2nthrt (problem 3.4.6)

Percentage Accurate: 53.8% → 86.2%

Time: 45.4s

Alternatives: 18

Speedup: 10.5×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% 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: 86.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 400000:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\frac{n}{0.5}} - \log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-17)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 400000.0)
     (/
      (-
       (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) (/ n 0.5))
       (log (/ x (+ x 1.0))))
      n)
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-17) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 400000.0) {
		tmp = (((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / (n / 0.5)) - log((x / (x + 1.0)))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-17) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 400000.0) {
		tmp = (((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) / (n / 0.5)) - Math.log((x / (x + 1.0)))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-17:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 400000.0:
		tmp = (((math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)) / (n / 0.5)) - math.log((x / (x + 1.0)))) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-17)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 400000.0)
		tmp = Float64(Float64(Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / Float64(n / 0.5)) - log(Float64(x / Float64(x + 1.0)))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-17], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 400000.0], N[(N[(N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(n / 0.5), $MachinePrecision]), $MachinePrecision] - N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $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 #s(literal 1 binary64) n) < -2.00000000000000014e-17

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 97.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 97.2%

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

      1. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}\right), \mathsf{*.f64}\left(\color{blue}{x}, n\right)\right) \]

      2. pow-to-exp N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log x \cdot \left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}\right), \mathsf{*.f64}\left(\color{blue}{x}, n\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log x \cdot \frac{\mathsf{neg}\left(-1\right)}{n}}\right), \mathsf{*.f64}\left(x, n\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log x \cdot \frac{1}{n}}\right), \mathsf{*.f64}\left(x, n\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\log x \cdot \frac{1}{n}\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, n\right)\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\frac{\log x}{n}\right)\right), \mathsf{*.f64}\left(x, n\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\log x, n\right)\right), \mathsf{*.f64}\left(x, n\right)\right) \]

      8. log-lowering-log.f64 97.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right), \mathsf{*.f64}\left(x, n\right)\right) \]

   7. Applied egg-rr 97.3%

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

   if -2.00000000000000014e-17 < (/.f64 #s(literal 1 binary64) n) < 4e5

   1. Initial program 30.2%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 77.8%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) - \log x\right) + \log \left(1 + x\right)\right), n\right) \]

      2. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) - \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) \cdot \frac{\frac{1}{2}}{n}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) \cdot \frac{1}{\frac{n}{\frac{1}{2}}}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{\frac{n}{\frac{1}{2}}}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left({\log \left(1 + x\right)}^{2}\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\log \left(1 + x\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      10. log1p-define N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      11. log1p-lowering-log1p.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\log x, 2\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      13. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      15. diff-log N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \log \left(\frac{x}{1 + x}\right)\right), n\right) \]

      16. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right)\right), n\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 78.3%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\frac{n}{0.5}} - \log \left(\frac{x}{1 + x}\right)}}{n} \]

   if 4e5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.8%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(e^{\log \left(x + 1\right) \cdot \frac{1}{n}}\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(x + 1\right) \cdot \frac{1}{n}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\left(\frac{\log \left(x + 1\right)}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\log \left(x + 1\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\log \left(1 + x\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{log1p.f64}\left(x\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 400000:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\frac{n}{0.5}} - \log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 400000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-17)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 400000.0)
     (/ (log (/ x (+ x 1.0))) (- 0.0 n))
     (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-17) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 400000.0) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-17) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 400000.0) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-17:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 400000.0:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-17)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 400000.0)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-17], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 400000.0], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $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 #s(literal 1 binary64) n) < -2.00000000000000014e-17

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 97.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 97.2%

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

      1. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}\right), \mathsf{*.f64}\left(\color{blue}{x}, n\right)\right) \]

      2. pow-to-exp N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log x \cdot \left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}\right), \mathsf{*.f64}\left(\color{blue}{x}, n\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log x \cdot \frac{\mathsf{neg}\left(-1\right)}{n}}\right), \mathsf{*.f64}\left(x, n\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\log x \cdot \frac{1}{n}}\right), \mathsf{*.f64}\left(x, n\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\log x \cdot \frac{1}{n}\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, n\right)\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\left(\frac{\log x}{n}\right)\right), \mathsf{*.f64}\left(x, n\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\log x, n\right)\right), \mathsf{*.f64}\left(x, n\right)\right) \]

      8. log-lowering-log.f64 97.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right), \mathsf{*.f64}\left(x, n\right)\right) \]

   7. Applied egg-rr 97.3%

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

   if -2.00000000000000014e-17 < (/.f64 #s(literal 1 binary64) n) < 4e5

   1. Initial program 30.2%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 77.8%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) - \log x\right) + \log \left(1 + x\right)\right), n\right) \]

      2. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) - \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) \cdot \frac{\frac{1}{2}}{n}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) \cdot \frac{1}{\frac{n}{\frac{1}{2}}}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{\frac{n}{\frac{1}{2}}}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left({\log \left(1 + x\right)}^{2}\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\log \left(1 + x\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      10. log1p-define N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      11. log1p-lowering-log1p.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\log x, 2\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      13. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      15. diff-log N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \log \left(\frac{x}{1 + x}\right)\right), n\right) \]

      16. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right)\right), n\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 78.3%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\frac{n}{0.5}} - \log \left(\frac{x}{1 + x}\right)}}{n} \]

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{x}{1 + x}\right), \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right), \left(\color{blue}{-1} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(\mathsf{neg}\left(n\right)\right)\right) \]

      10. neg-lowering-neg.f64 78.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{neg.f64}\left(n\right)\right) \]

   10. Simplified 78.1%

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

   if 4e5 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 52.8%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(e^{\log \left(x + 1\right) \cdot \frac{1}{n}}\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(x + 1\right) \cdot \frac{1}{n}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\left(\frac{\log \left(x + 1\right)}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\log \left(x + 1\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\log \left(1 + x\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{log1p.f64}\left(x\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 400000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 0.3:\\ \;\;\;\;x \cdot \left(\frac{1}{n} - \frac{t\_0}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{t\_0}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= x 0.3) (* x (- (/ 1.0 n) (/ t_0 x))) (/ (/ (exp t_0) n) x))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if (x <= 0.3) {
		tmp = x * ((1.0 / n) - (t_0 / x));
	} else {
		tmp = (exp(t_0) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / n
    if (x <= 0.3d0) then
        tmp = x * ((1.0d0 / n) - (t_0 / x))
    else
        tmp = (exp(t_0) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / n;
	double tmp;
	if (x <= 0.3) {
		tmp = x * ((1.0 / n) - (t_0 / x));
	} else {
		tmp = (Math.exp(t_0) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / n
	tmp = 0
	if x <= 0.3:
		tmp = x * ((1.0 / n) - (t_0 / x))
	else:
		tmp = (math.exp(t_0) / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (x <= 0.3)
		tmp = Float64(x * Float64(Float64(1.0 / n) - Float64(t_0 / x)));
	else
		tmp = Float64(Float64(exp(t_0) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / n;
	tmp = 0.0;
	if (x <= 0.3)
		tmp = x * ((1.0 / n) - (t_0 / x));
	else
		tmp = (exp(t_0) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 0.3], N[(x * N[(N[(1.0 / n), $MachinePrecision] - N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[t$95$0], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.299999999999999989

   1. Initial program 40.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{x \cdot 1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + x \cdot \frac{1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{n}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x \cdot 1}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. /-lowering-/.f64 39.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   5. Simplified 39.6%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - \log x\right), \color{blue}{n}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \log x\right), n\right) \]

      3. log-lowering-log.f64 52.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(x\right)\right), n\right) \]

   8. Simplified 52.5%

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

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n \cdot x}\right)}\right) \]

       2. log-rec N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{n} + \frac{\mathsf{neg}\left(\log x\right)}{\color{blue}{n} \cdot x}\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{n} + \frac{-1 \cdot \log x}{\color{blue}{n} \cdot x}\right)\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{n} + -1 \cdot \color{blue}{\frac{\log x}{n \cdot x}}\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{n} + \left(\mathsf{neg}\left(\frac{\log x}{n \cdot x}\right)\right)\right)\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{n} - \color{blue}{\frac{\log x}{n \cdot x}}\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(\frac{\log x}{n \cdot x}\right)}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\color{blue}{\log x}}{n \cdot x}\right)\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{\log x}{n}}{\color{blue}{x}}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(\frac{\log x}{n}\right), \color{blue}{x}\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\log x, n\right), x\right)\right)\right) \]

       12. log-lowering-log.f64 70.2%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right), x\right)\right)\right) \]

   11. Simplified 70.2%

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

   if 0.299999999999999989 < x

   1. Initial program 65.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 77.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{n \cdot e^{\frac{\log \left(\frac{1}{x}\right)}{n}}}\right)}, x\right) \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{\frac{\log \left(\frac{1}{x}\right)}{n}}}\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{\frac{\mathsf{neg}\left(\log x\right)}{n}}}\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{\frac{-1 \cdot \log x}{n}}}\right), x\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{-1 \cdot \frac{\log x}{n}}}\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{n}\right), \left(e^{-1 \cdot \frac{\log x}{n}}\right)\right), x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(e^{-1 \cdot \frac{\log x}{n}}\right)\right), x\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\left(-1 \cdot \frac{\log x}{n}\right)\right)\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\log x}{n}\right)\right)\right)\right), x\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\log x, n\right)\right)\right)\right), x\right) \]

      10. log-lowering-log.f64 97.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right)\right)\right), x\right) \]

   8. Simplified 97.3%

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

   9. Taylor expanded in n around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{n \cdot e^{-1 \cdot \frac{\log x}{n}}}\right)}, x\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{-1 \cdot \frac{\log x}{n}} \cdot n}\right), x\right) \]

       2. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{e^{-1 \cdot \frac{\log x}{n}}}}{n}\right), x\right) \]

       3. exp-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{\mathsf{neg}\left(-1 \cdot \frac{\log x}{n}\right)}}{n}\right), x\right) \]

       4. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\log x}{n}\right)\right)\right)}}{n}\right), x\right) \]

       5. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{e^{\frac{\log x}{n}}}{n}\right), x\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(e^{\frac{\log x}{n}}\right), n\right), x\right) \]

   11. Simplified 97.3%

       \[\leadsto \frac{\color{blue}{\frac{e^{\frac{\log x}{n}}}{n}}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 400000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+165}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-17)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 400000.0)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (if (<= (/ 1.0 n) 5e+165)
         (- (+ (/ x n) 1.0) t_0)
         (/ (/ (+ (/ (+ (/ 0.3333333333333333 x) -0.5) x) 1.0) x) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-17) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 400000.0) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 5e+165) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 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) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-2d-17)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 400000.0d0) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else if ((1.0d0 / n) <= 5d+165) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = (((((0.3333333333333333d0 / x) + (-0.5d0)) / x) + 1.0d0) / x) / 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) <= -2e-17) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 400000.0) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 5e+165) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-17:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 400000.0:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	elif (1.0 / n) <= 5e+165:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-17)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 400000.0)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 5e+165)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / 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) <= -2e-17)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 400000.0)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	elseif ((1.0 / n) <= 5e+165)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / 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], -2e-17], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 400000.0], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+165], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] + -0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000014e-17

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 97.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 97.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}\right), \color{blue}{\left(x \cdot n\right)}\right) \]

      2. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}\right), \left(\color{blue}{x} \cdot n\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}\right), \left(x \cdot n\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), \left(x \cdot n\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), \left(\color{blue}{x} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), \left(x \cdot n\right)\right) \]

      7. *-lowering-*.f64 97.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   7. Applied egg-rr 97.2%

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

   if -2.00000000000000014e-17 < (/.f64 #s(literal 1 binary64) n) < 4e5

   1. Initial program 30.2%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 77.8%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) - \log x\right) + \log \left(1 + x\right)\right), n\right) \]

      2. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) - \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) \cdot \frac{\frac{1}{2}}{n}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) \cdot \frac{1}{\frac{n}{\frac{1}{2}}}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{\frac{n}{\frac{1}{2}}}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left({\log \left(1 + x\right)}^{2}\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\log \left(1 + x\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      10. log1p-define N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      11. log1p-lowering-log1p.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\log x, 2\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      13. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      15. diff-log N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \log \left(\frac{x}{1 + x}\right)\right), n\right) \]

      16. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right)\right), n\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 78.3%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\frac{n}{0.5}} - \log \left(\frac{x}{1 + x}\right)}}{n} \]

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{x}{1 + x}\right), \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right), \left(\color{blue}{-1} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(\mathsf{neg}\left(n\right)\right)\right) \]

      10. neg-lowering-neg.f64 78.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{neg.f64}\left(n\right)\right) \]

   10. Simplified 78.1%

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

   if 4e5 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e165

   1. Initial program 76.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{x \cdot 1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + x \cdot \frac{1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{n}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x \cdot 1}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. /-lowering-/.f64 69.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   5. Simplified 69.3%

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

   if 4.9999999999999997e165 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 19.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 0.0%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 76.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   8. Simplified 76.1%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(1 + \frac{\frac{1}{3}}{x \cdot x}\right) - \frac{\frac{1}{2}}{x}}{x}\right), \color{blue}{n}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{\frac{1}{3}}{x \cdot x}\right) - \frac{\frac{1}{2}}{x}\right), x\right), n\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(\frac{\frac{1}{3}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\frac{1}{3}}{x}}{x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      7. sub-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\frac{1}{3}}{x} - \frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} - \frac{1}{2}\right), x\right)\right), x\right), n\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), x\right)\right), x\right), n\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} + \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{3}}{x}\right), \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

      12. /-lowering-/.f64 76.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, x\right), \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

   10. Applied egg-rr 76.1%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 400000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+165}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 400000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+144}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-17)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 400000.0)
       (/ (log (/ x (+ x 1.0))) (- 0.0 n))
       (if (<= (/ 1.0 n) 1e+144)
         (- 1.0 t_0)
         (/ (/ (+ (/ (+ (/ 0.3333333333333333 x) -0.5) x) 1.0) x) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-17) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 400000.0) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 1e+144) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 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) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-2d-17)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 400000.0d0) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else if ((1.0d0 / n) <= 1d+144) then
        tmp = 1.0d0 - t_0
    else
        tmp = (((((0.3333333333333333d0 / x) + (-0.5d0)) / x) + 1.0d0) / x) / 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) <= -2e-17) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 400000.0) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 1e+144) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-17:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 400000.0:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	elif (1.0 / n) <= 1e+144:
		tmp = 1.0 - t_0
	else:
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-17)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 400000.0)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 1e+144)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / 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) <= -2e-17)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 400000.0)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	elseif ((1.0 / n) <= 1e+144)
		tmp = 1.0 - t_0;
	else
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / 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], -2e-17], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 400000.0], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+144], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] + -0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000014e-17

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 97.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 97.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}\right), \color{blue}{\left(x \cdot n\right)}\right) \]

      2. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}\right), \left(\color{blue}{x} \cdot n\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}\right), \left(x \cdot n\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), \left(x \cdot n\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), \left(\color{blue}{x} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), \left(x \cdot n\right)\right) \]

      7. *-lowering-*.f64 97.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   7. Applied egg-rr 97.2%

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

   if -2.00000000000000014e-17 < (/.f64 #s(literal 1 binary64) n) < 4e5

   1. Initial program 30.2%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 77.8%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) - \log x\right) + \log \left(1 + x\right)\right), n\right) \]

      2. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) - \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) \cdot \frac{\frac{1}{2}}{n}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) \cdot \frac{1}{\frac{n}{\frac{1}{2}}}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{\frac{n}{\frac{1}{2}}}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left({\log \left(1 + x\right)}^{2}\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\log \left(1 + x\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      10. log1p-define N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      11. log1p-lowering-log1p.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\log x, 2\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      13. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      15. diff-log N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \log \left(\frac{x}{1 + x}\right)\right), n\right) \]

      16. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right)\right), n\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 78.3%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\frac{n}{0.5}} - \log \left(\frac{x}{1 + x}\right)}}{n} \]

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{x}{1 + x}\right), \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right), \left(\color{blue}{-1} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(\mathsf{neg}\left(n\right)\right)\right) \]

      10. neg-lowering-neg.f64 78.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{neg.f64}\left(n\right)\right) \]

   10. Simplified 78.1%

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

   if 4e5 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000002e144

   1. Initial program 79.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 69.5%

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

   if 1.00000000000000002e144 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 23.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 0.0%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 73.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   8. Simplified 73.4%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(1 + \frac{\frac{1}{3}}{x \cdot x}\right) - \frac{\frac{1}{2}}{x}}{x}\right), \color{blue}{n}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{\frac{1}{3}}{x \cdot x}\right) - \frac{\frac{1}{2}}{x}\right), x\right), n\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(\frac{\frac{1}{3}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\frac{1}{3}}{x}}{x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      7. sub-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\frac{1}{3}}{x} - \frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} - \frac{1}{2}\right), x\right)\right), x\right), n\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), x\right)\right), x\right), n\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} + \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{3}}{x}\right), \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

      12. /-lowering-/.f64 73.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, x\right), \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

   10. Applied egg-rr 73.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 400000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+144}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 400000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+144}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-17)
   (/ (pow x (+ (/ 1.0 n) -1.0)) n)
   (if (<= (/ 1.0 n) 400000.0)
     (/ (log (/ x (+ x 1.0))) (- 0.0 n))
     (if (<= (/ 1.0 n) 1e+144)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (/ (+ (/ (+ (/ 0.3333333333333333 x) -0.5) x) 1.0) x) n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-17) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 400000.0) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 1e+144) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 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) <= (-2d-17)) then
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    else if ((1.0d0 / n) <= 400000.0d0) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else if ((1.0d0 / n) <= 1d+144) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (((((0.3333333333333333d0 / x) + (-0.5d0)) / x) + 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) <= -2e-17) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 400000.0) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 1e+144) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-17:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 400000.0:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	elif (1.0 / n) <= 1e+144:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-17)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 400000.0)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 1e+144)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-17)
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	elseif ((1.0 / n) <= 400000.0)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	elseif ((1.0 / n) <= 1e+144)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-17], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 400000.0], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+144], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] + -0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.00000000000000014e-17

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 97.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 97.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x}\right), \color{blue}{n}\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{{x}^{\left(\frac{-1}{n}\right)}} \cdot \frac{1}{x}\right), n\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)} \cdot \frac{1}{x}\right), n\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)} \cdot \frac{1}{x}\right), n\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x}\right), n\right) \]

      7. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)} \cdot {x}^{-1}\right), n\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n} + -1\right)}\right), n\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n} + -1\right)\right), n\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), -1\right)\right), n\right) \]

      11. /-lowering-/.f64 97.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), -1\right)\right), n\right) \]

   7. Applied egg-rr 97.0%

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

   if -2.00000000000000014e-17 < (/.f64 #s(literal 1 binary64) n) < 4e5

   1. Initial program 30.2%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 77.8%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) - \log x\right) + \log \left(1 + x\right)\right), n\right) \]

      2. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) - \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) \cdot \frac{\frac{1}{2}}{n}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) \cdot \frac{1}{\frac{n}{\frac{1}{2}}}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{\frac{n}{\frac{1}{2}}}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left({\log \left(1 + x\right)}^{2}\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\log \left(1 + x\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      10. log1p-define N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      11. log1p-lowering-log1p.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\log x, 2\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      13. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      15. diff-log N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \log \left(\frac{x}{1 + x}\right)\right), n\right) \]

      16. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right)\right), n\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 78.3%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\frac{n}{0.5}} - \log \left(\frac{x}{1 + x}\right)}}{n} \]

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{x}{1 + x}\right), \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right), \left(\color{blue}{-1} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(\mathsf{neg}\left(n\right)\right)\right) \]

      10. neg-lowering-neg.f64 78.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{neg.f64}\left(n\right)\right) \]

   10. Simplified 78.1%

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

   if 4e5 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000002e144

   1. Initial program 79.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 69.5%

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

   if 1.00000000000000002e144 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 23.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 0.0%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 73.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   8. Simplified 73.4%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(1 + \frac{\frac{1}{3}}{x \cdot x}\right) - \frac{\frac{1}{2}}{x}}{x}\right), \color{blue}{n}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{\frac{1}{3}}{x \cdot x}\right) - \frac{\frac{1}{2}}{x}\right), x\right), n\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(\frac{\frac{1}{3}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\frac{1}{3}}{x}}{x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      7. sub-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\frac{1}{3}}{x} - \frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} - \frac{1}{2}\right), x\right)\right), x\right), n\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), x\right)\right), x\right), n\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} + \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{3}}{x}\right), \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

      12. /-lowering-/.f64 73.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, x\right), \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

   10. Applied egg-rr 73.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 400000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+144}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.5:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 400000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+144}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -0.5)
   (/ 0.3333333333333333 (* n (* x (* x x))))
   (if (<= (/ 1.0 n) 400000.0)
     (/ (log (/ x (+ x 1.0))) (- 0.0 n))
     (if (<= (/ 1.0 n) 1e+144)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (/ (+ (/ (+ (/ 0.3333333333333333 x) -0.5) x) 1.0) x) n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.5) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else if ((1.0 / n) <= 400000.0) {
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 1e+144) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 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) <= (-0.5d0)) then
        tmp = 0.3333333333333333d0 / (n * (x * (x * x)))
    else if ((1.0d0 / n) <= 400000.0d0) then
        tmp = log((x / (x + 1.0d0))) / (0.0d0 - n)
    else if ((1.0d0 / n) <= 1d+144) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (((((0.3333333333333333d0 / x) + (-0.5d0)) / x) + 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) <= -0.5) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else if ((1.0 / n) <= 400000.0) {
		tmp = Math.log((x / (x + 1.0))) / (0.0 - n);
	} else if ((1.0 / n) <= 1e+144) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.5:
		tmp = 0.3333333333333333 / (n * (x * (x * x)))
	elif (1.0 / n) <= 400000.0:
		tmp = math.log((x / (x + 1.0))) / (0.0 - n)
	elif (1.0 / n) <= 1e+144:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.5)
		tmp = Float64(0.3333333333333333 / Float64(n * Float64(x * Float64(x * x))));
	elseif (Float64(1.0 / n) <= 400000.0)
		tmp = Float64(log(Float64(x / Float64(x + 1.0))) / Float64(0.0 - n));
	elseif (Float64(1.0 / n) <= 1e+144)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -0.5)
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	elseif ((1.0 / n) <= 400000.0)
		tmp = log((x / (x + 1.0))) / (0.0 - n);
	elseif ((1.0 / n) <= 1e+144)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.5], N[(0.3333333333333333 / N[(n * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 400000.0], N[(N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+144], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] + -0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.5

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 43.8%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 47.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   8. Simplified 47.2%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(n \cdot {x}^{3}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 76.6%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   11. Simplified 76.6%

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

   if -0.5 < (/.f64 #s(literal 1 binary64) n) < 4e5

   1. Initial program 30.5%

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

   3. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 76.6%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) - \log x\right) + \log \left(1 + x\right)\right), n\right) \]

      2. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) - \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\frac{1}{2}}{n} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) \cdot \frac{\frac{1}{2}}{n}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right) \cdot \frac{1}{\frac{n}{\frac{1}{2}}}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{{\log \left(1 + x\right)}^{2} - {\log x}^{2}}{\frac{n}{\frac{1}{2}}}\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left({\log \left(1 + x\right)}^{2}\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\log \left(1 + x\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      10. log1p-define N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      11. log1p-lowering-log1p.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \left({\log x}^{2}\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\log x, 2\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      13. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \left(\frac{n}{\frac{1}{2}}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \left(\log x - \log \left(1 + x\right)\right)\right), n\right) \]

      15. diff-log N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \log \left(\frac{x}{1 + x}\right)\right), n\right) \]

      16. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right)\right), n\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{log1p.f64}\left(x\right), 2\right), \mathsf{pow.f64}\left(\mathsf{log.f64}\left(x\right), 2\right)\right), \mathsf{/.f64}\left(n, \frac{1}{2}\right)\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right)\right), n\right) \]

   7. Applied egg-rr 77.1%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{\frac{n}{0.5}} - \log \left(\frac{x}{1 + x}\right)}}{n} \]

   8. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{x}{1 + x}\right), \color{blue}{\left(-1 \cdot n\right)}\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{1 + x}\right)\right), \left(\color{blue}{-1} \cdot n\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(1 + x\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(x + 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(-1 \cdot n\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \left(\mathsf{neg}\left(n\right)\right)\right) \]

      10. neg-lowering-neg.f64 76.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{neg.f64}\left(n\right)\right) \]

   10. Simplified 76.6%

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

   if 4e5 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000002e144

   1. Initial program 79.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 69.5%

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

   if 1.00000000000000002e144 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 23.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 0.0%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 73.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   8. Simplified 73.4%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(1 + \frac{\frac{1}{3}}{x \cdot x}\right) - \frac{\frac{1}{2}}{x}}{x}\right), \color{blue}{n}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{\frac{1}{3}}{x \cdot x}\right) - \frac{\frac{1}{2}}{x}\right), x\right), n\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(\frac{\frac{1}{3}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\frac{1}{3}}{x}}{x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      7. sub-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\frac{1}{3}}{x} - \frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} - \frac{1}{2}\right), x\right)\right), x\right), n\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), x\right)\right), x\right), n\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} + \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{3}}{x}\right), \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

      12. /-lowering-/.f64 73.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, x\right), \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

   10. Applied egg-rr 73.4%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.5:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 400000:\\ \;\;\;\;\frac{\log \left(\frac{x}{x + 1}\right)}{0 - n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+144}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.35 \cdot 10^{-283}:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-252}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 3.35e-283)
   (- 0.0 (/ (log x) n))
   (if (<= x 2.7e-252)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 0.85)
       (/ (- x (log x)) n)
       (if (<= x 4.4e+101)
         (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) n) x)
         0.0)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 3.35e-283) {
		tmp = 0.0 - (log(x) / n);
	} else if (x <= 2.7e-252) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.85) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4.4e+101) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 3.35d-283) then
        tmp = 0.0d0 - (log(x) / n)
    else if (x <= 2.7d-252) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.85d0) then
        tmp = (x - log(x)) / n
    else if (x <= 4.4d+101) then
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 3.35e-283) {
		tmp = 0.0 - (Math.log(x) / n);
	} else if (x <= 2.7e-252) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.85) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 4.4e+101) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 3.35e-283:
		tmp = 0.0 - (math.log(x) / n)
	elif x <= 2.7e-252:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.85:
		tmp = (x - math.log(x)) / n
	elif x <= 4.4e+101:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 3.35e-283)
		tmp = Float64(0.0 - Float64(log(x) / n));
	elseif (x <= 2.7e-252)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.85)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4.4e+101)
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 3.35e-283)
		tmp = 0.0 - (log(x) / n);
	elseif (x <= 2.7e-252)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.85)
		tmp = (x - log(x)) / n;
	elseif (x <= 4.4e+101)
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 3.35e-283], N[(0.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e-252], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.85], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4.4e+101], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 5 regimes

2. if x < 3.35000000000000023e-283

   1. Initial program 26.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 26.9%

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

   6. Taylor expanded in n around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-1, \color{blue}{\left(\frac{\log x}{n}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\log x, \color{blue}{n}\right)\right) \]

      3. log-lowering-log.f64 78.9%

         \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right) \]

   8. Simplified 78.9%

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

   if 3.35000000000000023e-283 < x < 2.69999999999999981e-252

   1. Initial program 68.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 68.3%

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

   if 2.69999999999999981e-252 < x < 0.849999999999999978

   1. Initial program 36.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{x \cdot 1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + x \cdot \frac{1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{n}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x \cdot 1}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. /-lowering-/.f64 35.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   5. Simplified 35.1%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - \log x\right), \color{blue}{n}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \log x\right), n\right) \]

      3. log-lowering-log.f64 54.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(x\right)\right), n\right) \]

   8. Simplified 54.9%

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

   if 0.849999999999999978 < x < 4.4000000000000001e101

   1. Initial program 43.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 75.9%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n}\right)}, x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), n\right), x\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), n\right), x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), n\right), x\right) \]

      11. /-lowering-/.f64 62.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), n\right), x\right) \]

   8. Simplified 62.2%

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

   if 4.4000000000000001e101 < x

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 50.8%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 79.8%

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

      1. metadata-eval 79.8%

         \[\leadsto 0 \]

   10. Applied egg-rr 79.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.35 \cdot 10^{-283}:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-252}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.3:\\ \;\;\;\;x \cdot \left(\frac{1}{n} - \frac{\frac{\log x}{n}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.3)
   (* x (- (/ 1.0 n) (/ (/ (log x) n) x)))
   (/ (pow x (+ (/ 1.0 n) -1.0)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.3) {
		tmp = x * ((1.0 / n) - ((log(x) / n) / x));
	} else {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.3d0) then
        tmp = x * ((1.0d0 / n) - ((log(x) / n) / x))
    else
        tmp = (x ** ((1.0d0 / n) + (-1.0d0))) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.3) {
		tmp = x * ((1.0 / n) - ((Math.log(x) / n) / x));
	} else {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.3:
		tmp = x * ((1.0 / n) - ((math.log(x) / n) / x))
	else:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.3)
		tmp = Float64(x * Float64(Float64(1.0 / n) - Float64(Float64(log(x) / n) / x)));
	else
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.3)
		tmp = x * ((1.0 / n) - ((log(x) / n) / x));
	else
		tmp = (x ^ ((1.0 / n) + -1.0)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.3], N[(x * N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.299999999999999989

   1. Initial program 40.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{x \cdot 1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + x \cdot \frac{1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{n}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x \cdot 1}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. /-lowering-/.f64 39.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   5. Simplified 39.6%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - \log x\right), \color{blue}{n}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \log x\right), n\right) \]

      3. log-lowering-log.f64 52.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(x\right)\right), n\right) \]

   8. Simplified 52.5%

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

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n \cdot x}\right)}\right) \]

       2. log-rec N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{n} + \frac{\mathsf{neg}\left(\log x\right)}{\color{blue}{n} \cdot x}\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{n} + \frac{-1 \cdot \log x}{\color{blue}{n} \cdot x}\right)\right) \]

       4. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{n} + -1 \cdot \color{blue}{\frac{\log x}{n \cdot x}}\right)\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{n} + \left(\mathsf{neg}\left(\frac{\log x}{n \cdot x}\right)\right)\right)\right) \]

       6. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{n} - \color{blue}{\frac{\log x}{n \cdot x}}\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{1}{n}\right), \color{blue}{\left(\frac{\log x}{n \cdot x}\right)}\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\color{blue}{\log x}}{n \cdot x}\right)\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{\log x}{n}}{\color{blue}{x}}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(\frac{\log x}{n}\right), \color{blue}{x}\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\log x, n\right), x\right)\right)\right) \]

       12. log-lowering-log.f64 70.2%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right), x\right)\right)\right) \]

   11. Simplified 70.2%

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

   if 0.299999999999999989 < x

   1. Initial program 65.3%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 96.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 96.7%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x}\right), \color{blue}{n}\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{{x}^{\left(\frac{-1}{n}\right)}} \cdot \frac{1}{x}\right), n\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)} \cdot \frac{1}{x}\right), n\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)} \cdot \frac{1}{x}\right), n\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x}\right), n\right) \]

      7. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)} \cdot {x}^{-1}\right), n\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n} + -1\right)}\right), n\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n} + -1\right)\right), n\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), -1\right)\right), n\right) \]

      11. /-lowering-/.f64 97.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), -1\right)\right), n\right) \]

   7. Applied egg-rr 97.1%

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

Alternative 10: 60.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.85)
   (/ (- x (log x)) n)
   (if (<= x 4.5e+104)
     (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) n) x)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.85) {
		tmp = (x - log(x)) / n;
	} else if (x <= 4.5e+104) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.85d0) then
        tmp = (x - log(x)) / n
    else if (x <= 4.5d+104) then
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.85) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 4.5e+104) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.85:
		tmp = (x - math.log(x)) / n
	elif x <= 4.5e+104:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.85)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 4.5e+104)
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.85)
		tmp = (x - log(x)) / n;
	elseif (x <= 4.5e+104)
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.85], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4.5e+104], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.849999999999999978

   1. Initial program 40.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{x \cdot 1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + x \cdot \frac{1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{n}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x \cdot 1}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. /-lowering-/.f64 39.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   5. Simplified 39.6%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - \log x\right), \color{blue}{n}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \log x\right), n\right) \]

      3. log-lowering-log.f64 52.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(x\right)\right), n\right) \]

   8. Simplified 52.5%

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

   if 0.849999999999999978 < x < 4.4999999999999998e104

   1. Initial program 43.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 75.9%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n}\right)}, x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), n\right), x\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), n\right), x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), n\right), x\right) \]

      11. /-lowering-/.f64 62.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), n\right), x\right) \]

   8. Simplified 62.2%

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

   if 4.4999999999999998e104 < x

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 50.8%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 79.8%

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

      1. metadata-eval 79.8%

         \[\leadsto 0 \]

   10. Applied egg-rr 79.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.85:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.58)
   (- 0.0 (/ (log x) n))
   (if (<= x 2.6e+114)
     (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) n) x)
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.58) {
		tmp = 0.0 - (log(x) / n);
	} else if (x <= 2.6e+114) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.58d0) then
        tmp = 0.0d0 - (log(x) / n)
    else if (x <= 2.6d+114) then
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / n) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.58) {
		tmp = 0.0 - (Math.log(x) / n);
	} else if (x <= 2.6e+114) {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.58:
		tmp = 0.0 - (math.log(x) / n)
	elif x <= 2.6e+114:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.58)
		tmp = Float64(0.0 - Float64(log(x) / n));
	elseif (x <= 2.6e+114)
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / n) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.58)
		tmp = 0.0 - (log(x) / n);
	elseif (x <= 2.6e+114)
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.58], N[(0.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e+114], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.57999999999999996

   1. Initial program 40.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 39.1%

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

   6. Taylor expanded in n around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-1, \color{blue}{\left(\frac{\log x}{n}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\log x, \color{blue}{n}\right)\right) \]

      3. log-lowering-log.f64 51.6%

         \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right) \]

   8. Simplified 51.6%

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

   if 0.57999999999999996 < x < 2.6e114

   1. Initial program 43.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 75.9%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n}\right)}, x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), n\right), x\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), n\right), x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), n\right), x\right) \]

      11. /-lowering-/.f64 62.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), n\right), x\right) \]

   8. Simplified 62.2%

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

   if 2.6e114 < x

   1. Initial program 79.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 50.8%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 79.8%

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

      1. metadata-eval 79.8%

         \[\leadsto 0 \]

   10. Applied egg-rr 79.8%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.0% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{0.3333333333333333}{x} - 0.5\right)}{n \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -20000000.0)
   (/ 0.3333333333333333 (* n (* x (* x x))))
   (if (<= (/ 1.0 n) 2e-19)
     (/ (/ 1.0 n) x)
     (/ (+ x (- (/ 0.3333333333333333 x) 0.5)) (* n (* x x))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else if ((1.0 / n) <= 2e-19) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = (x + ((0.3333333333333333 / x) - 0.5)) / (n * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-20000000.0d0)) then
        tmp = 0.3333333333333333d0 / (n * (x * (x * x)))
    else if ((1.0d0 / n) <= 2d-19) then
        tmp = (1.0d0 / n) / x
    else
        tmp = (x + ((0.3333333333333333d0 / x) - 0.5d0)) / (n * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else if ((1.0 / n) <= 2e-19) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = (x + ((0.3333333333333333 / x) - 0.5)) / (n * (x * x));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -20000000.0:
		tmp = 0.3333333333333333 / (n * (x * (x * x)))
	elif (1.0 / n) <= 2e-19:
		tmp = (1.0 / n) / x
	else:
		tmp = (x + ((0.3333333333333333 / x) - 0.5)) / (n * (x * x))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -20000000.0)
		tmp = Float64(0.3333333333333333 / Float64(n * Float64(x * Float64(x * x))));
	elseif (Float64(1.0 / n) <= 2e-19)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = Float64(Float64(x + Float64(Float64(0.3333333333333333 / x) - 0.5)) / Float64(n * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -20000000.0)
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	elseif ((1.0 / n) <= 2e-19)
		tmp = (1.0 / n) / x;
	else
		tmp = (x + ((0.3333333333333333 / x) - 0.5)) / (n * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000000.0], N[(0.3333333333333333 / N[(n * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-19], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[(x + N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / N[(n * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 43.1%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 47.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   8. Simplified 47.9%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(n \cdot {x}^{3}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 77.7%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   11. Simplified 77.7%

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

   if -2e7 < (/.f64 #s(literal 1 binary64) n) < 2e-19

   1. Initial program 30.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 51.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{n \cdot e^{\frac{\log \left(\frac{1}{x}\right)}{n}}}\right)}, x\right) \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{\frac{\log \left(\frac{1}{x}\right)}{n}}}\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{\frac{\mathsf{neg}\left(\log x\right)}{n}}}\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{\frac{-1 \cdot \log x}{n}}}\right), x\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{-1 \cdot \frac{\log x}{n}}}\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{n}\right), \left(e^{-1 \cdot \frac{\log x}{n}}\right)\right), x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(e^{-1 \cdot \frac{\log x}{n}}\right)\right), x\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\left(-1 \cdot \frac{\log x}{n}\right)\right)\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\log x}{n}\right)\right)\right)\right), x\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\log x, n\right)\right)\right)\right), x\right) \]

      10. log-lowering-log.f64 51.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right)\right)\right), x\right) \]

   8. Simplified 51.6%

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

   9. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{n}\right)}, x\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 50.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), x\right) \]

   11. Simplified 50.6%

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

   if 2e-19 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 48.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 9.6%

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

   7. Applied egg-rr 9.6%

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

   8. Taylor expanded in n around -inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + -1 \cdot \left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)\right), \color{blue}{\left(n \cdot {x}^{2}\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)\right)\right)\right), \left(n \cdot {x}^{2}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - \left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)\right), \left(\color{blue}{n} \cdot {x}^{2}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{x}\right)\right), \left(\color{blue}{n} \cdot {x}^{2}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{3} \cdot \frac{1}{x}\right)\right)\right), \left(n \cdot {x}^{2}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{3} \cdot 1}{x}\right)\right)\right), \left(n \cdot {x}^{2}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{3}}{x}\right)\right)\right), \left(n \cdot {x}^{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{3}, x\right)\right)\right), \left(n \cdot {x}^{2}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{3}, x\right)\right)\right), \mathsf{*.f64}\left(n, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{3}, x\right)\right)\right), \mathsf{*.f64}\left(n, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      11. *-lowering-*.f64 37.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{3}, x\right)\right)\right), \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   10. Simplified 37.3%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{0.3333333333333333}{x} - 0.5\right)}{n \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.2% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -20000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 0.3333333333333333 (* n (* x (* x x))))))
   (if (<= (/ 1.0 n) -20000000.0)
     t_0
     (if (<= (/ 1.0 n) 4e+78) (/ (/ 1.0 n) x) t_0))))

C

double code(double x, double n) {
	double t_0 = 0.3333333333333333 / (n * (x * (x * x)));
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e+78) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 / (n * (x * (x * x)))
    if ((1.0d0 / n) <= (-20000000.0d0)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 4d+78) then
        tmp = (1.0d0 / n) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 0.3333333333333333 / (n * (x * (x * x)));
	double tmp;
	if ((1.0 / n) <= -20000000.0) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e+78) {
		tmp = (1.0 / n) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 0.3333333333333333 / (n * (x * (x * x)))
	tmp = 0
	if (1.0 / n) <= -20000000.0:
		tmp = t_0
	elif (1.0 / n) <= 4e+78:
		tmp = (1.0 / n) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(0.3333333333333333 / Float64(n * Float64(x * Float64(x * x))))
	tmp = 0.0
	if (Float64(1.0 / n) <= -20000000.0)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e+78)
		tmp = Float64(Float64(1.0 / n) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 0.3333333333333333 / (n * (x * (x * x)));
	tmp = 0.0;
	if ((1.0 / n) <= -20000000.0)
		tmp = t_0;
	elseif ((1.0 / n) <= 4e+78)
		tmp = (1.0 / n) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(0.3333333333333333 / N[(n * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -20000000.0], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e+78], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e7 or 4.00000000000000003e78 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 82.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 30.1%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 49.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   8. Simplified 49.1%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(n \cdot {x}^{3}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 70.0%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   11. Simplified 70.0%

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

   if -2e7 < (/.f64 #s(literal 1 binary64) n) < 4.00000000000000003e78

   1. Initial program 33.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 49.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{n \cdot e^{\frac{\log \left(\frac{1}{x}\right)}{n}}}\right)}, x\right) \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{\frac{\log \left(\frac{1}{x}\right)}{n}}}\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{\frac{\mathsf{neg}\left(\log x\right)}{n}}}\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{\frac{-1 \cdot \log x}{n}}}\right), x\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{-1 \cdot \frac{\log x}{n}}}\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{n}\right), \left(e^{-1 \cdot \frac{\log x}{n}}\right)\right), x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(e^{-1 \cdot \frac{\log x}{n}}\right)\right), x\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\left(-1 \cdot \frac{\log x}{n}\right)\right)\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\log x}{n}\right)\right)\right)\right), x\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\log x, n\right)\right)\right)\right), x\right) \]

      10. log-lowering-log.f64 49.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right)\right)\right), x\right) \]

   8. Simplified 49.4%

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

   9. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{n}\right)}, x\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 46.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), x\right) \]

   11. Simplified 46.7%

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

Alternative 14: 55.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+16)
   (/ 0.3333333333333333 (* n (* x (* x x))))
   (/ (/ (- (+ (/ 0.3333333333333333 (* x x)) 1.0) (/ 0.5 x)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+16) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d+16)) then
        tmp = 0.3333333333333333d0 / (n * (x * (x * x)))
    else
        tmp = ((((0.3333333333333333d0 / (x * x)) + 1.0d0) - (0.5d0 / x)) / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+16) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+16:
		tmp = 0.3333333333333333 / (n * (x * (x * x)))
	else:
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+16)
		tmp = Float64(0.3333333333333333 / Float64(n * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) + 1.0) - Float64(0.5 / x)) / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e+16)
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	else
		tmp = ((((0.3333333333333333 / (x * x)) + 1.0) - (0.5 / x)) / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+16], N[(0.3333333333333333 / N[(n * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e16

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 42.2%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 47.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   8. Simplified 47.0%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(n \cdot {x}^{3}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 77.4%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   11. Simplified 77.4%

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

   if -1e16 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 42.2%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}}{n}\right)}, x\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), n\right), x\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), n\right), x\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), n\right), x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), n\right), x\right) \]

      11. /-lowering-/.f64 47.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), n\right), x\right) \]

   8. Simplified 47.6%

      \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{0.3333333333333333}{x \cdot x}\right) - \frac{0.5}{x}}{n}}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} + 1\right) - \frac{0.5}{x}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.9% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+16)
   (/ 0.3333333333333333 (* n (* x (* x x))))
   (/ (/ (+ (/ (+ (/ 0.3333333333333333 x) -0.5) x) 1.0) x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+16) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else {
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 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+16)) then
        tmp = 0.3333333333333333d0 / (n * (x * (x * x)))
    else
        tmp = (((((0.3333333333333333d0 / x) + (-0.5d0)) / x) + 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+16) {
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	} else {
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+16:
		tmp = 0.3333333333333333 / (n * (x * (x * x)))
	else:
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+16)
		tmp = Float64(0.3333333333333333 / Float64(n * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e+16)
		tmp = 0.3333333333333333 / (n * (x * (x * x)));
	else
		tmp = (((((0.3333333333333333 / x) + -0.5) / x) + 1.0) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+16], N[(0.3333333333333333 / N[(n * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] + -0.5), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e16

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 42.2%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 47.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   8. Simplified 47.0%

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

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(n \cdot {x}^{3}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \color{blue}{\left({x}^{3}\right)}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \left(x \cdot {x}^{\color{blue}{2}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 77.4%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(n, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   11. Simplified 77.4%

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

   if -1e16 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 42.2%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{x}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{1}{3} \cdot \frac{1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3} \cdot 1}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(1 + \frac{\frac{1}{3}}{{x}^{2}}\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{{x}^{2}}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right), \left(n \cdot x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 47.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   8. Simplified 47.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(1 + \frac{\frac{1}{3}}{x \cdot x}\right) - \frac{\frac{1}{2}}{x}}{x}\right), \color{blue}{n}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \frac{\frac{1}{3}}{x \cdot x}\right) - \frac{\frac{1}{2}}{x}\right), x\right), n\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(\frac{\frac{1}{3}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\frac{1}{3}}{x}}{x} - \frac{\frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      7. sub-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\frac{1}{3}}{x} - \frac{1}{2}}{x}\right)\right), x\right), n\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} - \frac{1}{2}\right), x\right)\right), x\right), n\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), x\right)\right), x\right), n\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\frac{1}{3}}{x} + \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{3}}{x}\right), \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

      12. /-lowering-/.f64 47.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, x\right), \frac{-1}{2}\right), x\right)\right), x\right), n\right) \]

   10. Applied egg-rr 47.5%

       \[\leadsto \color{blue}{\frac{\frac{1 + \frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{x}}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.3333333333333333}{x} + -0.5}{x} + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.5% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ \mathbf{if}\;n \leq -2.6 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -2.4 \cdot 10^{-241}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x))))
   (if (<= n -2.6e-8) t_0 (if (<= n -2.4e-241) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (n <= -2.6e-8) {
		tmp = t_0;
	} else if (n <= -2.4e-241) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (n * x)
    if (n <= (-2.6d-8)) then
        tmp = t_0
    else if (n <= (-2.4d-241)) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (n <= -2.6e-8) {
		tmp = t_0;
	} else if (n <= -2.4e-241) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	tmp = 0
	if n <= -2.6e-8:
		tmp = t_0
	elif n <= -2.4e-241:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (n <= -2.6e-8)
		tmp = t_0;
	elseif (n <= -2.4e-241)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (n * x);
	tmp = 0.0;
	if (n <= -2.6e-8)
		tmp = t_0;
	elseif (n <= -2.4e-241)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.6e-8], t$95$0, If[LessEqual[n, -2.4e-241], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -2.6000000000000001e-8 or -2.4e-241 < n

   1. Initial program 40.1%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 46.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 46.6%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(n \cdot x\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{n}\right)\right) \]

      3. *-lowering-*.f64 45.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   8. Simplified 45.5%

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

   if -2.6000000000000001e-8 < n < -2.4e-241

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 45.5%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 56.9%

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

      1. metadata-eval 56.9%

         \[\leadsto 0 \]

   10. Applied egg-rr 56.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq -2.4 \cdot 10^{-241}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.2% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+16) 0.0 (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+16) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d+16)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+16) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+16:
		tmp = 0.0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+16)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e+16)
		tmp = 0.0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+16], 0.0, N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e16

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 50.0%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 52.3%

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

      1. metadata-eval 52.3%

         \[\leadsto 0 \]

   10. Applied egg-rr 52.3%

       \[\leadsto \color{blue}{0} \]

   if -1e16 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 35.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n} + \left(\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)}{x} + \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}} \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)}{{x}^{2}}\right)}{x}} \]

   5. Simplified 42.2%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{n \cdot e^{\frac{\log \left(\frac{1}{x}\right)}{n}}}\right)}, x\right) \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{\frac{\log \left(\frac{1}{x}\right)}{n}}}\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{\frac{\mathsf{neg}\left(\log x\right)}{n}}}\right), x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{\frac{-1 \cdot \log x}{n}}}\right), x\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{n}}{e^{-1 \cdot \frac{\log x}{n}}}\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{n}\right), \left(e^{-1 \cdot \frac{\log x}{n}}\right)\right), x\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(e^{-1 \cdot \frac{\log x}{n}}\right)\right), x\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\left(-1 \cdot \frac{\log x}{n}\right)\right)\right), x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\log x}{n}\right)\right)\right)\right), x\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\log x, n\right)\right)\right)\right), x\right) \]

      10. log-lowering-log.f64 42.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right)\right)\right), x\right) \]

   8. Simplified 42.8%

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

   9. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{n}\right)}, x\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 44.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, n\right), x\right) \]

   11. Simplified 44.8%

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

Alternative 18: 30.8% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 51.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
4. Step-by-step derivation

5. Simplified 38.1%

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

6. Taylor expanded in n around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
7. Step-by-step derivation

8. Simplified 30.7%

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

   1. metadata-eval 30.7%

      \[\leadsto 0 \]

10. Applied egg-rr 30.7%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))