2nthrt (problem 3.4.6)

Percentage Accurate: 54.2% → 86.5%

Time: 22.3s

Alternatives: 19

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-47)
   (* (/ 1.0 x) (/ (pow x (pow n -1.0)) n))
   (if (<= (/ 1.0 n) 1e-10)
     (+
      (fma 0.5 (/ (pow (log1p x) 2.0) (* n n)) (/ (- (log1p x) (log x)) n))
      (* (/ (pow (log x) 2.0) (* n n)) -0.5))
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-47) {
		tmp = (1.0 / x) * (pow(x, pow(n, -1.0)) / n);
	} else if ((1.0 / n) <= 1e-10) {
		tmp = fma(0.5, (pow(log1p(x), 2.0) / (n * n)), ((log1p(x) - log(x)) / n)) + ((pow(log(x), 2.0) / (n * n)) * -0.5);
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-47)
		tmp = Float64(Float64(1.0 / x) * Float64((x ^ (n ^ -1.0)) / n));
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = Float64(fma(0.5, Float64((log1p(x) ^ 2.0) / Float64(n * n)), Float64(Float64(log1p(x) - log(x)) / n)) + Float64(Float64((log(x) ^ 2.0) / Float64(n * n)) * -0.5));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-47], N[(N[(1.0 / x), $MachinePrecision] * N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -5.00000000000000011e-47

   1. Initial program 90.0%

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

   2. Taylor expanded in x around inf 97.4%

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

      1. log-rec 97.4%

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

      2. mul-1-neg 97.4%

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

      3. mul-1-neg 97.4%

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

      4. distribute-frac-neg 97.4%

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

      5. neg-mul-1 97.4%

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

      6. remove-double-neg 97.4%

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

      7. *-rgt-identity 97.4%

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

      8. associate-*r/ 97.4%

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

      9. unpow-1 97.4%

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

      10. exp-to-pow 97.4%

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

      11. unpow-1 97.4%

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

      12. *-commutative 97.4%

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

   4. Simplified 97.4%

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

      1. *-un-lft-identity 97.4%

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

      2. times-frac 97.9%

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

      3. inv-pow 97.9%

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

   6. Applied egg-rr 97.9%

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

   if -5.00000000000000011e-47 < (/.f64 1 n) < 1.00000000000000004e-10

   1. Initial program 29.3%

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

   2. Taylor expanded in n around inf 86.3%

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

      1. associate--r+ 81.5%

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

      2. sub-neg 81.5%

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

   4. Simplified 86.2%

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

   if 1.00000000000000004e-10 < (/.f64 1 n)

   1. Initial program 58.1%

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

   2. Taylor expanded in n around 0 58.1%

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

      1. log1p-def 96.7%

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

      2. *-rgt-identity 96.7%

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

      3. associate-*r/ 96.7%

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

      4. unpow-1 96.7%

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

      5. exp-to-pow 96.7%

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

      6. /-rgt-identity 96.7%

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

      7. metadata-eval 96.7%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{\frac{2}{2}}}\right)} \]

      8. associate-/l* 96.7%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1} \cdot 2}{2}\right)}} \]

      9. *-commutative 96.7%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{2 \cdot {n}^{-1}}}{2}\right)} \]

      10. *-commutative 96.7%

          \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{{n}^{-1} \cdot 2}}{2}\right)} \]

      11. associate-/l* 96.7%

          \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{\frac{2}{2}}\right)}} \]

      12. metadata-eval 96.7%

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

      13. /-rgt-identity 96.7%

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

      14. unpow-1 96.7%

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

   4. Simplified 96.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n}\right) + \frac{{\log x}^{2}}{n \cdot n} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-47)
   (* (/ 1.0 x) (/ (pow x (pow n -1.0)) n))
   (if (<= (/ 1.0 n) 1e-22)
     (- (/ (log (/ x (+ 1.0 x))) n))
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-47) {
		tmp = (1.0 / x) * (pow(x, pow(n, -1.0)) / n);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-47) {
		tmp = (1.0 / x) * (Math.pow(x, Math.pow(n, -1.0)) / n);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-47:
		tmp = (1.0 / x) * (math.pow(x, math.pow(n, -1.0)) / n)
	elif (1.0 / n) <= 1e-22:
		tmp = -(math.log((x / (1.0 + x))) / n)
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-47)
		tmp = Float64(Float64(1.0 / x) * Float64((x ^ (n ^ -1.0)) / n));
	elseif (Float64(1.0 / n) <= 1e-22)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-47], N[(N[(1.0 / x), $MachinePrecision] * N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-22], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -5.00000000000000011e-47

   1. Initial program 90.0%

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

   2. Taylor expanded in x around inf 97.4%

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

      1. log-rec 97.4%

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

      2. mul-1-neg 97.4%

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

      3. mul-1-neg 97.4%

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

      4. distribute-frac-neg 97.4%

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

      5. neg-mul-1 97.4%

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

      6. remove-double-neg 97.4%

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

      7. *-rgt-identity 97.4%

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

      8. associate-*r/ 97.4%

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

      9. unpow-1 97.4%

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

      10. exp-to-pow 97.4%

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

      11. unpow-1 97.4%

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

      12. *-commutative 97.4%

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

   4. Simplified 97.4%

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

      1. *-un-lft-identity 97.4%

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

      2. times-frac 97.9%

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

      3. inv-pow 97.9%

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

   6. Applied egg-rr 97.9%

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

   if -5.00000000000000011e-47 < (/.f64 1 n) < 1e-22

   1. Initial program 29.0%

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

   2. Taylor expanded in n around inf 86.1%

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

      1. log1p-def 86.1%

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

   4. Simplified 86.1%

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

      1. log1p-udef 86.1%

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

      2. diff-log 86.3%

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

      3. +-commutative 86.3%

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

   6. Applied egg-rr 86.3%

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

      1. clear-num 86.3%

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

      2. log-div 86.4%

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

      3. metadata-eval 86.4%

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

   8. Applied egg-rr 86.4%

      \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   9. Step-by-step derivation

      1. neg-sub0 86.4%

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

   10. Simplified 86.4%

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

   if 1e-22 < (/.f64 1 n)

   1. Initial program 58.3%

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

   2. Taylor expanded in n around 0 58.3%

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

      1. log1p-def 95.6%

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

      2. *-rgt-identity 95.6%

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

      3. associate-*r/ 95.6%

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

      4. unpow-1 95.6%

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

      5. exp-to-pow 95.6%

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

      6. /-rgt-identity 95.6%

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

      7. metadata-eval 95.6%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{{n}^{-1}}{\color{blue}{\frac{2}{2}}}\right)} \]

      8. associate-/l* 95.6%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1} \cdot 2}{2}\right)}} \]

      9. *-commutative 95.6%

         \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{2 \cdot {n}^{-1}}}{2}\right)} \]

      10. *-commutative 95.6%

          \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{\color{blue}{{n}^{-1} \cdot 2}}{2}\right)} \]

      11. associate-/l* 95.6%

          \[\leadsto e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\color{blue}{\left(\frac{{n}^{-1}}{\frac{2}{2}}\right)}} \]

      12. metadata-eval 95.6%

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

      13. /-rgt-identity 95.6%

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

      14. unpow-1 95.6%

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

   4. Simplified 95.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 3: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+195}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-47)
   (* (/ 1.0 x) (/ (pow x (pow n -1.0)) n))
   (if (<= (/ 1.0 n) 1e-22)
     (- (/ (log (/ x (+ 1.0 x))) n))
     (if (<= (/ 1.0 n) 1e+195)
       (- (pow (+ 1.0 x) (/ 1.0 n)) (pow x (/ 1.0 n)))
       (/ (/ n x) (* n n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-47) {
		tmp = (1.0 / x) * (pow(x, pow(n, -1.0)) / n);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 1e+195) {
		tmp = pow((1.0 + x), (1.0 / n)) - pow(x, (1.0 / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-5d-47)) then
        tmp = (1.0d0 / x) * ((x ** (n ** (-1.0d0))) / n)
    else if ((1.0d0 / n) <= 1d-22) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else if ((1.0d0 / n) <= 1d+195) then
        tmp = ((1.0d0 + x) ** (1.0d0 / n)) - (x ** (1.0d0 / n))
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-47) {
		tmp = (1.0 / x) * (Math.pow(x, Math.pow(n, -1.0)) / n);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 1e+195) {
		tmp = Math.pow((1.0 + x), (1.0 / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-47:
		tmp = (1.0 / x) * (math.pow(x, math.pow(n, -1.0)) / n)
	elif (1.0 / n) <= 1e-22:
		tmp = -(math.log((x / (1.0 + x))) / n)
	elif (1.0 / n) <= 1e+195:
		tmp = math.pow((1.0 + x), (1.0 / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-47)
		tmp = Float64(Float64(1.0 / x) * Float64((x ^ (n ^ -1.0)) / n));
	elseif (Float64(1.0 / n) <= 1e-22)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 1e+195)
		tmp = Float64((Float64(1.0 + x) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -5e-47)
		tmp = (1.0 / x) * ((x ^ (n ^ -1.0)) / n);
	elseif ((1.0 / n) <= 1e-22)
		tmp = -(log((x / (1.0 + x))) / n);
	elseif ((1.0 / n) <= 1e+195)
		tmp = ((1.0 + x) ^ (1.0 / n)) - (x ^ (1.0 / n));
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-47], N[(N[(1.0 / x), $MachinePrecision] * N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-22], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+195], N[(N[Power[N[(1.0 + x), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000011e-47

   1. Initial program 90.0%

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

   2. Taylor expanded in x around inf 97.4%

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

      1. log-rec 97.4%

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

      2. mul-1-neg 97.4%

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

      3. mul-1-neg 97.4%

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

      4. distribute-frac-neg 97.4%

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

      5. neg-mul-1 97.4%

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

      6. remove-double-neg 97.4%

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

      7. *-rgt-identity 97.4%

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

      8. associate-*r/ 97.4%

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

      9. unpow-1 97.4%

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

      10. exp-to-pow 97.4%

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

      11. unpow-1 97.4%

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

      12. *-commutative 97.4%

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

   4. Simplified 97.4%

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

      1. *-un-lft-identity 97.4%

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

      2. times-frac 97.9%

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

      3. inv-pow 97.9%

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

   6. Applied egg-rr 97.9%

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

   if -5.00000000000000011e-47 < (/.f64 1 n) < 1e-22

   1. Initial program 29.0%

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

   2. Taylor expanded in n around inf 86.1%

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

      1. log1p-def 86.1%

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

   4. Simplified 86.1%

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

      1. log1p-udef 86.1%

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

      2. diff-log 86.3%

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

      3. +-commutative 86.3%

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

   6. Applied egg-rr 86.3%

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

      1. clear-num 86.3%

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

      2. log-div 86.4%

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

      3. metadata-eval 86.4%

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

   8. Applied egg-rr 86.4%

      \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   9. Step-by-step derivation

      1. neg-sub0 86.4%

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

   10. Simplified 86.4%

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

   if 1e-22 < (/.f64 1 n) < 9.99999999999999977e194

   1. Initial program 81.0%

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

   if 9.99999999999999977e194 < (/.f64 1 n)

   1. Initial program 26.2%

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

   2. Taylor expanded in n around inf 7.3%

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

      1. log1p-def 7.3%

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

   4. Simplified 7.3%

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

      1. add-log-exp 83.9%

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

      2. div-inv 83.9%

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

      3. exp-prod 83.9%

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

      4. exp-diff 83.9%

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

      5. log1p-udef 83.9%

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

      6. add-exp-log 83.9%

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

      7. add-exp-log 83.9%

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

      8. +-commutative 83.9%

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

   6. Applied egg-rr 83.9%

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

      1. log-pow 7.3%

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

      2. associate-*l/ 7.3%

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

      3. *-un-lft-identity 7.3%

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

      4. log-div 7.3%

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

      5. +-commutative 7.3%

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

      6. log1p-udef 7.3%

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

      7. div-sub 7.3%

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

      8. frac-sub 83.9%

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

   8. Applied egg-rr 83.9%

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

      1. *-commutative 83.9%

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

      2. distribute-lft-out-- 83.9%

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

   10. Simplified 83.9%

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

   11. Taylor expanded in x around inf 83.9%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+195}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]

Alternative 4: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+195}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-47)
   (* (/ 1.0 x) (/ (pow x (pow n -1.0)) n))
   (if (<= (/ 1.0 n) 1e-22)
     (- (/ (log (/ x (+ 1.0 x))) n))
     (if (<= (/ 1.0 n) 1e+195)
       (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
       (/ (/ n x) (* n n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-47) {
		tmp = (1.0 / x) * (pow(x, pow(n, -1.0)) / n);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 1e+195) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-5d-47)) then
        tmp = (1.0d0 / x) * ((x ** (n ** (-1.0d0))) / n)
    else if ((1.0d0 / n) <= 1d-22) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else if ((1.0d0 / n) <= 1d+195) then
        tmp = (1.0d0 + (x / n)) - (x ** (1.0d0 / n))
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-47) {
		tmp = (1.0 / x) * (Math.pow(x, Math.pow(n, -1.0)) / n);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 1e+195) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-47:
		tmp = (1.0 / x) * (math.pow(x, math.pow(n, -1.0)) / n)
	elif (1.0 / n) <= 1e-22:
		tmp = -(math.log((x / (1.0 + x))) / n)
	elif (1.0 / n) <= 1e+195:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-47)
		tmp = Float64(Float64(1.0 / x) * Float64((x ^ (n ^ -1.0)) / n));
	elseif (Float64(1.0 / n) <= 1e-22)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 1e+195)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -5e-47)
		tmp = (1.0 / x) * ((x ^ (n ^ -1.0)) / n);
	elseif ((1.0 / n) <= 1e-22)
		tmp = -(log((x / (1.0 + x))) / n);
	elseif ((1.0 / n) <= 1e+195)
		tmp = (1.0 + (x / n)) - (x ^ (1.0 / n));
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-47], N[(N[(1.0 / x), $MachinePrecision] * N[(N[Power[x, N[Power[n, -1.0], $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-22], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+195], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000011e-47

   1. Initial program 90.0%

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

   2. Taylor expanded in x around inf 97.4%

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

      1. log-rec 97.4%

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

      2. mul-1-neg 97.4%

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

      3. mul-1-neg 97.4%

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

      4. distribute-frac-neg 97.4%

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

      5. neg-mul-1 97.4%

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

      6. remove-double-neg 97.4%

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

      7. *-rgt-identity 97.4%

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

      8. associate-*r/ 97.4%

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

      9. unpow-1 97.4%

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

      10. exp-to-pow 97.4%

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

      11. unpow-1 97.4%

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

      12. *-commutative 97.4%

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

   4. Simplified 97.4%

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

      1. *-un-lft-identity 97.4%

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

      2. times-frac 97.9%

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

      3. inv-pow 97.9%

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

   6. Applied egg-rr 97.9%

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

   if -5.00000000000000011e-47 < (/.f64 1 n) < 1e-22

   1. Initial program 29.0%

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

   2. Taylor expanded in n around inf 86.1%

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

      1. log1p-def 86.1%

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

   4. Simplified 86.1%

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

      1. log1p-udef 86.1%

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

      2. diff-log 86.3%

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

      3. +-commutative 86.3%

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

   6. Applied egg-rr 86.3%

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

      1. clear-num 86.3%

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

      2. log-div 86.4%

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

      3. metadata-eval 86.4%

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

   8. Applied egg-rr 86.4%

      \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   9. Step-by-step derivation

      1. neg-sub0 86.4%

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

   10. Simplified 86.4%

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

   if 1e-22 < (/.f64 1 n) < 9.99999999999999977e194

   1. Initial program 81.0%

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

   2. Taylor expanded in x around 0 75.1%

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

   if 9.99999999999999977e194 < (/.f64 1 n)

   1. Initial program 26.2%

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

   2. Taylor expanded in n around inf 7.3%

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

      1. log1p-def 7.3%

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

   4. Simplified 7.3%

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

      1. add-log-exp 83.9%

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

      2. div-inv 83.9%

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

      3. exp-prod 83.9%

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

      4. exp-diff 83.9%

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

      5. log1p-udef 83.9%

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

      6. add-exp-log 83.9%

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

      7. add-exp-log 83.9%

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

      8. +-commutative 83.9%

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

   6. Applied egg-rr 83.9%

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

      1. log-pow 7.3%

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

      2. associate-*l/ 7.3%

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

      3. *-un-lft-identity 7.3%

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

      4. log-div 7.3%

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

      5. +-commutative 7.3%

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

      6. log1p-udef 7.3%

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

      7. div-sub 7.3%

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

      8. frac-sub 83.9%

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

   8. Applied egg-rr 83.9%

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

      1. *-commutative 83.9%

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

      2. distribute-lft-out-- 83.9%

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

   10. Simplified 83.9%

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

   11. Taylor expanded in x around inf 83.9%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left({n}^{-1}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+195}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]

Alternative 5: 67.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+224}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -50000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+195}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -1e+224)
     t_0
     (if (<= (/ 1.0 n) -5e+206)
       t_1
       (if (<= (/ 1.0 n) -50000.0)
         t_0
         (if (<= (/ 1.0 n) 1e-22)
           t_1
           (if (<= (/ 1.0 n) 1e+195) t_0 (/ (/ n x) (* n n)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e+224) {
		tmp = t_0;
	} else if ((1.0 / n) <= -5e+206) {
		tmp = t_1;
	} else if ((1.0 / n) <= -50000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-22) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+195) {
		tmp = t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = log(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-1d+224)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-5d+206)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-50000.0d0)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d-22) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+195) then
        tmp = t_0
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -1e+224) {
		tmp = t_0;
	} else if ((1.0 / n) <= -5e+206) {
		tmp = t_1;
	} else if ((1.0 / n) <= -50000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-22) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+195) {
		tmp = t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -1e+224:
		tmp = t_0
	elif (1.0 / n) <= -5e+206:
		tmp = t_1
	elif (1.0 / n) <= -50000.0:
		tmp = t_0
	elif (1.0 / n) <= 1e-22:
		tmp = t_1
	elif (1.0 / n) <= 1e+195:
		tmp = t_0
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+224)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -5e+206)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -50000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-22)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+195)
		tmp = t_0;
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -1e+224)
		tmp = t_0;
	elseif ((1.0 / n) <= -5e+206)
		tmp = t_1;
	elseif ((1.0 / n) <= -50000.0)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e-22)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+195)
		tmp = t_0;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+224], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+206], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -50000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-22], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+195], t$95$0, N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -9.9999999999999997e223 or -5.0000000000000002e206 < (/.f64 1 n) < -5e4 or 1e-22 < (/.f64 1 n) < 9.99999999999999977e194

   1. Initial program 96.4%

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

   2. Taylor expanded in x around 0 64.8%

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

      1. *-rgt-identity 64.8%

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

      2. associate-*r/ 64.8%

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

      3. unpow-1 64.8%

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

      4. exp-to-pow 64.8%

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

      5. unpow-1 64.8%

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

   4. Simplified 64.8%

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

   if -9.9999999999999997e223 < (/.f64 1 n) < -5.0000000000000002e206 or -5e4 < (/.f64 1 n) < 1e-22

   1. Initial program 31.5%

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

   2. Taylor expanded in n around inf 82.6%

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

      1. log1p-def 82.6%

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

   4. Simplified 82.6%

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

      1. log1p-udef 82.6%

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

      2. diff-log 82.7%

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

      3. +-commutative 82.7%

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

   6. Applied egg-rr 82.7%

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

   if 9.99999999999999977e194 < (/.f64 1 n)

   1. Initial program 26.2%

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

   2. Taylor expanded in n around inf 7.3%

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

      1. log1p-def 7.3%

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

   4. Simplified 7.3%

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

      1. add-log-exp 83.9%

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

      2. div-inv 83.9%

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

      3. exp-prod 83.9%

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

      4. exp-diff 83.9%

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

      5. log1p-udef 83.9%

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

      6. add-exp-log 83.9%

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

      7. add-exp-log 83.9%

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

      8. +-commutative 83.9%

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

   6. Applied egg-rr 83.9%

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

      1. log-pow 7.3%

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

      2. associate-*l/ 7.3%

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

      3. *-un-lft-identity 7.3%

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

      4. log-div 7.3%

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

      5. +-commutative 7.3%

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

      6. log1p-udef 7.3%

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

      7. div-sub 7.3%

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

      8. frac-sub 83.9%

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

   8. Applied egg-rr 83.9%

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

      1. *-commutative 83.9%

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

      2. distribute-lft-out-- 83.9%

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

   10. Simplified 83.9%

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

   11. Taylor expanded in x around inf 83.9%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+224}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+206}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -50000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+195}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]

Alternative 6: 67.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+224}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+206}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -50000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+195}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -1e+224)
     t_0
     (if (<= (/ 1.0 n) -5e+206)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) -50000.0)
         t_0
         (if (<= (/ 1.0 n) 1e-22)
           (- (/ (log (/ x (+ 1.0 x))) n))
           (if (<= (/ 1.0 n) 1e+195) t_0 (/ (/ n x) (* n n)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+224) {
		tmp = t_0;
	} else if ((1.0 / n) <= -5e+206) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= -50000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-22) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 1e+195) {
		tmp = t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-1d+224)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-5d+206)) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= (-50000.0d0)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 1d-22) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else if ((1.0d0 / n) <= 1d+195) then
        tmp = t_0
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+224) {
		tmp = t_0;
	} else if ((1.0 / n) <= -5e+206) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= -50000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 1e-22) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 1e+195) {
		tmp = t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e+224:
		tmp = t_0
	elif (1.0 / n) <= -5e+206:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= -50000.0:
		tmp = t_0
	elif (1.0 / n) <= 1e-22:
		tmp = -(math.log((x / (1.0 + x))) / n)
	elif (1.0 / n) <= 1e+195:
		tmp = t_0
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+224)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -5e+206)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= -50000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 1e-22)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 1e+195)
		tmp = t_0;
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -1e+224)
		tmp = t_0;
	elseif ((1.0 / n) <= -5e+206)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= -50000.0)
		tmp = t_0;
	elseif ((1.0 / n) <= 1e-22)
		tmp = -(log((x / (1.0 + x))) / n);
	elseif ((1.0 / n) <= 1e+195)
		tmp = t_0;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+224], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+206], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -50000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-22], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+195], t$95$0, N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -9.9999999999999997e223 or -5.0000000000000002e206 < (/.f64 1 n) < -5e4 or 1e-22 < (/.f64 1 n) < 9.99999999999999977e194

   1. Initial program 96.4%

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

   2. Taylor expanded in x around 0 64.8%

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

      1. *-rgt-identity 64.8%

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

      2. associate-*r/ 64.8%

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

      3. unpow-1 64.8%

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

      4. exp-to-pow 64.8%

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

      5. unpow-1 64.8%

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

   4. Simplified 64.8%

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

   if -9.9999999999999997e223 < (/.f64 1 n) < -5.0000000000000002e206

   1. Initial program 100.0%

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

   2. Taylor expanded in n around inf 88.2%

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

      1. log1p-def 88.2%

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

   4. Simplified 88.2%

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

      1. log1p-udef 88.2%

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

      2. diff-log 88.2%

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

      3. +-commutative 88.2%

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

   6. Applied egg-rr 88.2%

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

   if -5e4 < (/.f64 1 n) < 1e-22

   1. Initial program 27.7%

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

   2. Taylor expanded in n around inf 82.3%

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

      1. log1p-def 82.3%

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

   4. Simplified 82.3%

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

      1. log1p-udef 82.3%

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

      2. diff-log 82.4%

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

      3. +-commutative 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. clear-num 82.4%

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

      2. log-div 82.5%

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

      3. metadata-eval 82.5%

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

   8. Applied egg-rr 82.5%

      \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   9. Step-by-step derivation

      1. neg-sub0 82.5%

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

   10. Simplified 82.5%

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

   if 9.99999999999999977e194 < (/.f64 1 n)

   1. Initial program 26.2%

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

   2. Taylor expanded in n around inf 7.3%

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

      1. log1p-def 7.3%

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

   4. Simplified 7.3%

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

      1. add-log-exp 83.9%

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

      2. div-inv 83.9%

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

      3. exp-prod 83.9%

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

      4. exp-diff 83.9%

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

      5. log1p-udef 83.9%

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

      6. add-exp-log 83.9%

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

      7. add-exp-log 83.9%

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

      8. +-commutative 83.9%

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

   6. Applied egg-rr 83.9%

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

      1. log-pow 7.3%

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

      2. associate-*l/ 7.3%

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

      3. *-un-lft-identity 7.3%

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

      4. log-div 7.3%

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

      5. +-commutative 7.3%

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

      6. log1p-udef 7.3%

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

      7. div-sub 7.3%

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

      8. frac-sub 83.9%

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

   8. Applied egg-rr 83.9%

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

      1. *-commutative 83.9%

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

      2. distribute-lft-out-- 83.9%

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

   10. Simplified 83.9%

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

   11. Taylor expanded in x around inf 83.9%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+224}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{+206}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -50000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+195}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]

Alternative 7: 82.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+195}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-47)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e-22)
       (- (/ (log (/ x (+ 1.0 x))) n))
       (if (<= (/ 1.0 n) 1e+195)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (/ n x) (* n n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-47) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 1e+195) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-47)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 1d-22) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else if ((1.0d0 / n) <= 1d+195) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (n / x) / (n * n)
    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) <= -5e-47) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 1e+195) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-47:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e-22:
		tmp = -(math.log((x / (1.0 + x))) / n)
	elif (1.0 / n) <= 1e+195:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-47)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-22)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 1e+195)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-47)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 1e-22)
		tmp = -(log((x / (1.0 + x))) / n);
	elseif ((1.0 / n) <= 1e+195)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = (n / x) / (n * n);
	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], -5e-47], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-22], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+195], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000011e-47

   1. Initial program 90.0%

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

   2. Taylor expanded in x around inf 97.4%

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

      1. log-rec 97.4%

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

      2. mul-1-neg 97.4%

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

      3. mul-1-neg 97.4%

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

      4. distribute-frac-neg 97.4%

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

      5. neg-mul-1 97.4%

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

      6. remove-double-neg 97.4%

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

      7. *-rgt-identity 97.4%

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

      8. associate-*r/ 97.4%

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

      9. unpow-1 97.4%

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

      10. exp-to-pow 97.4%

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

      11. unpow-1 97.4%

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

      12. *-commutative 97.4%

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

   4. Simplified 97.4%

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

   if -5.00000000000000011e-47 < (/.f64 1 n) < 1e-22

   1. Initial program 29.0%

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

   2. Taylor expanded in n around inf 86.1%

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

      1. log1p-def 86.1%

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

   4. Simplified 86.1%

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

      1. log1p-udef 86.1%

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

      2. diff-log 86.3%

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

      3. +-commutative 86.3%

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

   6. Applied egg-rr 86.3%

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

      1. clear-num 86.3%

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

      2. log-div 86.4%

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

      3. metadata-eval 86.4%

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

   8. Applied egg-rr 86.4%

      \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   9. Step-by-step derivation

      1. neg-sub0 86.4%

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

   10. Simplified 86.4%

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

   if 1e-22 < (/.f64 1 n) < 9.99999999999999977e194

   1. Initial program 81.0%

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

   2. Taylor expanded in x around 0 75.1%

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

   if 9.99999999999999977e194 < (/.f64 1 n)

   1. Initial program 26.2%

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

   2. Taylor expanded in n around inf 7.3%

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

      1. log1p-def 7.3%

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

   4. Simplified 7.3%

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

      1. add-log-exp 83.9%

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

      2. div-inv 83.9%

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

      3. exp-prod 83.9%

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

      4. exp-diff 83.9%

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

      5. log1p-udef 83.9%

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

      6. add-exp-log 83.9%

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

      7. add-exp-log 83.9%

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

      8. +-commutative 83.9%

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

   6. Applied egg-rr 83.9%

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

      1. log-pow 7.3%

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

      2. associate-*l/ 7.3%

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

      3. *-un-lft-identity 7.3%

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

      4. log-div 7.3%

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

      5. +-commutative 7.3%

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

      6. log1p-udef 7.3%

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

      7. div-sub 7.3%

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

      8. frac-sub 83.9%

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

   8. Applied egg-rr 83.9%

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

      1. *-commutative 83.9%

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

      2. distribute-lft-out-- 83.9%

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

   10. Simplified 83.9%

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

   11. Taylor expanded in x around inf 83.9%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+195}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]

Alternative 8: 82.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+195}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-47)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e-22)
       (- (/ (log (/ x (+ 1.0 x))) n))
       (if (<= (/ 1.0 n) 1e+195) (- 1.0 t_0) (/ (/ n x) (* n n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-47) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = -(log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 1e+195) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-47)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 1d-22) then
        tmp = -(log((x / (1.0d0 + x))) / n)
    else if ((1.0d0 / n) <= 1d+195) then
        tmp = 1.0d0 - t_0
    else
        tmp = (n / x) / (n * n)
    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) <= -5e-47) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-22) {
		tmp = -(Math.log((x / (1.0 + x))) / n);
	} else if ((1.0 / n) <= 1e+195) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-47:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e-22:
		tmp = -(math.log((x / (1.0 + x))) / n)
	elif (1.0 / n) <= 1e+195:
		tmp = 1.0 - t_0
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-47)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-22)
		tmp = Float64(-Float64(log(Float64(x / Float64(1.0 + x))) / n));
	elseif (Float64(1.0 / n) <= 1e+195)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-47)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 1e-22)
		tmp = -(log((x / (1.0 + x))) / n);
	elseif ((1.0 / n) <= 1e+195)
		tmp = 1.0 - t_0;
	else
		tmp = (n / x) / (n * n);
	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], -5e-47], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-22], (-N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]), If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+195], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000011e-47

   1. Initial program 90.0%

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

   2. Taylor expanded in x around inf 97.4%

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

      1. log-rec 97.4%

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

      2. mul-1-neg 97.4%

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

      3. mul-1-neg 97.4%

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

      4. distribute-frac-neg 97.4%

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

      5. neg-mul-1 97.4%

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

      6. remove-double-neg 97.4%

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

      7. *-rgt-identity 97.4%

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

      8. associate-*r/ 97.4%

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

      9. unpow-1 97.4%

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

      10. exp-to-pow 97.4%

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

      11. unpow-1 97.4%

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

      12. *-commutative 97.4%

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

   4. Simplified 97.4%

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

   if -5.00000000000000011e-47 < (/.f64 1 n) < 1e-22

   1. Initial program 29.0%

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

   2. Taylor expanded in n around inf 86.1%

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

      1. log1p-def 86.1%

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

   4. Simplified 86.1%

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

      1. log1p-udef 86.1%

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

      2. diff-log 86.3%

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

      3. +-commutative 86.3%

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

   6. Applied egg-rr 86.3%

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

      1. clear-num 86.3%

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

      2. log-div 86.4%

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

      3. metadata-eval 86.4%

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

   8. Applied egg-rr 86.4%

      \[\leadsto \frac{\color{blue}{0 - \log \left(\frac{x}{x + 1}\right)}}{n} \]
   9. Step-by-step derivation

      1. neg-sub0 86.4%

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

   10. Simplified 86.4%

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

   if 1e-22 < (/.f64 1 n) < 9.99999999999999977e194

   1. Initial program 81.0%

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

   2. Taylor expanded in x around 0 75.1%

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

      1. *-rgt-identity 75.1%

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

      2. associate-*r/ 75.1%

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

      3. unpow-1 75.1%

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

      4. exp-to-pow 75.1%

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

      5. unpow-1 75.1%

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

   4. Simplified 75.1%

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

   if 9.99999999999999977e194 < (/.f64 1 n)

   1. Initial program 26.2%

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

   2. Taylor expanded in n around inf 7.3%

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

      1. log1p-def 7.3%

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

   4. Simplified 7.3%

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

      1. add-log-exp 83.9%

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

      2. div-inv 83.9%

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

      3. exp-prod 83.9%

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

      4. exp-diff 83.9%

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

      5. log1p-udef 83.9%

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

      6. add-exp-log 83.9%

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

      7. add-exp-log 83.9%

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

      8. +-commutative 83.9%

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

   6. Applied egg-rr 83.9%

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

      1. log-pow 7.3%

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

      2. associate-*l/ 7.3%

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

      3. *-un-lft-identity 7.3%

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

      4. log-div 7.3%

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

      5. +-commutative 7.3%

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

      6. log1p-udef 7.3%

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

      7. div-sub 7.3%

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

      8. frac-sub 83.9%

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

   8. Applied egg-rr 83.9%

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

      1. *-commutative 83.9%

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

      2. distribute-lft-out-- 83.9%

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

   10. Simplified 83.9%

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

   11. Taylor expanded in x around inf 83.9%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-22}:\\ \;\;\;\;-\frac{\log \left(\frac{x}{1 + x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+195}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]

Alternative 9: 58.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.4 \cdot 10^{-284}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-233}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 10^{+89}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 5.4e-284)
   (/ (- (log x)) n)
   (if (<= x 2.3e-233)
     (/ 1.0 (* n x))
     (if (<= x 0.98)
       (/ (- x (log x)) n)
       (if (<= x 1e+89) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5.4e-284) {
		tmp = -log(x) / n;
	} else if (x <= 2.3e-233) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.98) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1e+89) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 5.4d-284) then
        tmp = -log(x) / n
    else if (x <= 2.3d-233) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 0.98d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1d+89) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 5.4e-284) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.3e-233) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.98) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1e+89) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 5.4e-284:
		tmp = -math.log(x) / n
	elif x <= 2.3e-233:
		tmp = 1.0 / (n * x)
	elif x <= 0.98:
		tmp = (x - math.log(x)) / n
	elif x <= 1e+89:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 5.4e-284)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.3e-233)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 0.98)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1e+89)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 5.4e-284)
		tmp = -log(x) / n;
	elseif (x <= 2.3e-233)
		tmp = 1.0 / (n * x);
	elseif (x <= 0.98)
		tmp = (x - log(x)) / n;
	elseif (x <= 1e+89)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 5.4e-284], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.3e-233], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.98], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1e+89], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 5.39999999999999969e-284

   1. Initial program 42.5%

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

   2. Taylor expanded in n around inf 61.6%

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

      1. log1p-def 61.6%

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

   4. Simplified 61.6%

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

   5. Taylor expanded in x around 0 61.6%

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

      1. neg-mul-1 61.6%

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

   7. Simplified 61.6%

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

   if 5.39999999999999969e-284 < x < 2.3000000000000002e-233

   1. Initial program 68.1%

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

   2. Taylor expanded in n around inf 26.8%

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

      1. log1p-def 26.8%

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

   4. Simplified 26.8%

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

   5. Taylor expanded in x around inf 52.2%

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

      1. *-commutative 52.2%

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

   7. Simplified 52.2%

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

   if 2.3000000000000002e-233 < x < 0.97999999999999998

   1. Initial program 37.4%

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

   2. Taylor expanded in n around inf 61.2%

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

      1. log1p-def 61.2%

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

   4. Simplified 61.2%

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

   5. Taylor expanded in x around 0 61.1%

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

      1. neg-mul-1 61.1%

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

      2. sub-neg 61.1%

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

   7. Simplified 61.1%

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

   if 0.97999999999999998 < x < 9.99999999999999995e88

   1. Initial program 45.4%

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

   2. Taylor expanded in n around inf 50.9%

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

      1. log1p-def 50.9%

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

   4. Simplified 50.9%

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

   5. Taylor expanded in x around inf 68.4%

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

      1. associate-*r/ 68.4%

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

      2. metadata-eval 68.4%

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

      3. unpow2 68.4%

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

   7. Simplified 68.4%

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

   if 9.99999999999999995e88 < x

   1. Initial program 83.2%

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

   2. Taylor expanded in n around inf 83.2%

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

      1. log1p-def 83.2%

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

   4. Simplified 83.2%

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

      1. log1p-udef 83.2%

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

      2. diff-log 83.2%

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

      3. +-commutative 83.2%

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

   6. Applied egg-rr 83.2%

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

   7. Taylor expanded in x around inf 83.2%

      \[\leadsto \frac{\log \color{blue}{1}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.4 \cdot 10^{-284}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-233}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 10^{+89}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 10: 58.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-284}:\\ \;\;\;\;\log x \cdot \frac{1}{-n}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-233}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.5e-284)
   (* (log x) (/ 1.0 (- n)))
   (if (<= x 2.3e-233)
     (/ 1.0 (* n x))
     (if (<= x 0.96)
       (/ (- x (log x)) n)
       (if (<= x 3.5e+90) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.5e-284) {
		tmp = log(x) * (1.0 / -n);
	} else if (x <= 2.3e-233) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.96) {
		tmp = (x - log(x)) / n;
	} else if (x <= 3.5e+90) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3.5d-284) then
        tmp = log(x) * (1.0d0 / -n)
    else if (x <= 2.3d-233) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 0.96d0) then
        tmp = (x - log(x)) / n
    else if (x <= 3.5d+90) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.5e-284) {
		tmp = Math.log(x) * (1.0 / -n);
	} else if (x <= 2.3e-233) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.96) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 3.5e+90) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.5e-284:
		tmp = math.log(x) * (1.0 / -n)
	elif x <= 2.3e-233:
		tmp = 1.0 / (n * x)
	elif x <= 0.96:
		tmp = (x - math.log(x)) / n
	elif x <= 3.5e+90:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.5e-284)
		tmp = Float64(log(x) * Float64(1.0 / Float64(-n)));
	elseif (x <= 2.3e-233)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 0.96)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 3.5e+90)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.5e-284)
		tmp = log(x) * (1.0 / -n);
	elseif (x <= 2.3e-233)
		tmp = 1.0 / (n * x);
	elseif (x <= 0.96)
		tmp = (x - log(x)) / n;
	elseif (x <= 3.5e+90)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.5e-284], N[(N[Log[x], $MachinePrecision] * N[(1.0 / (-n)), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-233], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.96], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 3.5e+90], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 3.49999999999999975e-284

   1. Initial program 42.5%

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

   2. Taylor expanded in n around inf 61.6%

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

      1. log1p-def 61.6%

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

   4. Simplified 61.6%

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

      1. log1p-udef 61.6%

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

      2. diff-log 61.6%

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

      3. +-commutative 61.6%

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

   6. Applied egg-rr 61.6%

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

      1. frac-2neg 61.6%

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

      2. div-inv 61.9%

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

      3. neg-log 61.9%

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

      4. clear-num 61.9%

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

   8. Applied egg-rr 61.9%

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

   9. Taylor expanded in x around 0 61.9%

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

   if 3.49999999999999975e-284 < x < 2.3000000000000002e-233

   1. Initial program 68.1%

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

   2. Taylor expanded in n around inf 26.8%

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

      1. log1p-def 26.8%

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

   4. Simplified 26.8%

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

   5. Taylor expanded in x around inf 52.2%

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

      1. *-commutative 52.2%

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

   7. Simplified 52.2%

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

   if 2.3000000000000002e-233 < x < 0.95999999999999996

   1. Initial program 37.4%

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

   2. Taylor expanded in n around inf 61.2%

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

      1. log1p-def 61.2%

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

   4. Simplified 61.2%

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

   5. Taylor expanded in x around 0 61.1%

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

      1. neg-mul-1 61.1%

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

      2. sub-neg 61.1%

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

   7. Simplified 61.1%

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

   if 0.95999999999999996 < x < 3.4999999999999998e90

   1. Initial program 45.4%

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

   2. Taylor expanded in n around inf 50.9%

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

      1. log1p-def 50.9%

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

   4. Simplified 50.9%

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

   5. Taylor expanded in x around inf 68.4%

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

      1. associate-*r/ 68.4%

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

      2. metadata-eval 68.4%

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

      3. unpow2 68.4%

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

   7. Simplified 68.4%

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

   if 3.4999999999999998e90 < x

   1. Initial program 83.2%

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

   2. Taylor expanded in n around inf 83.2%

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

      1. log1p-def 83.2%

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

   4. Simplified 83.2%

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

      1. log1p-udef 83.2%

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

      2. diff-log 83.2%

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

      3. +-commutative 83.2%

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

   6. Applied egg-rr 83.2%

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

   7. Taylor expanded in x around inf 83.2%

      \[\leadsto \frac{\log \color{blue}{1}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-284}:\\ \;\;\;\;\log x \cdot \frac{1}{-n}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-233}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 11: 58.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 5.4 \cdot 10^{-284}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-233}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log x)) n)))
   (if (<= x 5.4e-284)
     t_0
     (if (<= x 2.1e-233)
       (/ 1.0 (* n x))
       (if (<= x 0.68)
         t_0
         (if (<= x 1.15e+89)
           (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n)
           (/ 0.0 n)))))))

C

double code(double x, double n) {
	double t_0 = -log(x) / n;
	double tmp;
	if (x <= 5.4e-284) {
		tmp = t_0;
	} else if (x <= 2.1e-233) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.68) {
		tmp = t_0;
	} else if (x <= 1.15e+89) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -log(x) / n
    if (x <= 5.4d-284) then
        tmp = t_0
    else if (x <= 2.1d-233) then
        tmp = 1.0d0 / (n * x)
    else if (x <= 0.68d0) then
        tmp = t_0
    else if (x <= 1.15d+89) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log(x) / n;
	double tmp;
	if (x <= 5.4e-284) {
		tmp = t_0;
	} else if (x <= 2.1e-233) {
		tmp = 1.0 / (n * x);
	} else if (x <= 0.68) {
		tmp = t_0;
	} else if (x <= 1.15e+89) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log(x) / n
	tmp = 0
	if x <= 5.4e-284:
		tmp = t_0
	elif x <= 2.1e-233:
		tmp = 1.0 / (n * x)
	elif x <= 0.68:
		tmp = t_0
	elif x <= 1.15e+89:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 5.4e-284)
		tmp = t_0;
	elseif (x <= 2.1e-233)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (x <= 0.68)
		tmp = t_0;
	elseif (x <= 1.15e+89)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log(x) / n;
	tmp = 0.0;
	if (x <= 5.4e-284)
		tmp = t_0;
	elseif (x <= 2.1e-233)
		tmp = 1.0 / (n * x);
	elseif (x <= 0.68)
		tmp = t_0;
	elseif (x <= 1.15e+89)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 5.4e-284], t$95$0, If[LessEqual[x, 2.1e-233], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.68], t$95$0, If[LessEqual[x, 1.15e+89], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 5.39999999999999969e-284 or 2.0999999999999999e-233 < x < 0.680000000000000049

   1. Initial program 38.0%

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

   2. Taylor expanded in n around inf 61.2%

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

      1. log1p-def 61.2%

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

   4. Simplified 61.2%

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

   5. Taylor expanded in x around 0 60.4%

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

      1. neg-mul-1 60.4%

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

   7. Simplified 60.4%

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

   if 5.39999999999999969e-284 < x < 2.0999999999999999e-233

   1. Initial program 68.1%

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

   2. Taylor expanded in n around inf 26.8%

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

      1. log1p-def 26.8%

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

   4. Simplified 26.8%

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

   5. Taylor expanded in x around inf 52.2%

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

      1. *-commutative 52.2%

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

   7. Simplified 52.2%

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

   if 0.680000000000000049 < x < 1.1499999999999999e89

   1. Initial program 45.4%

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

   2. Taylor expanded in n around inf 50.9%

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

      1. log1p-def 50.9%

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

   4. Simplified 50.9%

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

   5. Taylor expanded in x around inf 68.4%

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

      1. associate-*r/ 68.4%

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

      2. metadata-eval 68.4%

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

      3. unpow2 68.4%

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

   7. Simplified 68.4%

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

   if 1.1499999999999999e89 < x

   1. Initial program 83.2%

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

   2. Taylor expanded in n around inf 83.2%

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

      1. log1p-def 83.2%

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

   4. Simplified 83.2%

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

      1. log1p-udef 83.2%

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

      2. diff-log 83.2%

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

      3. +-commutative 83.2%

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

   6. Applied egg-rr 83.2%

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

   7. Taylor expanded in x around inf 83.2%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.4 \cdot 10^{-284}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-233}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;x \leq 0.68:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 12: 60.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4e-233)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 0.98)
     (/ (- x (log x)) n)
     (if (<= x 1.3e+90) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4e-233) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.98) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.3e+90) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4d-233) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.98d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.3d+90) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4e-233) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.98) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.3e+90) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4e-233:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.98:
		tmp = (x - math.log(x)) / n
	elif x <= 1.3e+90:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4e-233)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.98)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.3e+90)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4e-233)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.98)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.3e+90)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4e-233], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.98], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.3e+90], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 3.99999999999999983e-233

   1. Initial program 58.4%

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

   2. Taylor expanded in x around 0 58.4%

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

      1. *-rgt-identity 58.4%

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

      2. associate-*r/ 58.4%

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

      3. unpow-1 58.4%

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

      4. exp-to-pow 58.4%

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

      5. unpow-1 58.4%

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

   4. Simplified 58.4%

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

   if 3.99999999999999983e-233 < x < 0.97999999999999998

   1. Initial program 37.2%

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

   2. Taylor expanded in n around inf 61.2%

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

      1. log1p-def 61.2%

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

   4. Simplified 61.2%

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

   5. Taylor expanded in x around 0 61.1%

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

      1. neg-mul-1 61.1%

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

      2. sub-neg 61.1%

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

   7. Simplified 61.1%

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

   if 0.97999999999999998 < x < 1.2999999999999999e90

   1. Initial program 45.4%

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

   2. Taylor expanded in n around inf 50.9%

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

      1. log1p-def 50.9%

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

   4. Simplified 50.9%

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

   5. Taylor expanded in x around inf 68.4%

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

      1. associate-*r/ 68.4%

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

      2. metadata-eval 68.4%

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

      3. unpow2 68.4%

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

   7. Simplified 68.4%

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

   if 1.2999999999999999e90 < x

   1. Initial program 83.2%

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

   2. Taylor expanded in n around inf 83.2%

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

      1. log1p-def 83.2%

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

   4. Simplified 83.2%

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

      1. log1p-udef 83.2%

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

      2. diff-log 83.2%

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

      3. +-commutative 83.2%

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

   6. Applied egg-rr 83.2%

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

   7. Taylor expanded in x around inf 83.2%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 13: 44.6% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+24} \lor \neg \left(\frac{1}{n} \leq 10^{-88}\right):\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= (/ 1.0 n) -1e+24) (not (<= (/ 1.0 n) 1e-88)))
   (/ (/ n x) (* n n))
   (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if (((1.0 / n) <= -1e+24) || !((1.0 / n) <= 1e-88)) {
		tmp = (n / x) / (n * n);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (((1.0d0 / n) <= (-1d+24)) .or. (.not. ((1.0d0 / n) <= 1d-88))) then
        tmp = (n / x) / (n * n)
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (((1.0 / n) <= -1e+24) || !((1.0 / n) <= 1e-88)) {
		tmp = (n / x) / (n * n);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if ((1.0 / n) <= -1e+24) or not ((1.0 / n) <= 1e-88):
		tmp = (n / x) / (n * n)
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((Float64(1.0 / n) <= -1e+24) || !(Float64(1.0 / n) <= 1e-88))
		tmp = Float64(Float64(n / x) / Float64(n * n));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (((1.0 / n) <= -1e+24) || ~(((1.0 / n) <= 1e-88)))
		tmp = (n / x) / (n * n);
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+24], N[Not[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-88]], $MachinePrecision]], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -9.9999999999999998e23 or 9.99999999999999934e-89 < (/.f64 1 n)

   1. Initial program 81.5%

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

   2. Taylor expanded in n around inf 41.1%

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

      1. log1p-def 41.1%

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

   4. Simplified 41.1%

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

      1. add-log-exp 78.3%

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

      2. div-inv 78.3%

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

      3. exp-prod 78.3%

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

      4. exp-diff 78.3%

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

      5. log1p-udef 78.3%

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

      6. add-exp-log 51.4%

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

      7. add-exp-log 78.3%

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

      8. +-commutative 78.3%

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

   6. Applied egg-rr 78.3%

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

      1. log-pow 41.1%

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

      2. associate-*l/ 41.1%

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

      3. *-un-lft-identity 41.1%

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

      4. log-div 41.1%

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

      5. +-commutative 41.1%

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

      6. log1p-udef 41.1%

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

      7. div-sub 41.1%

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

      8. frac-sub 51.7%

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

   8. Applied egg-rr 51.7%

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

      1. *-commutative 51.7%

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

      2. distribute-lft-out-- 51.7%

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

   10. Simplified 51.7%

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

   11. Taylor expanded in x around inf 42.4%

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

   if -9.9999999999999998e23 < (/.f64 1 n) < 9.99999999999999934e-89

   1. Initial program 30.5%

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

   2. Taylor expanded in n around inf 80.5%

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

      1. log1p-def 80.5%

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

   4. Simplified 80.5%

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

   5. Taylor expanded in x around inf 44.4%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+24} \lor \neg \left(\frac{1}{n} \leq 10^{-88}\right):\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 14: 44.6% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\frac{n \cdot \frac{1}{x}}{n \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-88}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-35)
   (/ (* n (/ 1.0 x)) (* n n))
   (if (<= (/ 1.0 n) 1e-88) (/ (/ 1.0 x) n) (/ (/ n x) (* n n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-35) {
		tmp = (n * (1.0 / x)) / (n * n);
	} else if ((1.0 / n) <= 1e-88) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d-35)) then
        tmp = (n * (1.0d0 / x)) / (n * n)
    else if ((1.0d0 / n) <= 1d-88) then
        tmp = (1.0d0 / x) / n
    else
        tmp = (n / x) / (n * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-35) {
		tmp = (n * (1.0 / x)) / (n * n);
	} else if ((1.0 / n) <= 1e-88) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = (n / x) / (n * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-35:
		tmp = (n * (1.0 / x)) / (n * n)
	elif (1.0 / n) <= 1e-88:
		tmp = (1.0 / x) / n
	else:
		tmp = (n / x) / (n * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-35)
		tmp = Float64(Float64(n * Float64(1.0 / x)) / Float64(n * n));
	elseif (Float64(1.0 / n) <= 1e-88)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(Float64(n / x) / Float64(n * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-35)
		tmp = (n * (1.0 / x)) / (n * n);
	elseif ((1.0 / n) <= 1e-88)
		tmp = (1.0 / x) / n;
	else
		tmp = (n / x) / (n * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-35], N[(N[(n * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-88], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2.00000000000000002e-35

   1. Initial program 91.0%

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

   2. Taylor expanded in n around inf 44.4%

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

      1. log1p-def 44.4%

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

   4. Simplified 44.4%

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

      1. add-log-exp 90.0%

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

      2. div-inv 90.0%

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

      3. exp-prod 90.0%

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

      4. exp-diff 90.0%

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

      5. log1p-udef 90.0%

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

      6. add-exp-log 53.5%

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

      7. add-exp-log 90.0%

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

      8. +-commutative 90.0%

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

   6. Applied egg-rr 90.0%

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

      1. log-pow 44.4%

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

      2. associate-*l/ 44.4%

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

      3. *-un-lft-identity 44.4%

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

      4. log-div 44.4%

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

      5. +-commutative 44.4%

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

      6. log1p-udef 44.4%

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

      7. div-sub 44.4%

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

      8. frac-sub 48.0%

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

   8. Applied egg-rr 48.0%

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

      1. *-commutative 48.0%

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

      2. distribute-lft-out-- 48.0%

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

   10. Simplified 48.0%

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

   11. Taylor expanded in x around inf 45.0%

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

   if -2.00000000000000002e-35 < (/.f64 1 n) < 9.99999999999999934e-89

   1. Initial program 29.7%

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

   2. Taylor expanded in n around inf 85.7%

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

      1. log1p-def 85.7%

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

   4. Simplified 85.7%

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

   5. Taylor expanded in x around inf 44.3%

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

   if 9.99999999999999934e-89 < (/.f64 1 n)

   1. Initial program 47.1%

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

   2. Taylor expanded in n around inf 30.7%

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

      1. log1p-def 30.7%

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

   4. Simplified 30.7%

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

      1. add-log-exp 40.5%

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

      2. div-inv 40.5%

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

      3. exp-prod 40.5%

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

      4. exp-diff 40.5%

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

      5. log1p-udef 40.5%

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

      6. add-exp-log 40.5%

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

      7. add-exp-log 40.5%

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

      8. +-commutative 40.5%

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

   6. Applied egg-rr 40.5%

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

      1. log-pow 30.7%

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

      2. associate-*l/ 30.7%

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

      3. *-un-lft-identity 30.7%

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

      4. log-div 30.7%

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

      5. +-commutative 30.7%

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

      6. log1p-udef 30.7%

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

      7. div-sub 30.7%

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

      8. frac-sub 53.1%

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

   8. Applied egg-rr 53.1%

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

      1. *-commutative 53.1%

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

      2. distribute-lft-out-- 53.1%

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

   10. Simplified 53.1%

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

   11. Taylor expanded in x around inf 37.8%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\frac{n \cdot \frac{1}{x}}{n \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-88}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \end{array} \]

Alternative 15: 46.8% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (/ n x) (* n n))
   (if (<= x 9.6e+89) (/ (- (/ 1.0 x) (/ 0.5 (* x x))) n) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (n / x) / (n * n);
	} else if (x <= 9.6e+89) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (n / x) / (n * n)
    else if (x <= 9.6d+89) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x * x))) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (n / x) / (n * n);
	} else if (x <= 9.6e+89) {
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = (n / x) / (n * n)
	elif x <= 9.6e+89:
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(n / x) / Float64(n * n));
	elseif (x <= 9.6e+89)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = (n / x) / (n * n);
	elseif (x <= 9.6e+89)
		tmp = ((1.0 / x) - (0.5 / (x * x))) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(n / x), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.6e+89], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 42.7%

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

   2. Taylor expanded in n around inf 55.9%

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

      1. log1p-def 55.9%

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

   4. Simplified 55.9%

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

      1. add-log-exp 41.0%

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

      2. div-inv 41.0%

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

      3. exp-prod 41.0%

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

      4. exp-diff 41.0%

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

      5. log1p-udef 41.0%

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

      6. add-exp-log 41.0%

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

      7. add-exp-log 41.0%

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

      8. +-commutative 41.0%

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

   6. Applied egg-rr 41.0%

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

      1. log-pow 55.9%

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

      2. associate-*l/ 55.9%

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

      3. *-un-lft-identity 55.9%

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

      4. log-div 55.9%

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

      5. +-commutative 55.9%

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

      6. log1p-udef 55.9%

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

      7. div-sub 55.9%

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

      8. frac-sub 48.4%

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

   8. Applied egg-rr 48.4%

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

      1. *-commutative 48.4%

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

      2. distribute-lft-out-- 48.4%

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

   10. Simplified 48.4%

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

   11. Taylor expanded in x around inf 28.4%

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

   if 1 < x < 9.60000000000000018e89

   1. Initial program 45.4%

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

   2. Taylor expanded in n around inf 50.9%

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

      1. log1p-def 50.9%

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

   4. Simplified 50.9%

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

   5. Taylor expanded in x around inf 68.4%

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

      1. associate-*r/ 68.4%

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

      2. metadata-eval 68.4%

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

      3. unpow2 68.4%

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

   7. Simplified 68.4%

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

   if 9.60000000000000018e89 < x

   1. Initial program 83.2%

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

   2. Taylor expanded in n around inf 83.2%

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

      1. log1p-def 83.2%

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

   4. Simplified 83.2%

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

      1. log1p-udef 83.2%

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

      2. diff-log 83.2%

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

      3. +-commutative 83.2%

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

   6. Applied egg-rr 83.2%

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

   7. Taylor expanded in x around inf 83.2%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 16: 42.7% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 42.7%

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

   2. Taylor expanded in n around inf 55.9%

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

      1. log1p-def 55.9%

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

   4. Simplified 55.9%

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

      1. add-log-exp 41.0%

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

      2. div-inv 41.0%

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

      3. exp-prod 41.0%

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

      4. exp-diff 41.0%

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

      5. log1p-udef 41.0%

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

      6. add-exp-log 41.0%

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

      7. add-exp-log 41.0%

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

      8. +-commutative 41.0%

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

   6. Applied egg-rr 41.0%

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

      1. log-pow 55.9%

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

      2. associate-*l/ 55.9%

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

      3. *-un-lft-identity 55.9%

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

      4. log-div 55.9%

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

      5. +-commutative 55.9%

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

      6. log1p-udef 55.9%

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

      7. div-sub 55.9%

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

      8. frac-sub 48.4%

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

   8. Applied egg-rr 48.4%

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

      1. *-commutative 48.4%

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

      2. distribute-lft-out-- 48.4%

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

   10. Simplified 48.4%

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

   11. Taylor expanded in x around inf 28.4%

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

   if 1 < x

   1. Initial program 70.8%

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

   2. Taylor expanded in n around inf 72.6%

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

      1. log1p-def 72.6%

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

   4. Simplified 72.6%

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

   5. Taylor expanded in x around inf 63.9%

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

      1. associate-*r/ 63.9%

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

      2. metadata-eval 63.9%

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

      3. unpow2 63.9%

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

   7. Simplified 63.9%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} - \frac{0.5}{x \cdot x}}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\frac{n}{x}}{n \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{x \cdot x}}{n}\\ \end{array} \]

Alternative 17: 39.8% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

2. Taylor expanded in n around inf 62.5%

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

   1. log1p-def 62.5%

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

4. Simplified 62.5%

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

5. Taylor expanded in x around inf 37.1%

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

   1. *-commutative 37.1%

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

7. Simplified 37.1%

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

8. Final simplification 37.1%

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

Alternative 18: 40.3% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

2. Taylor expanded in n around inf 62.5%

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

   1. log1p-def 62.5%

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

4. Simplified 62.5%

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

5. Taylor expanded in x around inf 37.6%

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

6. Final simplification 37.6%

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

Alternative 19: 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. Taylor expanded in x around 0 33.3%

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

3. Taylor expanded in x around inf 4.6%

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

4. Final simplification 4.6%

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

Reproduce

herbie shell --seed 2023227 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))