}{n} \]

11. Final simplification 48.9%

    \[\leadsto \frac{\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{x}}{n} \]
12. Add Preprocessing

Alternative 19: 40.1% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* x n)))

C

double code(double x, double n) {
	return 1.0 / (x * n);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 1.0d0 / (x * n)
end function

Java

public static double code(double x, double n) {
	return 1.0 / (x * n);
}

Python

def code(x, n):
	return 1.0 / (x * n)

Julia

function code(x, n)
	return Float64(1.0 / Float64(x * n))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (x * n);
end

Wolfram

code[x_, n_] := N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 54.9%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 54.9%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 54.9%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]

6. Taylor expanded in x around inf 42.6%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
7. Step-by-step derivation

   1. *-commutative 42.6%

      \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

8. Simplified 42.6%

   \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

9. Final simplification 42.6%

   \[\leadsto \frac{1}{x \cdot n} \]
10. Add Preprocessing

Alternative 20: 40.7% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n}}{x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 n) x))

C

double code(double x, double n) {
	return (1.0 / n) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 / n) / x
end function

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

function code(x, n)
	return Float64(Float64(1.0 / n) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 / n) / x;
end

Wolfram

code[x_, n_] := N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 62.0%

   \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
4. Step-by-step derivation

   1. associate-/r* 62.5%

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n}}{x}} \]

   2. mul-1-neg 62.5%

      \[\leadsto \frac{\frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n}}{x} \]

   3. log-rec 62.5%

      \[\leadsto \frac{\frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n}}{x} \]

   4. mul-1-neg 62.5%

      \[\leadsto \frac{\frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n}}{x} \]

   5. distribute-neg-frac 62.5%

      \[\leadsto \frac{\frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n}}{x} \]

   6. mul-1-neg 62.5%

      \[\leadsto \frac{\frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n}}{x} \]

   7. remove-double-neg 62.5%

      \[\leadsto \frac{\frac{e^{\frac{\color{blue}{\log x}}{n}}}{n}}{x} \]

   8. *-rgt-identity 62.5%

      \[\leadsto \frac{\frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n}}{x} \]

   9. associate-/l* 62.5%

      \[\leadsto \frac{\frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n}}{x} \]

   10. exp-to-pow 62.5%

       \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n}}{x} \]

5. Simplified 62.5%

   \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}} \]

6. Taylor expanded in n around inf 42.9%

   \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]

7. Final simplification 42.9%

   \[\leadsto \frac{\frac{1}{n}}{x} \]
8. Add Preprocessing

Alternative 21: 4.5% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ x n))

C

double code(double x, double n) {
	return x / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = x / n
end function

Java

public static double code(double x, double n) {
	return x / n;
}

Python

def code(x, n):
	return x / n

Julia

function code(x, n)
	return Float64(x / n)
end

MATLAB

function tmp = code(x, n)
	tmp = x / n;
end

Wolfram

code[x_, n_] := N[(x / n), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
2. Add Preprocessing

3. Taylor expanded in n around inf 54.9%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 54.9%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

5. Simplified 54.9%

   \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
6. Step-by-step derivation

   1. clear-num 54.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

   2. inv-pow 54.8%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]

7. Applied egg-rr 54.8%

   \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 54.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

9. Simplified 54.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{log1p}\left(x\right) - \log x}}} \]

10. Taylor expanded in x around 0 27.8%

    \[\leadsto \frac{1}{\frac{n}{\color{blue}{x - \log x}}} \]

11. Taylor expanded in x around inf 4.4%

    \[\leadsto \color{blue}{\frac{x}{n}} \]

12. Final simplification 4.4%

    \[\leadsto \frac{x}{n} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))