2nthrt (problem 3.4.6)

Percentage Accurate: 53.9% → 85.7%

Time: 42.3s

Alternatives: 17

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;\frac{\frac{{\left({\left(e^{\log x}\right)}^{\left({n}^{-0.5}\right)}\right)}^{\left({n}^{-0.5}\right)}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2.5e-20)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 2e-51)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5.0)
         (/ (/ (pow (pow (exp (log x)) (pow n -0.5)) (pow n -0.5)) x) n)
         (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2.5e-20) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = (pow(pow(exp(log(x)), pow(n, -0.5)), pow(n, -0.5)) / x) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2.5e-20) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = (Math.pow(Math.pow(Math.exp(Math.log(x)), Math.pow(n, -0.5)), Math.pow(n, -0.5)) / x) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2.5e-20:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 2e-51:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5.0:
		tmp = (math.pow(math.pow(math.exp(math.log(x)), math.pow(n, -0.5)), math.pow(n, -0.5)) / x) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.5e-20)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-51)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5.0)
		tmp = Float64(Float64(((exp(log(x)) ^ (n ^ -0.5)) ^ (n ^ -0.5)) / x) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.5e-20], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-51], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5.0], N[(N[(N[Power[N[Power[N[Exp[N[Log[x], $MachinePrecision]], $MachinePrecision], N[Power[n, -0.5], $MachinePrecision]], $MachinePrecision], N[Power[n, -0.5], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.4999999999999999e-20

   1. Initial program 92.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 96.4%

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

   5. Simplified 96.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]
   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. pow-flip N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. /-lowering-/.f64 96.5%

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

   7. Applied egg-rr 96.5%

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

   if -2.4999999999999999e-20 < (/.f64 #s(literal 1 binary64) n) < 2e-51

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 82.4%

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

   5. Simplified 82.4%

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

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

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

      2. diff-log N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. +-lowering-+.f64 82.6%

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

   7. Applied egg-rr 82.6%

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

   if 2e-51 < (/.f64 #s(literal 1 binary64) n) < 5

   1. Initial program 4.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 77.1%

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

   5. Simplified 77.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]
   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. pow-flip N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. /-lowering-/.f64 77.3%

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

   7. Applied egg-rr 77.3%

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

      1. pow-to-exp N/A

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

      2. exp-prod N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow-unpow N/A

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

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

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

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

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

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

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

      13. metadata-eval 77.5%

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

   9. Applied egg-rr 77.5%

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

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

   1. Initial program 62.0%

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

Alternative 2: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-269}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 18:\\ \;\;\;\;\frac{\frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n} - \log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{1}{x}\right)}^{\left(\frac{-1}{n}\right)}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 8e-269)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 18.0)
     (/
      (-
       (/
        (+
         (* 0.5 (* (log (* x (+ x 1.0))) (log (/ (+ x 1.0) x))))
         (*
          (- (pow (log1p x) 3.0) (pow (log x) 3.0))
          (/ 0.16666666666666666 n)))
        n)
       (log (/ x (+ x 1.0))))
      n)
     (/ (/ (pow (/ 1.0 x) (/ -1.0 n)) x) n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 8e-269) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 18.0) {
		tmp = ((((0.5 * (log((x * (x + 1.0))) * log(((x + 1.0) / x)))) + ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) * (0.16666666666666666 / n))) / n) - log((x / (x + 1.0)))) / n;
	} else {
		tmp = (pow((1.0 / x), (-1.0 / n)) / x) / n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 8e-269) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 18.0) {
		tmp = ((((0.5 * (Math.log((x * (x + 1.0))) * Math.log(((x + 1.0) / x)))) + ((Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0)) * (0.16666666666666666 / n))) / n) - Math.log((x / (x + 1.0)))) / n;
	} else {
		tmp = (Math.pow((1.0 / x), (-1.0 / n)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 8e-269:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 18.0:
		tmp = ((((0.5 * (math.log((x * (x + 1.0))) * math.log(((x + 1.0) / x)))) + ((math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0)) * (0.16666666666666666 / n))) / n) - math.log((x / (x + 1.0)))) / n
	else:
		tmp = (math.pow((1.0 / x), (-1.0 / n)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 8e-269)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 18.0)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 * Float64(log(Float64(x * Float64(x + 1.0))) * log(Float64(Float64(x + 1.0) / x)))) + Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) * Float64(0.16666666666666666 / n))) / n) - log(Float64(x / Float64(x + 1.0)))) / n);
	else
		tmp = Float64(Float64((Float64(1.0 / x) ^ Float64(-1.0 / n)) / x) / n);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 8e-269], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 18.0], N[(N[(N[(N[(N[(0.5 * N[(N[Log[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Power[N[(1.0 / x), $MachinePrecision], N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.9999999999999997e-269

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.1%

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

   if 7.9999999999999997e-269 < x < 18

   1. Initial program 35.2%

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

   3. Taylor expanded in n around -inf

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

   5. Simplified 84.4%

      \[\leadsto \color{blue}{\frac{\left(\log x - \mathsf{log1p}\left(x\right)\right) - \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) - \frac{-0.16666666666666666}{n} \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{0 - n}} \]

   7. Applied egg-rr 84.4%

      \[\leadsto \color{blue}{\frac{\left(-\log \left(\frac{x}{x + 1}\right)\right) + \frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n}}{n}} \]

   if 18 < x

   1. Initial program 66.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 97.9%

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

   5. Simplified 97.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]
   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. pow-flip N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. /-lowering-/.f64 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. frac-2neg N/A

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

      2. metadata-eval N/A

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

      3. div-inv N/A

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

      4. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(-1 \cdot \frac{1}{0 - n}\right)}\right), x\right), n\right) \]

      5. pow-unpow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\left({x}^{-1}\right)}^{\left(\frac{1}{0 - n}\right)}\right), x\right), n\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\left(\frac{1}{x}\right)}^{\left(\frac{1}{0 - n}\right)}\right), x\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{1}{x}\right), \left(\frac{1}{0 - n}\right)\right), x\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{1}{0 - n}\right)\right), x\right), n\right) \]

      9. sub0-neg N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. remove-double-neg N/A

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

      13. /-lowering-/.f64 99.1%

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

   9. Applied egg-rr 99.1%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-269}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 18:\\ \;\;\;\;\frac{\frac{0.5 \cdot \left(\log \left(x \cdot \left(x + 1\right)\right) \cdot \log \left(\frac{x + 1}{x}\right)\right) + \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right) \cdot \frac{0.16666666666666666}{n}}{n} - \log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{1}{x}\right)}^{\left(\frac{-1}{n}\right)}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;\frac{\frac{{\left({x}^{\left({n}^{-0.5}\right)}\right)}^{\left({n}^{-0.5}\right)}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2.5e-20)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 2e-51)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5.0)
         (/ (/ (pow (pow x (pow n -0.5)) (pow n -0.5)) x) n)
         (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2.5e-20) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = (pow(pow(x, pow(n, -0.5)), pow(n, -0.5)) / x) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2.5e-20) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = (Math.pow(Math.pow(x, Math.pow(n, -0.5)), Math.pow(n, -0.5)) / x) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2.5e-20:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 2e-51:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5.0:
		tmp = (math.pow(math.pow(x, math.pow(n, -0.5)), math.pow(n, -0.5)) / x) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.5e-20)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-51)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5.0)
		tmp = Float64(Float64(((x ^ (n ^ -0.5)) ^ (n ^ -0.5)) / x) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.5e-20], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-51], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5.0], N[(N[(N[Power[N[Power[x, N[Power[n, -0.5], $MachinePrecision]], $MachinePrecision], N[Power[n, -0.5], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.4999999999999999e-20

   1. Initial program 92.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 96.4%

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

   5. Simplified 96.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]
   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. pow-flip N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. /-lowering-/.f64 96.5%

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

   7. Applied egg-rr 96.5%

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

   if -2.4999999999999999e-20 < (/.f64 #s(literal 1 binary64) n) < 2e-51

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 82.4%

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

   5. Simplified 82.4%

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

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

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

      2. diff-log N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. +-lowering-+.f64 82.6%

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

   7. Applied egg-rr 82.6%

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

   if 2e-51 < (/.f64 #s(literal 1 binary64) n) < 5

   1. Initial program 4.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 77.1%

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

   5. Simplified 77.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]
   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. pow-flip N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. /-lowering-/.f64 77.3%

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

   7. Applied egg-rr 77.3%

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

      1. inv-pow N/A

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

      2. sqr-pow N/A

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

      3. pow-unpow N/A

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

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

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

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

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

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

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval 77.5%

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

   9. Applied egg-rr 77.5%

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

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

   1. Initial program 62.0%

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

Alternative 4: 85.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;\frac{\frac{1}{x \cdot n}}{{x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2.5e-20)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 2e-51)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5.0)
         (/ (/ 1.0 (* x n)) (pow x (/ -1.0 n)))
         (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2.5e-20) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = (1.0 / (x * n)) / pow(x, (-1.0 / n));
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2.5e-20) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = (1.0 / (x * n)) / Math.pow(x, (-1.0 / n));
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2.5e-20:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 2e-51:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5.0:
		tmp = (1.0 / (x * n)) / math.pow(x, (-1.0 / n))
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.5e-20)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-51)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5.0)
		tmp = Float64(Float64(1.0 / Float64(x * n)) / (x ^ Float64(-1.0 / n)));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.5e-20], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-51], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5.0], N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] / N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.4999999999999999e-20

   1. Initial program 92.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 96.4%

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

   5. Simplified 96.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]
   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. pow-flip N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. /-lowering-/.f64 96.5%

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

   7. Applied egg-rr 96.5%

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

   if -2.4999999999999999e-20 < (/.f64 #s(literal 1 binary64) n) < 2e-51

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 82.4%

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

   5. Simplified 82.4%

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

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

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

      2. diff-log N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. +-lowering-+.f64 82.6%

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

   7. Applied egg-rr 82.6%

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

   if 2e-51 < (/.f64 #s(literal 1 binary64) n) < 5

   1. Initial program 4.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 77.1%

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

   5. Simplified 77.1%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      7. /-lowering-/.f64 77.3%

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

   7. Applied egg-rr 77.3%

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

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

   1. Initial program 62.0%

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

Alternative 5: 81.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;\frac{\frac{1}{x \cdot n}}{{x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right) + 1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2.5e-20)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 2e-51)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5.0)
         (/ (/ 1.0 (* x n)) (pow x (/ -1.0 n)))
         (if (<= (/ 1.0 n) 2e+187)
           (- (+ (/ x n) 1.0) t_0)
           (-
            (+ (* x (+ (/ 1.0 n) (* x (+ (/ 0.5 (* n n)) (/ -0.5 n))))) 1.0)
            t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2.5e-20) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = (1.0 / (x * n)) / pow(x, (-1.0 / n));
	} else if ((1.0 / n) <= 2e+187) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))) + 1.0) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-2.5d-20)) then
        tmp = (t_0 / x) / n
    else if ((1.0d0 / n) <= 2d-51) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5.0d0) then
        tmp = (1.0d0 / (x * n)) / (x ** ((-1.0d0) / n))
    else if ((1.0d0 / n) <= 2d+187) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = ((x * ((1.0d0 / n) + (x * ((0.5d0 / (n * n)) + ((-0.5d0) / n))))) + 1.0d0) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2.5e-20) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = (1.0 / (x * n)) / Math.pow(x, (-1.0 / n));
	} else if ((1.0 / n) <= 2e+187) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))) + 1.0) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2.5e-20:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 2e-51:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5.0:
		tmp = (1.0 / (x * n)) / math.pow(x, (-1.0 / n))
	elif (1.0 / n) <= 2e+187:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))) + 1.0) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.5e-20)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-51)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5.0)
		tmp = Float64(Float64(1.0 / Float64(x * n)) / (x ^ Float64(-1.0 / n)));
	elseif (Float64(1.0 / n) <= 2e+187)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(0.5 / Float64(n * n)) + Float64(-0.5 / n))))) + 1.0) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -2.5e-20)
		tmp = (t_0 / x) / n;
	elseif ((1.0 / n) <= 2e-51)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5.0)
		tmp = (1.0 / (x * n)) / (x ^ (-1.0 / n));
	elseif ((1.0 / n) <= 2e+187)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = ((x * ((1.0 / n) + (x * ((0.5 / (n * n)) + (-0.5 / n))))) + 1.0) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.5e-20], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-51], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5.0], N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] / N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+187], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(x * N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.4999999999999999e-20

   1. Initial program 92.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 96.4%

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

   5. Simplified 96.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]
   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. pow-flip N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. /-lowering-/.f64 96.5%

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

   7. Applied egg-rr 96.5%

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

   if -2.4999999999999999e-20 < (/.f64 #s(literal 1 binary64) n) < 2e-51

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 82.4%

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

   5. Simplified 82.4%

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

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

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

      2. diff-log N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. +-lowering-+.f64 82.6%

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

   7. Applied egg-rr 82.6%

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

   if 2e-51 < (/.f64 #s(literal 1 binary64) n) < 5

   1. Initial program 4.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 77.1%

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

   5. Simplified 77.1%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      7. /-lowering-/.f64 77.3%

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

   7. Applied egg-rr 77.3%

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

   if 5 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999981e187

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   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 84.9%

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

   5. Simplified 84.9%

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

   if 1.99999999999999981e187 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 41.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 74.7%

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

   5. Simplified 74.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;\frac{\frac{1}{x \cdot n}}{{x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{0.5}{n \cdot n} + \frac{-0.5}{n}\right)\right) + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;\frac{\frac{1}{x \cdot n}}{{x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - \frac{0.5}{x}\right) + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2.5e-20)
     (/ (/ t_0 x) n)
     (if (<= (/ 1.0 n) 2e-51)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5.0)
         (/ (/ 1.0 (* x n)) (pow x (/ -1.0 n)))
         (if (<= (/ 1.0 n) 2e+187)
           (- (+ (/ x n) 1.0) t_0)
           (/
            (/ (+ (- (/ 0.3333333333333333 (* x 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) <= -2.5e-20) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = (1.0 / (x * n)) / pow(x, (-1.0 / n));
	} else if ((1.0 / n) <= 2e+187) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * 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) <= (-2.5d-20)) then
        tmp = (t_0 / x) / n
    else if ((1.0d0 / n) <= 2d-51) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5.0d0) then
        tmp = (1.0d0 / (x * n)) / (x ** ((-1.0d0) / n))
    else if ((1.0d0 / n) <= 2d+187) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = ((((0.3333333333333333d0 / (x * 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) <= -2.5e-20) {
		tmp = (t_0 / x) / n;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = (1.0 / (x * n)) / Math.pow(x, (-1.0 / n));
	} else if ((1.0 / n) <= 2e+187) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * 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) <= -2.5e-20:
		tmp = (t_0 / x) / n
	elif (1.0 / n) <= 2e-51:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5.0:
		tmp = (1.0 / (x * n)) / math.pow(x, (-1.0 / n))
	elif (1.0 / n) <= 2e+187:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = ((((0.3333333333333333 / (x * 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) <= -2.5e-20)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-51)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5.0)
		tmp = Float64(Float64(1.0 / Float64(x * n)) / (x ^ Float64(-1.0 / n)));
	elseif (Float64(1.0 / n) <= 2e+187)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) - Float64(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) <= -2.5e-20)
		tmp = (t_0 / x) / n;
	elseif ((1.0 / n) <= 2e-51)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5.0)
		tmp = (1.0 / (x * n)) / (x ^ (-1.0 / n));
	elseif ((1.0 / n) <= 2e+187)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = ((((0.3333333333333333 / (x * 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], -2.5e-20], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-51], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5.0], N[(N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision] / N[Power[x, N[(-1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+187], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.4999999999999999e-20

   1. Initial program 92.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 96.4%

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

   5. Simplified 96.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]
   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. pow-flip N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. /-lowering-/.f64 96.5%

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

   7. Applied egg-rr 96.5%

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

   if -2.4999999999999999e-20 < (/.f64 #s(literal 1 binary64) n) < 2e-51

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 82.4%

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

   5. Simplified 82.4%

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

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

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

      2. diff-log N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. +-lowering-+.f64 82.6%

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

   7. Applied egg-rr 82.6%

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

   if 2e-51 < (/.f64 #s(literal 1 binary64) n) < 5

   1. Initial program 4.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 77.1%

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

   5. Simplified 77.1%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      7. /-lowering-/.f64 77.3%

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

   7. Applied egg-rr 77.3%

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

   if 5 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999981e187

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   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 84.9%

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

   5. Simplified 84.9%

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

   if 1.99999999999999981e187 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 41.0%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 6.4%

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

   5. Simplified 6.4%

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

   6. Taylor expanded in x around inf

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

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

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 58.8%

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

   8. Simplified 58.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;\frac{\frac{1}{x \cdot n}}{{x}^{\left(\frac{-1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - \frac{0.5}{x}\right) + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{x}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - \frac{0.5}{x}\right) + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 x) n)))
   (if (<= (/ 1.0 n) -2.5e-20)
     t_1
     (if (<= (/ 1.0 n) 2e-51)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5.0)
         t_1
         (if (<= (/ 1.0 n) 2e+187)
           (- (+ (/ x n) 1.0) t_0)
           (/
            (/ (+ (- (/ 0.3333333333333333 (* x x)) (/ 0.5 x)) 1.0) x)
            n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -2.5e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+187) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * 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) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / x) / n
    if ((1.0d0 / n) <= (-2.5d-20)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-51) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d+187) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = ((((0.3333333333333333d0 / (x * 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 t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -2.5e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+187) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) - (0.5 / x)) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / x) / n
	tmp = 0
	if (1.0 / n) <= -2.5e-20:
		tmp = t_1
	elif (1.0 / n) <= 2e-51:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5.0:
		tmp = t_1
	elif (1.0 / n) <= 2e+187:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = ((((0.3333333333333333 / (x * x)) - (0.5 / x)) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.5e-20)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-51)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e+187)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) - Float64(0.5 / x)) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / x) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -2.5e-20)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-51)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e+187)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = ((((0.3333333333333333 / (x * 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]}, Block[{t$95$1 = N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.5e-20], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-51], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+187], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.4999999999999999e-20 or 2e-51 < (/.f64 #s(literal 1 binary64) n) < 5

   1. Initial program 78.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 93.5%

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

   5. Simplified 93.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]
   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. pow-flip N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. /-lowering-/.f64 93.6%

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

   7. Applied egg-rr 93.6%

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

   if -2.4999999999999999e-20 < (/.f64 #s(literal 1 binary64) n) < 2e-51

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 82.4%

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

   5. Simplified 82.4%

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

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

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

      2. diff-log N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. +-lowering-+.f64 82.6%

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

   7. Applied egg-rr 82.6%

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

   if 5 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999981e187

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   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 84.9%

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

   5. Simplified 84.9%

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

   if 1.99999999999999981e187 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 41.0%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 6.4%

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

   5. Simplified 6.4%

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

   6. Taylor expanded in x around inf

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

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

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 58.8%

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

   8. Simplified 58.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - \frac{0.5}{x}\right) + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{x}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - \frac{0.5}{x}\right) + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 x) n)))
   (if (<= (/ 1.0 n) -2.5e-20)
     t_1
     (if (<= (/ 1.0 n) 2e-51)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5.0)
         t_1
         (if (<= (/ 1.0 n) 2e+187)
           (- 1.0 t_0)
           (/
            (/ (+ (- (/ 0.3333333333333333 (* x x)) (/ 0.5 x)) 1.0) x)
            n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -2.5e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+187) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * 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) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / x) / n
    if ((1.0d0 / n) <= (-2.5d-20)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-51) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d+187) then
        tmp = 1.0d0 - t_0
    else
        tmp = ((((0.3333333333333333d0 / (x * 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 t_1 = (t_0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -2.5e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+187) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) - (0.5 / x)) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / x) / n
	tmp = 0
	if (1.0 / n) <= -2.5e-20:
		tmp = t_1
	elif (1.0 / n) <= 2e-51:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5.0:
		tmp = t_1
	elif (1.0 / n) <= 2e+187:
		tmp = 1.0 - t_0
	else:
		tmp = ((((0.3333333333333333 / (x * x)) - (0.5 / x)) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.5e-20)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-51)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e+187)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) - Float64(0.5 / x)) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / x) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -2.5e-20)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-51)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e+187)
		tmp = 1.0 - t_0;
	else
		tmp = ((((0.3333333333333333 / (x * 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]}, Block[{t$95$1 = N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.5e-20], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-51], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+187], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.4999999999999999e-20 or 2e-51 < (/.f64 #s(literal 1 binary64) n) < 5

   1. Initial program 78.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 93.5%

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

   5. Simplified 93.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]
   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. pow-flip N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. /-lowering-/.f64 93.6%

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

   7. Applied egg-rr 93.6%

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

   if -2.4999999999999999e-20 < (/.f64 #s(literal 1 binary64) n) < 2e-51

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 82.4%

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

   5. Simplified 82.4%

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

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

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

      2. diff-log N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. +-lowering-+.f64 82.6%

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

   7. Applied egg-rr 82.6%

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

   if 5 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999981e187

   1. Initial program 84.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 84.0%

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

   if 1.99999999999999981e187 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 41.0%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 6.4%

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

   5. Simplified 6.4%

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

   6. Taylor expanded in x around inf

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

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

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 58.8%

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

   8. Simplified 58.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - \frac{0.5}{x}\right) + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - \frac{0.5}{x}\right) + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -2.5e-20)
     t_1
     (if (<= (/ 1.0 n) 2e-51)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5.0)
         t_1
         (if (<= (/ 1.0 n) 2e+187)
           (- 1.0 t_0)
           (/
            (/ (+ (- (/ 0.3333333333333333 (* x x)) (/ 0.5 x)) 1.0) x)
            n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -2.5e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+187) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * 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) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / n) / x
    if ((1.0d0 / n) <= (-2.5d-20)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-51) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d+187) then
        tmp = 1.0d0 - t_0
    else
        tmp = ((((0.3333333333333333d0 / (x * 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 t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -2.5e-20) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-51) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e+187) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) - (0.5 / x)) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -2.5e-20:
		tmp = t_1
	elif (1.0 / n) <= 2e-51:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5.0:
		tmp = t_1
	elif (1.0 / n) <= 2e+187:
		tmp = 1.0 - t_0
	else:
		tmp = ((((0.3333333333333333 / (x * x)) - (0.5 / x)) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2.5e-20)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-51)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e+187)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) - Float64(0.5 / x)) + 1.0) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / n) / x;
	tmp = 0.0;
	if ((1.0 / n) <= -2.5e-20)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-51)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e+187)
		tmp = 1.0 - t_0;
	else
		tmp = ((((0.3333333333333333 / (x * 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]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2.5e-20], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-51], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+187], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.4999999999999999e-20 or 2e-51 < (/.f64 #s(literal 1 binary64) n) < 5

   1. Initial program 78.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 93.5%

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

   5. Simplified 93.5%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

      4. pow-flip N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. /-lowering-/.f64 93.5%

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

   7. Applied egg-rr 93.5%

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

   if -2.4999999999999999e-20 < (/.f64 #s(literal 1 binary64) n) < 2e-51

   1. Initial program 27.1%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 82.4%

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

   5. Simplified 82.4%

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

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

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

      2. diff-log N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. +-lowering-+.f64 82.6%

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

   7. Applied egg-rr 82.6%

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

   if 5 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999981e187

   1. Initial program 84.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 84.0%

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

   if 1.99999999999999981e187 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 41.0%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 6.4%

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

   5. Simplified 6.4%

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

   6. Taylor expanded in x around inf

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

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

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 58.8%

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

   8. Simplified 58.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - \frac{0.5}{x}\right) + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - \frac{0.5}{x}\right) + 1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 2e-51)
   (/ (log (/ (+ x 1.0) x)) n)
   (if (<= (/ 1.0 n) 5.0)
     (/ (/ 1.0 x) n)
     (if (<= (/ 1.0 n) 2e+187)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (/ (+ (- (/ 0.3333333333333333 (* x x)) (/ 0.5 x)) 1.0) x) n)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 2e-51) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e+187) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((((0.3333333333333333 / (x * 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-51) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5.0d0) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 2d+187) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = ((((0.3333333333333333d0 / (x * 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-51) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5.0) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e+187) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((((0.3333333333333333 / (x * x)) - (0.5 / x)) + 1.0) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 2e-51:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5.0:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e+187:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = ((((0.3333333333333333 / (x * x)) - (0.5 / x)) + 1.0) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 2e-51)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5.0)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e+187)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) - Float64(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-51)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5.0)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 2e+187)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = ((((0.3333333333333333 / (x * 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-51], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5.0], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+187], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < 2e-51

   1. Initial program 53.1%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 71.7%

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

   5. Simplified 71.7%

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

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

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

      2. diff-log N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. +-lowering-+.f64 71.9%

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

   7. Applied egg-rr 71.9%

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

   if 2e-51 < (/.f64 #s(literal 1 binary64) n) < 5

   1. Initial program 4.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 77.1%

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

   5. Simplified 77.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]
   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. pow-flip N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. /-lowering-/.f64 77.3%

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

   7. Applied egg-rr 77.3%

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

   8. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 57.6%

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

   10. Simplified 57.6%

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

   if 5 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999981e187

   1. Initial program 84.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 84.0%

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

   if 1.99999999999999981e187 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 41.0%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 6.4%

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

   5. Simplified 6.4%

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

   6. Taylor expanded in x around inf

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

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

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 58.8%

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

   8. Simplified 58.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - \frac{0.5}{x}\right) + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-267}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-40}:\\ \;\;\;\;\frac{\log x}{0 - n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right) + n \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot -0.5\right) + \left(x \cdot n\right) \cdot \left(x \cdot \left(-0.5 + x \cdot 0.3333333333333333\right) + 1\right)\right)}{n \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+147}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+196}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1e-267)
   (- 1.0 (pow x (/ 1.0 n)))
   (if (<= x 2.55e-40)
     (/ (log x) (- 0.0 n))
     (if (<= x 1.0)
       (/
        (+
         (* 0.16666666666666666 (* x (* x x)))
         (*
          n
          (+
           (* (* x x) (+ 0.5 (* x -0.5)))
           (* (* x n) (+ (* x (+ -0.5 (* x 0.3333333333333333))) 1.0)))))
        (* n (* n n)))
       (if (<= x 2.8e+53)
         (/
          (/ (+ 1.0 (/ (- -0.5 (/ (+ (/ 0.25 x) -0.3333333333333333) x)) x)) x)
          n)
         (if (<= x 4e+147)
           0.0
           (if (<= x 5.2e+196) (/ (/ (- 1.0 (/ 0.5 x)) x) n) 0.0)))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1e-267) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 2.55e-40) {
		tmp = log(x) / (0.0 - n);
	} else if (x <= 1.0) {
		tmp = ((0.16666666666666666 * (x * (x * x))) + (n * (((x * x) * (0.5 + (x * -0.5))) + ((x * n) * ((x * (-0.5 + (x * 0.3333333333333333))) + 1.0))))) / (n * (n * n));
	} else if (x <= 2.8e+53) {
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x)) / x) / n;
	} else if (x <= 4e+147) {
		tmp = 0.0;
	} else if (x <= 5.2e+196) {
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	} 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 <= 1d-267) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 2.55d-40) then
        tmp = log(x) / (0.0d0 - n)
    else if (x <= 1.0d0) then
        tmp = ((0.16666666666666666d0 * (x * (x * x))) + (n * (((x * x) * (0.5d0 + (x * (-0.5d0)))) + ((x * n) * ((x * ((-0.5d0) + (x * 0.3333333333333333d0))) + 1.0d0))))) / (n * (n * n))
    else if (x <= 2.8d+53) then
        tmp = ((1.0d0 + (((-0.5d0) - (((0.25d0 / x) + (-0.3333333333333333d0)) / x)) / x)) / x) / n
    else if (x <= 4d+147) then
        tmp = 0.0d0
    else if (x <= 5.2d+196) then
        tmp = ((1.0d0 - (0.5d0 / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1e-267) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 2.55e-40) {
		tmp = Math.log(x) / (0.0 - n);
	} else if (x <= 1.0) {
		tmp = ((0.16666666666666666 * (x * (x * x))) + (n * (((x * x) * (0.5 + (x * -0.5))) + ((x * n) * ((x * (-0.5 + (x * 0.3333333333333333))) + 1.0))))) / (n * (n * n));
	} else if (x <= 2.8e+53) {
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x)) / x) / n;
	} else if (x <= 4e+147) {
		tmp = 0.0;
	} else if (x <= 5.2e+196) {
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1e-267:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 2.55e-40:
		tmp = math.log(x) / (0.0 - n)
	elif x <= 1.0:
		tmp = ((0.16666666666666666 * (x * (x * x))) + (n * (((x * x) * (0.5 + (x * -0.5))) + ((x * n) * ((x * (-0.5 + (x * 0.3333333333333333))) + 1.0))))) / (n * (n * n))
	elif x <= 2.8e+53:
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x)) / x) / n
	elif x <= 4e+147:
		tmp = 0.0
	elif x <= 5.2e+196:
		tmp = ((1.0 - (0.5 / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1e-267)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 2.55e-40)
		tmp = Float64(log(x) / Float64(0.0 - n));
	elseif (x <= 1.0)
		tmp = Float64(Float64(Float64(0.16666666666666666 * Float64(x * Float64(x * x))) + Float64(n * Float64(Float64(Float64(x * x) * Float64(0.5 + Float64(x * -0.5))) + Float64(Float64(x * n) * Float64(Float64(x * Float64(-0.5 + Float64(x * 0.3333333333333333))) + 1.0))))) / Float64(n * Float64(n * n)));
	elseif (x <= 2.8e+53)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / x) + -0.3333333333333333) / x)) / x)) / x) / n);
	elseif (x <= 4e+147)
		tmp = 0.0;
	elseif (x <= 5.2e+196)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1e-267)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 2.55e-40)
		tmp = log(x) / (0.0 - n);
	elseif (x <= 1.0)
		tmp = ((0.16666666666666666 * (x * (x * x))) + (n * (((x * x) * (0.5 + (x * -0.5))) + ((x * n) * ((x * (-0.5 + (x * 0.3333333333333333))) + 1.0))))) / (n * (n * n));
	elseif (x <= 2.8e+53)
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x)) / x) / n;
	elseif (x <= 4e+147)
		tmp = 0.0;
	elseif (x <= 5.2e+196)
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1e-267], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55e-40], N[(N[Log[x], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * n), $MachinePrecision] * N[(N[(x * N[(-0.5 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e+53], N[(N[(N[(1.0 + N[(N[(-0.5 - N[(N[(N[(0.25 / x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4e+147], 0.0, If[LessEqual[x, 5.2e+196], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < 9.9999999999999998e-268

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.1%

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

   if 9.9999999999999998e-268 < x < 2.55000000000000019e-40

   1. Initial program 31.8%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 59.3%

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

   5. Simplified 59.3%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. log-lowering-log.f64 59.3%

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

   8. Simplified 59.3%

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

   if 2.55000000000000019e-40 < x < 1

   1. Initial program 64.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   5. Simplified 18.0%

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

   6. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{6} \cdot {x}^{3} + n \cdot \left(n \cdot \left(x \cdot \left(1 + x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right)\right) + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)\right)}{{n}^{3}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{6} \cdot {x}^{3} + n \cdot \left(n \cdot \left(x \cdot \left(1 + x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right)\right) + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)\right)\right), \color{blue}{\left({n}^{3}\right)}\right) \]

   8. Simplified 71.3%

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

   if 1 < x < 2.8e53

   1. Initial program 32.1%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 26.4%

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

   5. Simplified 26.4%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}\right)}, n\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1\right), x\right)\right), n\right) \]

   8. Simplified 64.0%

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

   if 2.8e53 < x < 3.9999999999999999e147 or 5.20000000000000024e196 < x

   1. Initial program 81.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 33.8%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 81.7%

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

      1. metadata-eval 81.7%

         \[\leadsto 0 \]

   10. Applied egg-rr 81.7%

       \[\leadsto \color{blue}{0} \]

   if 3.9999999999999999e147 < x < 5.20000000000000024e196

   1. Initial program 44.1%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 44.1%

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

   5. Simplified 44.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)}, n\right) \]
   7. Step-by-step derivation

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 78.9%

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

   8. Simplified 78.9%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-267}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-40}:\\ \;\;\;\;\frac{\log x}{0 - n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right) + n \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot -0.5\right) + \left(x \cdot n\right) \cdot \left(x \cdot \left(-0.5 + x \cdot 0.3333333333333333\right) + 1\right)\right)}{n \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+147}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+196}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log x}{0 - n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right) + n \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot -0.5\right) + \left(x \cdot n\right) \cdot \left(x \cdot \left(-0.5 + x \cdot 0.3333333333333333\right) + 1\right)\right)}{n \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+151}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+197}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.3e-41)
   (/ (log x) (- 0.0 n))
   (if (<= x 1.0)
     (/
      (+
       (* 0.16666666666666666 (* x (* x x)))
       (*
        n
        (+
         (* (* x x) (+ 0.5 (* x -0.5)))
         (* (* x n) (+ (* x (+ -0.5 (* x 0.3333333333333333))) 1.0)))))
      (* n (* n n)))
     (if (<= x 2.6e+56)
       (/
        (/ (+ 1.0 (/ (- -0.5 (/ (+ (/ 0.25 x) -0.3333333333333333) x)) x)) x)
        n)
       (if (<= x 5.2e+151)
         0.0
         (if (<= x 4.4e+197) (/ (/ (- 1.0 (/ 0.5 x)) x) n) 0.0))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.3e-41) {
		tmp = log(x) / (0.0 - n);
	} else if (x <= 1.0) {
		tmp = ((0.16666666666666666 * (x * (x * x))) + (n * (((x * x) * (0.5 + (x * -0.5))) + ((x * n) * ((x * (-0.5 + (x * 0.3333333333333333))) + 1.0))))) / (n * (n * n));
	} else if (x <= 2.6e+56) {
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x)) / x) / n;
	} else if (x <= 5.2e+151) {
		tmp = 0.0;
	} else if (x <= 4.4e+197) {
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	} 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 <= 4.3d-41) then
        tmp = log(x) / (0.0d0 - n)
    else if (x <= 1.0d0) then
        tmp = ((0.16666666666666666d0 * (x * (x * x))) + (n * (((x * x) * (0.5d0 + (x * (-0.5d0)))) + ((x * n) * ((x * ((-0.5d0) + (x * 0.3333333333333333d0))) + 1.0d0))))) / (n * (n * n))
    else if (x <= 2.6d+56) then
        tmp = ((1.0d0 + (((-0.5d0) - (((0.25d0 / x) + (-0.3333333333333333d0)) / x)) / x)) / x) / n
    else if (x <= 5.2d+151) then
        tmp = 0.0d0
    else if (x <= 4.4d+197) then
        tmp = ((1.0d0 - (0.5d0 / x)) / x) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.3e-41) {
		tmp = Math.log(x) / (0.0 - n);
	} else if (x <= 1.0) {
		tmp = ((0.16666666666666666 * (x * (x * x))) + (n * (((x * x) * (0.5 + (x * -0.5))) + ((x * n) * ((x * (-0.5 + (x * 0.3333333333333333))) + 1.0))))) / (n * (n * n));
	} else if (x <= 2.6e+56) {
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x)) / x) / n;
	} else if (x <= 5.2e+151) {
		tmp = 0.0;
	} else if (x <= 4.4e+197) {
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.3e-41:
		tmp = math.log(x) / (0.0 - n)
	elif x <= 1.0:
		tmp = ((0.16666666666666666 * (x * (x * x))) + (n * (((x * x) * (0.5 + (x * -0.5))) + ((x * n) * ((x * (-0.5 + (x * 0.3333333333333333))) + 1.0))))) / (n * (n * n))
	elif x <= 2.6e+56:
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x)) / x) / n
	elif x <= 5.2e+151:
		tmp = 0.0
	elif x <= 4.4e+197:
		tmp = ((1.0 - (0.5 / x)) / x) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.3e-41)
		tmp = Float64(log(x) / Float64(0.0 - n));
	elseif (x <= 1.0)
		tmp = Float64(Float64(Float64(0.16666666666666666 * Float64(x * Float64(x * x))) + Float64(n * Float64(Float64(Float64(x * x) * Float64(0.5 + Float64(x * -0.5))) + Float64(Float64(x * n) * Float64(Float64(x * Float64(-0.5 + Float64(x * 0.3333333333333333))) + 1.0))))) / Float64(n * Float64(n * n)));
	elseif (x <= 2.6e+56)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / x) + -0.3333333333333333) / x)) / x)) / x) / n);
	elseif (x <= 5.2e+151)
		tmp = 0.0;
	elseif (x <= 4.4e+197)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / x)) / x) / n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.3e-41)
		tmp = log(x) / (0.0 - n);
	elseif (x <= 1.0)
		tmp = ((0.16666666666666666 * (x * (x * x))) + (n * (((x * x) * (0.5 + (x * -0.5))) + ((x * n) * ((x * (-0.5 + (x * 0.3333333333333333))) + 1.0))))) / (n * (n * n));
	elseif (x <= 2.6e+56)
		tmp = ((1.0 + ((-0.5 - (((0.25 / x) + -0.3333333333333333) / x)) / x)) / x) / n;
	elseif (x <= 5.2e+151)
		tmp = 0.0;
	elseif (x <= 4.4e+197)
		tmp = ((1.0 - (0.5 / x)) / x) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.3e-41], N[(N[Log[x], $MachinePrecision] / N[(0.0 - n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * n), $MachinePrecision] * N[(N[(x * N[(-0.5 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e+56], N[(N[(N[(1.0 + N[(N[(-0.5 - N[(N[(N[(0.25 / x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 5.2e+151], 0.0, If[LessEqual[x, 4.4e+197], N[(N[(N[(1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 4.2999999999999999e-41

   1. Initial program 39.5%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 54.6%

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

   5. Simplified 54.6%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. log-lowering-log.f64 54.6%

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

   8. Simplified 54.6%

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

   if 4.2999999999999999e-41 < x < 1

   1. Initial program 64.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{{n}^{3}} + \frac{1}{3} \cdot \frac{1}{n}\right) - \frac{1}{2} \cdot \frac{1}{{n}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   5. Simplified 18.0%

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

   6. Taylor expanded in n around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{6} \cdot {x}^{3} + n \cdot \left(n \cdot \left(x \cdot \left(1 + x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right)\right) + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)\right)}{{n}^{3}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{6} \cdot {x}^{3} + n \cdot \left(n \cdot \left(x \cdot \left(1 + x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right)\right) + {x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot x\right)\right)\right), \color{blue}{\left({n}^{3}\right)}\right) \]

   8. Simplified 71.3%

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

   if 1 < x < 2.60000000000000011e56

   1. Initial program 32.1%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 26.4%

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

   5. Simplified 26.4%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}\right)}, n\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1}{x}\right)\right), n\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{x} - \frac{1}{3}}{x} - \frac{1}{2}}{x} - 1\right), x\right)\right), n\right) \]

   8. Simplified 64.0%

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

   if 2.60000000000000011e56 < x < 5.20000000000000026e151 or 4.39999999999999979e197 < x

   1. Initial program 81.7%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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 33.8%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 81.7%

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

      1. metadata-eval 81.7%

         \[\leadsto 0 \]

   10. Applied egg-rr 81.7%

       \[\leadsto \color{blue}{0} \]

   if 5.20000000000000026e151 < x < 4.39999999999999979e197

   1. Initial program 44.1%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 44.1%

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

   5. Simplified 44.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1 - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)}, n\right) \]
   7. Step-by-step derivation

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 78.9%

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

   8. Simplified 78.9%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log x}{0 - n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right) + n \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot -0.5\right) + \left(x \cdot n\right) \cdot \left(x \cdot \left(-0.5 + x \cdot 0.3333333333333333\right) + 1\right)\right)}{n \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 - \frac{\frac{0.25}{x} + -0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+151}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+197}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - \frac{0.5}{x}\right) + 1}{x}}{n}\\ \mathbf{if}\;n \leq -1.75 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -7.2 \cdot 10^{-306}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ (+ (- (/ 0.3333333333333333 (* x x)) (/ 0.5 x)) 1.0) x) n)))
   (if (<= n -1.75e-32) t_0 (if (<= n -7.2e-306) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = ((((0.3333333333333333 / (x * x)) - (0.5 / x)) + 1.0) / x) / n;
	double tmp;
	if (n <= -1.75e-32) {
		tmp = t_0;
	} else if (n <= -7.2e-306) {
		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 = ((((0.3333333333333333d0 / (x * x)) - (0.5d0 / x)) + 1.0d0) / x) / n
    if (n <= (-1.75d-32)) then
        tmp = t_0
    else if (n <= (-7.2d-306)) 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 = ((((0.3333333333333333 / (x * x)) - (0.5 / x)) + 1.0) / x) / n;
	double tmp;
	if (n <= -1.75e-32) {
		tmp = t_0;
	} else if (n <= -7.2e-306) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = ((((0.3333333333333333 / (x * x)) - (0.5 / x)) + 1.0) / x) / n
	tmp = 0
	if n <= -1.75e-32:
		tmp = t_0
	elif n <= -7.2e-306:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / Float64(x * x)) - Float64(0.5 / x)) + 1.0) / x) / n)
	tmp = 0.0
	if (n <= -1.75e-32)
		tmp = t_0;
	elseif (n <= -7.2e-306)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((((0.3333333333333333 / (x * x)) - (0.5 / x)) + 1.0) / x) / n;
	tmp = 0.0;
	if (n <= -1.75e-32)
		tmp = t_0;
	elseif (n <= -7.2e-306)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(N[(N[(0.3333333333333333 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -1.75e-32], t$95$0, If[LessEqual[n, -7.2e-306], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.7499999999999999e-32 or -7.19999999999999982e-306 < n

   1. Initial program 35.4%

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

   3. Taylor expanded in n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 56.9%

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

   5. Simplified 56.9%

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

   6. Taylor expanded in x around inf

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

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

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. /-lowering-/.f64 44.2%

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

   8. Simplified 44.2%

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

   if -1.7499999999999999e-32 < n < -7.19999999999999982e-306

   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 39.8%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 62.7%

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

      1. metadata-eval 62.7%

         \[\leadsto 0 \]

   10. Applied egg-rr 62.7%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - \frac{0.5}{x}\right) + 1}{x}}{n}\\ \mathbf{elif}\;n \leq -7.2 \cdot 10^{-306}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{0.3333333333333333}{x \cdot x} - \frac{0.5}{x}\right) + 1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.6% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.5:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -7.2 \cdot 10^{-306}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -9.5) (/ (/ 1.0 n) x) (if (<= n -7.2e-306) 0.0 (/ (/ 1.0 x) n))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -9.5) {
		tmp = (1.0 / n) / x;
	} else if (n <= -7.2e-306) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-9.5d0)) then
        tmp = (1.0d0 / n) / x
    else if (n <= (-7.2d-306)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -9.5) {
		tmp = (1.0 / n) / x;
	} else if (n <= -7.2e-306) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -9.5:
		tmp = (1.0 / n) / x
	elif n <= -7.2e-306:
		tmp = 0.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -9.5)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (n <= -7.2e-306)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -9.5)
		tmp = (1.0 / n) / x;
	elseif (n <= -7.2e-306)
		tmp = 0.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -9.5], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[n, -7.2e-306], 0.0, N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.5

   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 n around inf

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

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

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

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

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

      3. log1p-define N/A

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

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

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

      5. log-lowering-log.f64 74.6%

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

   5. Simplified 74.6%

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

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{n \cdot x} - \frac{1}{3} \cdot \frac{1}{n}}{x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{n \cdot x} - \frac{1}{3} \cdot \frac{1}{n}}{x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{n \cdot x} - \frac{1}{3} \cdot \frac{1}{n}}{x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{n \cdot x} - \frac{1}{3} \cdot \frac{1}{n}}{x} - \frac{1}{2} \cdot \frac{1}{n}}{x} - \frac{1}{n}\right), x\right)\right) \]

   8. Simplified 46.2%

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

   9. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 47.4%

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

   11. Simplified 47.4%

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

   if -9.5 < n < -7.19999999999999982e-306

   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 41.3%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 61.2%

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

      1. metadata-eval 61.2%

         \[\leadsto 0 \]

   10. Applied egg-rr 61.2%

       \[\leadsto \color{blue}{0} \]

   if -7.19999999999999982e-306 < n

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 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 29.5%

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

   5. Simplified 29.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]
   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. pow-flip N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}\right), n\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), x\right), n\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), x\right), n\right) \]

      8. /-lowering-/.f64 30.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), x\right), n\right) \]

   7. Applied egg-rr 30.7%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} \]

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{x}\right)}, n\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 39.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), n\right) \]

   10. Simplified 39.0%

       \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;n \leq -7.2 \cdot 10^{-306}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.6% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -9.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -4 \cdot 10^{-306}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)))
   (if (<= n -9.5) t_0 (if (<= n -4e-306) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if (n <= -9.5) {
		tmp = t_0;
	} else if (n <= -4e-306) {
		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 / x) / n
    if (n <= (-9.5d0)) then
        tmp = t_0
    else if (n <= (-4d-306)) 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 / x) / n;
	double tmp;
	if (n <= -9.5) {
		tmp = t_0;
	} else if (n <= -4e-306) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / x) / n
	tmp = 0
	if n <= -9.5:
		tmp = t_0
	elif n <= -4e-306:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (n <= -9.5)
		tmp = t_0;
	elseif (n <= -4e-306)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / x) / n;
	tmp = 0.0;
	if (n <= -9.5)
		tmp = t_0;
	elseif (n <= -4e-306)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -9.5], t$95$0, If[LessEqual[n, -4e-306], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -9.5 or -4.00000000000000011e-306 < n

   1. Initial program 33.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 36.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]
   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. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}}{x}\right), n\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}}{x}\right), n\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}\right), n\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), x\right), n\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), x\right), n\right) \]

      8. /-lowering-/.f64 37.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), x\right), n\right) \]

   7. Applied egg-rr 37.8%

      \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} \]

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{x}\right)}, n\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 42.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), n\right) \]

   10. Simplified 42.1%

       \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]

   if -9.5 < n < -4.00000000000000011e-306

   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 41.3%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 61.2%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 61.2%

         \[\leadsto 0 \]

   10. Applied egg-rr 61.2%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 46.1% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot n}\\ \mathbf{if}\;n \leq -1.55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -3.3 \cdot 10^{-308}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x n))))
   (if (<= n -1.55) t_0 (if (<= n -3.3e-308) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (x * n);
	double tmp;
	if (n <= -1.55) {
		tmp = t_0;
	} else if (n <= -3.3e-308) {
		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 / (x * n)
    if (n <= (-1.55d0)) then
        tmp = t_0
    else if (n <= (-3.3d-308)) 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 / (x * n);
	double tmp;
	if (n <= -1.55) {
		tmp = t_0;
	} else if (n <= -3.3e-308) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (x * n)
	tmp = 0
	if n <= -1.55:
		tmp = t_0
	elif n <= -3.3e-308:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(x * n))
	tmp = 0.0
	if (n <= -1.55)
		tmp = t_0;
	elseif (n <= -3.3e-308)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (x * n);
	tmp = 0.0;
	if (n <= -1.55)
		tmp = t_0;
	elseif (n <= -3.3e-308)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.55], t$95$0, If[LessEqual[n, -3.3e-308], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.55000000000000004 or -3.2999999999999998e-308 < n

   1. Initial program 33.6%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   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 36.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x \cdot n}} \]

   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 41.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   8. Simplified 41.5%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]

   if -1.55000000000000004 < n < -3.2999999999999998e-308

   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 41.3%

      \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 61.2%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 61.2%

         \[\leadsto 0 \]

   10. Applied egg-rr 61.2%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 30.7% 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 52.0%

   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
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 34.6%

   \[\leadsto \color{blue}{1} - {x}^{\left(\frac{1}{n}\right)} \]

6. Taylor expanded in n around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
7. Step-by-step derivation

8. Simplified 30.3%

   \[\leadsto 1 - \color{blue}{1} \]
9. Step-by-step derivation

   1. metadata-eval 30.3%

      \[\leadsto 0 \]

10. Applied egg-rr 30.3%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))