2nthrt (problem 3.4.6)

Percentage Accurate: 57.3% → 88.0%

Time: 38.1s

Alternatives: 10

Speedup: 2.0×

• could not determine a ground truth

  1. x = 2.3481411065296882e+194
  2. n = 5.658463429979717e-185

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: 57.3% 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: 88.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -2000:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-311}:\\ \;\;\;\;e^{\frac{x \cdot \left(1 + x \cdot -0.5\right)}{n}} - t\_0\\ \mathbf{elif}\;x \leq 450:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -2000.0)
     (/ (/ t_0 x) n)
     (if (<= x -2e-311)
       (- (exp (/ (* x (+ 1.0 (* x -0.5))) n)) t_0)
       (if (<= x 450.0)
         (/
          (-
           (+
            (log1p x)
            (/
             (+
              (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0)))
              (/
               (*
                0.16666666666666666
                (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
               n))
             n))
           (log x))
          n)
         (* t_0 (/ 1.0 (* x n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -2000.0) {
		tmp = (t_0 / x) / n;
	} else if (x <= -2e-311) {
		tmp = exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	} else if (x <= 450.0) {
		tmp = ((log1p(x) + (((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) + ((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / n)) / n)) - log(x)) / n;
	} else {
		tmp = t_0 * (1.0 / (x * n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= -2000.0) {
		tmp = (t_0 / x) / n;
	} else if (x <= -2e-311) {
		tmp = Math.exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	} else if (x <= 450.0) {
		tmp = ((Math.log1p(x) + (((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) + ((0.16666666666666666 * (Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0))) / n)) / n)) - Math.log(x)) / n;
	} else {
		tmp = t_0 * (1.0 / (x * n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -2000.0:
		tmp = (t_0 / x) / n
	elif x <= -2e-311:
		tmp = math.exp(((x * (1.0 + (x * -0.5))) / n)) - t_0
	elif x <= 450.0:
		tmp = ((math.log1p(x) + (((0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0))) + ((0.16666666666666666 * (math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0))) / n)) / n)) - math.log(x)) / n
	else:
		tmp = t_0 * (1.0 / (x * n))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -2000.0)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (x <= -2e-311)
		tmp = Float64(exp(Float64(Float64(x * Float64(1.0 + Float64(x * -0.5))) / n)) - t_0);
	elseif (x <= 450.0)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) + Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / n)) / n)) - log(x)) / n);
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(x * n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2000.0], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, -2e-311], N[(N[Exp[N[(N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 450.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. log-rec 0.0%

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

      3. mul-1-neg 0.0%

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

      4. associate-*r/ 0.0%

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

      5. associate-*r* 0.0%

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

      6. metadata-eval 0.0%

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

      7. *-commutative 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 0.0%

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

      10. *-commutative 0.0%

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

      11. log-rec 0.0%

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

      12. mul-1-neg 0.0%

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

      13. associate-*r/ 0.0%

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

      14. associate-*r* 0.0%

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

      15. metadata-eval 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-/l* 0.0%

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

   5. Simplified 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-inv 100.0%

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

      3. fma-define 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. fma-undefine 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. +-inverses 100.0%

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

   9. Simplified 100.0%

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

       1. +-rgt-identity 100.0%

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

       2. associate-/r* 100.0%

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

       3. pow1 100.0%

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

       4. pow-div 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. pow-sub 100.0%

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

       2. pow1 100.0%

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

       3. div-inv 100.0%

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

   13. Applied egg-rr 100.0%

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

       1. associate-*r/ 100.0%

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

       2. *-rgt-identity 100.0%

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

   15. Simplified 100.0%

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

   if -2e3 < x < -1.9999999999999e-311

   1. Initial program 46.2%

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

   3. Taylor expanded in n around 0 0.0%

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

      1. log1p-define 0.0%

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

      2. *-rgt-identity 0.0%

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

      3. associate-*l/ 0.0%

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

      4. associate-/l* 0.0%

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

      5. exp-to-pow 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in x around 0 93.9%

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

      1. *-commutative 93.9%

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

   8. Simplified 93.9%

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

   if -1.9999999999999e-311 < x < 450

   1. Initial program 43.1%

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

   3. Taylor expanded in n around -inf 80.3%

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

   5. Simplified 80.3%

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

   if 450 < x

   1. Initial program 65.4%

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

   3. Taylor expanded in x around inf 65.4%

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

      1. +-commutative 65.4%

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

      2. log-rec 65.4%

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

      3. mul-1-neg 65.4%

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

      4. associate-*r/ 65.4%

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

      5. associate-*r* 65.4%

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

      6. metadata-eval 65.4%

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

      7. *-commutative 65.4%

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

      8. associate-/l* 65.4%

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

      9. exp-to-pow 65.4%

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

      10. *-commutative 65.4%

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

      11. log-rec 65.4%

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

      12. mul-1-neg 65.4%

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

      13. associate-*r/ 65.4%

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

      14. associate-*r* 65.4%

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

      15. metadata-eval 65.4%

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

      16. *-commutative 65.4%

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

      17. associate-/l* 65.4%

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

   5. Simplified 65.4%

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

      1. associate--l+ 98.7%

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

      2. div-inv 98.7%

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

      3. fma-define 98.7%

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

   7. Applied egg-rr 98.7%

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

      1. fma-undefine 98.7%

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

      2. associate-*r/ 98.7%

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

      3. *-rgt-identity 98.7%

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

      4. +-inverses 99.9%

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

   9. Simplified 99.9%

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

       1. +-rgt-identity 99.9%

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

       2. div-inv 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-311}:\\ \;\;\;\;e^{\frac{x \cdot \left(1 + x \cdot -0.5\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 450:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + \frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -2000:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-308}:\\ \;\;\;\;e^{\frac{x \cdot \left(1 + x \cdot -0.5\right)}{n}} - t\_0\\ \mathbf{elif}\;x \leq 0.215:\\ \;\;\;\;\frac{\left(x + \frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + -0.5 \cdot {\log x}^{2}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -2000.0)
     (/ (/ t_0 x) n)
     (if (<= x 5.8e-308)
       (- (exp (/ (* x (+ 1.0 (* x -0.5))) n)) t_0)
       (if (<= x 0.215)
         (/
          (-
           (+
            x
            (/
             (+
              (* -0.16666666666666666 (/ (pow (log x) 3.0) n))
              (* -0.5 (pow (log x) 2.0)))
             n))
           (log x))
          n)
         (* t_0 (/ 1.0 (* x n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -2000.0) {
		tmp = (t_0 / x) / n;
	} else if (x <= 5.8e-308) {
		tmp = exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	} else if (x <= 0.215) {
		tmp = ((x + (((-0.16666666666666666 * (pow(log(x), 3.0) / n)) + (-0.5 * pow(log(x), 2.0))) / n)) - log(x)) / n;
	} else {
		tmp = t_0 * (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 (x <= (-2000.0d0)) then
        tmp = (t_0 / x) / n
    else if (x <= 5.8d-308) then
        tmp = exp(((x * (1.0d0 + (x * (-0.5d0)))) / n)) - t_0
    else if (x <= 0.215d0) then
        tmp = ((x + ((((-0.16666666666666666d0) * ((log(x) ** 3.0d0) / n)) + ((-0.5d0) * (log(x) ** 2.0d0))) / n)) - log(x)) / n
    else
        tmp = t_0 * (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 (x <= -2000.0) {
		tmp = (t_0 / x) / n;
	} else if (x <= 5.8e-308) {
		tmp = Math.exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	} else if (x <= 0.215) {
		tmp = ((x + (((-0.16666666666666666 * (Math.pow(Math.log(x), 3.0) / n)) + (-0.5 * Math.pow(Math.log(x), 2.0))) / n)) - Math.log(x)) / n;
	} else {
		tmp = t_0 * (1.0 / (x * n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -2000.0:
		tmp = (t_0 / x) / n
	elif x <= 5.8e-308:
		tmp = math.exp(((x * (1.0 + (x * -0.5))) / n)) - t_0
	elif x <= 0.215:
		tmp = ((x + (((-0.16666666666666666 * (math.pow(math.log(x), 3.0) / n)) + (-0.5 * math.pow(math.log(x), 2.0))) / n)) - math.log(x)) / n
	else:
		tmp = t_0 * (1.0 / (x * n))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -2000.0)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (x <= 5.8e-308)
		tmp = Float64(exp(Float64(Float64(x * Float64(1.0 + Float64(x * -0.5))) / n)) - t_0);
	elseif (x <= 0.215)
		tmp = Float64(Float64(Float64(x + Float64(Float64(Float64(-0.16666666666666666 * Float64((log(x) ^ 3.0) / n)) + Float64(-0.5 * (log(x) ^ 2.0))) / n)) - log(x)) / n);
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(x * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= -2000.0)
		tmp = (t_0 / x) / n;
	elseif (x <= 5.8e-308)
		tmp = exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	elseif (x <= 0.215)
		tmp = ((x + (((-0.16666666666666666 * ((log(x) ^ 3.0) / n)) + (-0.5 * (log(x) ^ 2.0))) / n)) - log(x)) / n;
	else
		tmp = t_0 * (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[x, -2000.0], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 5.8e-308], N[(N[Exp[N[(N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 0.215], N[(N[(N[(x + N[(N[(N[(-0.16666666666666666 * N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. log-rec 0.0%

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

      3. mul-1-neg 0.0%

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

      4. associate-*r/ 0.0%

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

      5. associate-*r* 0.0%

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

      6. metadata-eval 0.0%

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

      7. *-commutative 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 0.0%

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

      10. *-commutative 0.0%

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

      11. log-rec 0.0%

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

      12. mul-1-neg 0.0%

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

      13. associate-*r/ 0.0%

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

      14. associate-*r* 0.0%

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

      15. metadata-eval 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-/l* 0.0%

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

   5. Simplified 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-inv 100.0%

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

      3. fma-define 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. fma-undefine 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. +-inverses 100.0%

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

   9. Simplified 100.0%

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

       1. +-rgt-identity 100.0%

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

       2. associate-/r* 100.0%

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

       3. pow1 100.0%

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

       4. pow-div 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. pow-sub 100.0%

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

       2. pow1 100.0%

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

       3. div-inv 100.0%

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

   13. Applied egg-rr 100.0%

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

       1. associate-*r/ 100.0%

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

       2. *-rgt-identity 100.0%

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

   15. Simplified 100.0%

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

   if -2e3 < x < 5.8000000000000001e-308

   1. Initial program 46.2%

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

   3. Taylor expanded in n around 0 0.0%

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

      1. log1p-define 0.0%

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

      2. *-rgt-identity 0.0%

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

      3. associate-*l/ 0.0%

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

      4. associate-/l* 0.0%

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

      5. exp-to-pow 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in x around 0 93.9%

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

      1. *-commutative 93.9%

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

   8. Simplified 93.9%

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

   if 5.8000000000000001e-308 < x < 0.214999999999999997

   1. Initial program 43.1%

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

   3. Taylor expanded in x around 0 43.5%

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

   4. Taylor expanded in n around -inf 79.6%

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

      1. mul-1-neg 79.6%

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

   6. Simplified 79.6%

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

   if 0.214999999999999997 < x

   1. Initial program 65.4%

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

   3. Taylor expanded in x around inf 65.4%

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

      1. +-commutative 65.4%

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

      2. log-rec 65.4%

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

      3. mul-1-neg 65.4%

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

      4. associate-*r/ 65.4%

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

      5. associate-*r* 65.4%

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

      6. metadata-eval 65.4%

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

      7. *-commutative 65.4%

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

      8. associate-/l* 65.4%

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

      9. exp-to-pow 65.4%

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

      10. *-commutative 65.4%

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

      11. log-rec 65.4%

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

      12. mul-1-neg 65.4%

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

      13. associate-*r/ 65.4%

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

      14. associate-*r* 65.4%

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

      15. metadata-eval 65.4%

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

      16. *-commutative 65.4%

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

      17. associate-/l* 65.4%

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

   5. Simplified 65.4%

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

      1. associate--l+ 98.7%

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

      2. div-inv 98.7%

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

      3. fma-define 98.7%

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

   7. Applied egg-rr 98.7%

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

      1. fma-undefine 98.7%

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

      2. associate-*r/ 98.7%

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

      3. *-rgt-identity 98.7%

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

      4. +-inverses 99.9%

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

   9. Simplified 99.9%

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

       1. +-rgt-identity 99.9%

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

       2. div-inv 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-308}:\\ \;\;\;\;e^{\frac{x \cdot \left(1 + x \cdot -0.5\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.215:\\ \;\;\;\;\frac{\left(x + \frac{-0.16666666666666666 \cdot \frac{{\log x}^{3}}{n} + -0.5 \cdot {\log x}^{2}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -2000:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-311}:\\ \;\;\;\;e^{\frac{x \cdot \left(1 + x \cdot -0.5\right)}{n}} - t\_0\\ \mathbf{elif}\;x \leq 0.048:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\frac{{\left(\frac{\log x}{n}\right)}^{2}}{x}, -0.5, \frac{1}{n} - \frac{\log x}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -2000.0)
     (/ (/ t_0 x) n)
     (if (<= x -2e-311)
       (- (exp (/ (* x (+ 1.0 (* x -0.5))) n)) t_0)
       (if (<= x 0.048)
         (*
          x
          (fma
           (/ (pow (/ (log x) n) 2.0) x)
           -0.5
           (- (/ 1.0 n) (/ (log x) (* x n)))))
         (* t_0 (/ 1.0 (* x n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -2000.0) {
		tmp = (t_0 / x) / n;
	} else if (x <= -2e-311) {
		tmp = exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	} else if (x <= 0.048) {
		tmp = x * fma((pow((log(x) / n), 2.0) / x), -0.5, ((1.0 / n) - (log(x) / (x * n))));
	} else {
		tmp = t_0 * (1.0 / (x * n));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -2000.0)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (x <= -2e-311)
		tmp = Float64(exp(Float64(Float64(x * Float64(1.0 + Float64(x * -0.5))) / n)) - t_0);
	elseif (x <= 0.048)
		tmp = Float64(x * fma(Float64((Float64(log(x) / n) ^ 2.0) / x), -0.5, Float64(Float64(1.0 / n) - Float64(log(x) / Float64(x * n)))));
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(x * n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2000.0], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, -2e-311], N[(N[Exp[N[(N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 0.048], N[(x * N[(N[(N[Power[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision], 2.0], $MachinePrecision] / x), $MachinePrecision] * -0.5 + N[(N[(1.0 / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. log-rec 0.0%

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

      3. mul-1-neg 0.0%

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

      4. associate-*r/ 0.0%

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

      5. associate-*r* 0.0%

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

      6. metadata-eval 0.0%

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

      7. *-commutative 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 0.0%

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

      10. *-commutative 0.0%

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

      11. log-rec 0.0%

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

      12. mul-1-neg 0.0%

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

      13. associate-*r/ 0.0%

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

      14. associate-*r* 0.0%

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

      15. metadata-eval 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-/l* 0.0%

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

   5. Simplified 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-inv 100.0%

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

      3. fma-define 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. fma-undefine 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. +-inverses 100.0%

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

   9. Simplified 100.0%

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

       1. +-rgt-identity 100.0%

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

       2. associate-/r* 100.0%

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

       3. pow1 100.0%

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

       4. pow-div 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. pow-sub 100.0%

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

       2. pow1 100.0%

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

       3. div-inv 100.0%

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

   13. Applied egg-rr 100.0%

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

       1. associate-*r/ 100.0%

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

       2. *-rgt-identity 100.0%

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

   15. Simplified 100.0%

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

   if -2e3 < x < -1.9999999999999e-311

   1. Initial program 46.2%

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

   3. Taylor expanded in n around 0 0.0%

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

      1. log1p-define 0.0%

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

      2. *-rgt-identity 0.0%

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

      3. associate-*l/ 0.0%

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

      4. associate-/l* 0.0%

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

      5. exp-to-pow 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in x around 0 93.9%

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

      1. *-commutative 93.9%

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

   8. Simplified 93.9%

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

   if -1.9999999999999e-311 < x < 0.048000000000000001

   1. Initial program 43.1%

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

   3. Taylor expanded in x around 0 43.5%

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

   4. Taylor expanded in n around inf 65.8%

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

   5. Taylor expanded in x around inf 71.5%

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

      1. associate--l+ 71.5%

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

      2. *-commutative 71.5%

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

      3. fma-define 71.5%

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

      4. associate-/r* 71.5%

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

      5. log-rec 71.5%

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

      6. unpow2 71.5%

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

      7. sqr-neg 71.5%

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

      8. unpow2 71.5%

         \[\leadsto x \cdot \mathsf{fma}\left(\frac{\frac{\log x \cdot \log x}{\color{blue}{n \cdot n}}}{x}, -0.5, \frac{1}{n} - -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n \cdot x}\right) \]

      9. times-frac 71.5%

         \[\leadsto x \cdot \mathsf{fma}\left(\frac{\color{blue}{\frac{\log x}{n} \cdot \frac{\log x}{n}}}{x}, -0.5, \frac{1}{n} - -1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n \cdot x}\right) \]

      10. unpow2 71.5%

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

      11. log-rec 71.5%

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

   7. Simplified 71.5%

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

   if 0.048000000000000001 < x

   1. Initial program 65.4%

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

   3. Taylor expanded in x around inf 65.4%

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

      1. +-commutative 65.4%

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

      2. log-rec 65.4%

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

      3. mul-1-neg 65.4%

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

      4. associate-*r/ 65.4%

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

      5. associate-*r* 65.4%

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

      6. metadata-eval 65.4%

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

      7. *-commutative 65.4%

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

      8. associate-/l* 65.4%

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

      9. exp-to-pow 65.4%

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

      10. *-commutative 65.4%

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

      11. log-rec 65.4%

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

      12. mul-1-neg 65.4%

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

      13. associate-*r/ 65.4%

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

      14. associate-*r* 65.4%

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

      15. metadata-eval 65.4%

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

      16. *-commutative 65.4%

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

      17. associate-/l* 65.4%

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

   5. Simplified 65.4%

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

      1. associate--l+ 98.7%

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

      2. div-inv 98.7%

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

      3. fma-define 98.7%

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

   7. Applied egg-rr 98.7%

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

      1. fma-undefine 98.7%

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

      2. associate-*r/ 98.7%

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

      3. *-rgt-identity 98.7%

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

      4. +-inverses 99.9%

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

   9. Simplified 99.9%

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

       1. +-rgt-identity 99.9%

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

       2. div-inv 99.9%

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

   11. Applied egg-rr 99.9%

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

Alternative 4: 81.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-172}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-169}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(\frac{\log x}{x \cdot n} - \log x\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000000:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -1e-172)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-169)
       (/ (+ (log1p x) (- (/ (log x) (* x n)) (log x))) n)
       (if (<= (/ 1.0 n) 1000000.0)
         (/ (/ t_0 x) n)
         (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-172) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-169) {
		tmp = (log1p(x) + ((log(x) / (x * n)) - log(x))) / n;
	} else if ((1.0 / n) <= 1000000.0) {
		tmp = (t_0 / x) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e-172) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-169) {
		tmp = (Math.log1p(x) + ((Math.log(x) / (x * n)) - Math.log(x))) / n;
	} else if ((1.0 / n) <= 1000000.0) {
		tmp = (t_0 / x) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e-172:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 2e-169:
		tmp = (math.log1p(x) + ((math.log(x) / (x * n)) - math.log(x))) / n
	elif (1.0 / n) <= 1000000.0:
		tmp = (t_0 / x) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-172)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-169)
		tmp = Float64(Float64(log1p(x) + Float64(Float64(log(x) / Float64(x * n)) - log(x))) / n);
	elseif (Float64(1.0 / n) <= 1000000.0)
		tmp = Float64(Float64(t_0 / x) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-172], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-169], N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1000000.0], N[(N[(t$95$0 / 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) < -1e-172

   1. Initial program 77.1%

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

   3. Taylor expanded in x around inf 29.8%

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

      1. +-commutative 29.8%

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

      2. log-rec 29.8%

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

      3. mul-1-neg 29.8%

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

      4. associate-*r/ 29.8%

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

      5. associate-*r* 29.8%

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

      6. metadata-eval 29.8%

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

      7. *-commutative 29.8%

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

      8. associate-/l* 29.8%

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

      9. exp-to-pow 29.8%

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

      10. *-commutative 29.8%

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

      11. log-rec 29.8%

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

      12. mul-1-neg 29.8%

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

      13. associate-*r/ 29.8%

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

      14. associate-*r* 29.8%

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

      15. metadata-eval 29.8%

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

      16. *-commutative 29.8%

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

      17. associate-/l* 29.8%

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

   5. Simplified 49.5%

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

      1. associate--l+ 60.7%

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

      2. div-inv 60.7%

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

      3. fma-define 60.7%

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

   7. Applied egg-rr 60.7%

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

      1. fma-undefine 60.7%

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

      2. associate-*r/ 60.7%

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

      3. *-rgt-identity 60.7%

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

      4. +-inverses 85.9%

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

   9. Simplified 85.9%

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

   10. Taylor expanded in x around 0 66.3%

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

       1. *-rgt-identity 66.3%

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

       2. associate-*r/ 66.3%

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

       3. exp-to-pow 85.9%

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

       4. *-commutative 85.9%

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

   12. Simplified 85.9%

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

   if -1e-172 < (/.f64 #s(literal 1 binary64) n) < 2.00000000000000004e-169

   1. Initial program 34.9%

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

   3. Taylor expanded in n around inf 95.5%

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

   5. Simplified 93.4%

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

   6. Taylor expanded in x around inf 78.3%

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

      1. distribute-lft-out-- 78.3%

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

      2. log-rec 78.3%

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

      3. *-commutative 78.3%

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

      4. log-rec 78.3%

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

   8. Simplified 78.3%

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

   if 2.00000000000000004e-169 < (/.f64 #s(literal 1 binary64) n) < 1e6

   1. Initial program 20.7%

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

   3. Taylor expanded in x around inf 18.2%

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

      1. +-commutative 18.2%

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

      2. log-rec 18.2%

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

      3. mul-1-neg 18.2%

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

      4. associate-*r/ 18.2%

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

      5. associate-*r* 18.2%

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

      6. metadata-eval 18.2%

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

      7. *-commutative 18.2%

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

      8. associate-/l* 18.2%

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

      9. exp-to-pow 18.2%

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

      10. *-commutative 18.2%

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

      11. log-rec 18.2%

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

      12. mul-1-neg 18.2%

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

      13. associate-*r/ 18.2%

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

      14. associate-*r* 18.2%

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

      15. metadata-eval 18.2%

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

      16. *-commutative 18.2%

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

      17. associate-/l* 18.2%

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

   5. Simplified 18.2%

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

      1. associate--l+ 60.3%

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

      2. div-inv 60.3%

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

      3. fma-define 60.3%

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

   7. Applied egg-rr 60.3%

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

      1. fma-undefine 60.3%

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

      2. associate-*r/ 60.3%

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

      3. *-rgt-identity 60.3%

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

      4. +-inverses 63.7%

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

   9. Simplified 63.7%

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

       1. +-rgt-identity 63.7%

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

       2. associate-/r* 63.8%

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

       3. pow1 63.8%

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

       4. pow-div 63.4%

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

   11. Applied egg-rr 63.4%

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

       1. pow-sub 63.8%

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

       2. pow1 63.8%

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

       3. div-inv 63.8%

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

   13. Applied egg-rr 63.8%

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

       1. associate-*r/ 63.8%

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

       2. *-rgt-identity 63.8%

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

   15. Simplified 63.8%

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

   if 1e6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 44.6%

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

   3. Taylor expanded in n around 0 17.9%

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

      1. log1p-define 41.9%

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

      2. *-rgt-identity 41.9%

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

      3. associate-*l/ 41.9%

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

      4. associate-/l* 41.9%

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

      5. exp-to-pow 96.3%

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

   5. Simplified 96.3%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-172}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-169}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(\frac{\log x}{x \cdot n} - \log x\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 1000000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -2000:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-307}:\\ \;\;\;\;e^{\frac{x \cdot \left(1 + x \cdot -0.5\right)}{n}} - t\_0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{\frac{n}{\left(x + -0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -2000.0)
     (/ (/ t_0 x) n)
     (if (<= x 2.4e-307)
       (- (exp (/ (* x (+ 1.0 (* x -0.5))) n)) t_0)
       (if (<= x 1.95e-28)
         (/ 1.0 (/ n (- (+ x (* -0.5 (/ (pow (log x) 2.0) n))) (log x))))
         (* t_0 (/ 1.0 (* x n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -2000.0) {
		tmp = (t_0 / x) / n;
	} else if (x <= 2.4e-307) {
		tmp = exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	} else if (x <= 1.95e-28) {
		tmp = 1.0 / (n / ((x + (-0.5 * (pow(log(x), 2.0) / n))) - log(x)));
	} else {
		tmp = t_0 * (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 (x <= (-2000.0d0)) then
        tmp = (t_0 / x) / n
    else if (x <= 2.4d-307) then
        tmp = exp(((x * (1.0d0 + (x * (-0.5d0)))) / n)) - t_0
    else if (x <= 1.95d-28) then
        tmp = 1.0d0 / (n / ((x + ((-0.5d0) * ((log(x) ** 2.0d0) / n))) - log(x)))
    else
        tmp = t_0 * (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 (x <= -2000.0) {
		tmp = (t_0 / x) / n;
	} else if (x <= 2.4e-307) {
		tmp = Math.exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	} else if (x <= 1.95e-28) {
		tmp = 1.0 / (n / ((x + (-0.5 * (Math.pow(Math.log(x), 2.0) / n))) - Math.log(x)));
	} else {
		tmp = t_0 * (1.0 / (x * n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -2000.0:
		tmp = (t_0 / x) / n
	elif x <= 2.4e-307:
		tmp = math.exp(((x * (1.0 + (x * -0.5))) / n)) - t_0
	elif x <= 1.95e-28:
		tmp = 1.0 / (n / ((x + (-0.5 * (math.pow(math.log(x), 2.0) / n))) - math.log(x)))
	else:
		tmp = t_0 * (1.0 / (x * n))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -2000.0)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (x <= 2.4e-307)
		tmp = Float64(exp(Float64(Float64(x * Float64(1.0 + Float64(x * -0.5))) / n)) - t_0);
	elseif (x <= 1.95e-28)
		tmp = Float64(1.0 / Float64(n / Float64(Float64(x + Float64(-0.5 * Float64((log(x) ^ 2.0) / n))) - log(x))));
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(x * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= -2000.0)
		tmp = (t_0 / x) / n;
	elseif (x <= 2.4e-307)
		tmp = exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	elseif (x <= 1.95e-28)
		tmp = 1.0 / (n / ((x + (-0.5 * ((log(x) ^ 2.0) / n))) - log(x)));
	else
		tmp = t_0 * (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[x, -2000.0], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.4e-307], N[(N[Exp[N[(N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 1.95e-28], N[(1.0 / N[(n / N[(N[(x + N[(-0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. log-rec 0.0%

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

      3. mul-1-neg 0.0%

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

      4. associate-*r/ 0.0%

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

      5. associate-*r* 0.0%

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

      6. metadata-eval 0.0%

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

      7. *-commutative 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 0.0%

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

      10. *-commutative 0.0%

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

      11. log-rec 0.0%

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

      12. mul-1-neg 0.0%

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

      13. associate-*r/ 0.0%

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

      14. associate-*r* 0.0%

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

      15. metadata-eval 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-/l* 0.0%

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

   5. Simplified 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-inv 100.0%

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

      3. fma-define 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. fma-undefine 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. +-inverses 100.0%

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

   9. Simplified 100.0%

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

       1. +-rgt-identity 100.0%

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

       2. associate-/r* 100.0%

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

       3. pow1 100.0%

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

       4. pow-div 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. pow-sub 100.0%

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

       2. pow1 100.0%

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

       3. div-inv 100.0%

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

   13. Applied egg-rr 100.0%

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

       1. associate-*r/ 100.0%

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

       2. *-rgt-identity 100.0%

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

   15. Simplified 100.0%

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

   if -2e3 < x < 2.40000000000000018e-307

   1. Initial program 46.2%

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

   3. Taylor expanded in n around 0 0.0%

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

      1. log1p-define 0.0%

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

      2. *-rgt-identity 0.0%

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

      3. associate-*l/ 0.0%

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

      4. associate-/l* 0.0%

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

      5. exp-to-pow 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in x around 0 93.9%

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

      1. *-commutative 93.9%

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

   8. Simplified 93.9%

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

   if 2.40000000000000018e-307 < x < 1.94999999999999999e-28

   1. Initial program 40.2%

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

   3. Taylor expanded in x around 0 40.4%

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

   4. Taylor expanded in n around inf 67.5%

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

      1. clear-num 67.5%

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

      2. inv-pow 67.5%

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

      3. +-commutative 67.5%

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

      4. fma-define 67.5%

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

   6. Applied egg-rr 67.5%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{fma}\left(-0.5, \frac{{\log x}^{2}}{n}, x\right) - \log x}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 67.5%

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

   8. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{n}{\mathsf{fma}\left(-0.5, \frac{{\log x}^{2}}{n}, x\right) - \log x}}} \]
   9. Step-by-step derivation

      1. fma-undefine 67.5%

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

   10. Applied egg-rr 67.5%

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

   if 1.94999999999999999e-28 < x

   1. Initial program 65.2%

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

   3. Taylor expanded in x around inf 56.4%

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

      1. +-commutative 56.4%

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

      2. log-rec 56.4%

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

      3. mul-1-neg 56.4%

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

      4. associate-*r/ 56.4%

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

      5. associate-*r* 56.4%

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

      6. metadata-eval 56.4%

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

      7. *-commutative 56.4%

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

      8. associate-/l* 56.4%

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

      9. exp-to-pow 56.4%

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

      10. *-commutative 56.4%

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

      11. log-rec 56.4%

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

      12. mul-1-neg 56.4%

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

      13. associate-*r/ 56.4%

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

      14. associate-*r* 56.4%

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

      15. metadata-eval 56.4%

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

      16. *-commutative 56.4%

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

      17. associate-/l* 56.4%

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

   5. Simplified 56.4%

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

      1. associate--l+ 85.3%

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

      2. div-inv 85.3%

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

      3. fma-define 85.3%

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

   7. Applied egg-rr 85.3%

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

      1. fma-undefine 85.3%

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

      2. associate-*r/ 85.3%

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

      3. *-rgt-identity 85.3%

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

      4. +-inverses 93.4%

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

   9. Simplified 93.4%

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

       1. +-rgt-identity 93.4%

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

       2. div-inv 93.4%

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

   11. Applied egg-rr 93.4%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-307}:\\ \;\;\;\;e^{\frac{x \cdot \left(1 + x \cdot -0.5\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{\frac{n}{\left(x + -0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -2000:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-308}:\\ \;\;\;\;e^{\frac{x \cdot \left(1 + x \cdot -0.5\right)}{n}} - t\_0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-28}:\\ \;\;\;\;\frac{\left(x + -0.5 \cdot \frac{{\log x}^{2}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -2000.0)
     (/ (/ t_0 x) n)
     (if (<= x 1.6e-308)
       (- (exp (/ (* x (+ 1.0 (* x -0.5))) n)) t_0)
       (if (<= x 1.95e-28)
         (/ (- (+ x (* -0.5 (/ (pow (log x) 2.0) n))) (log x)) n)
         (* t_0 (/ 1.0 (* x n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -2000.0) {
		tmp = (t_0 / x) / n;
	} else if (x <= 1.6e-308) {
		tmp = exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	} else if (x <= 1.95e-28) {
		tmp = ((x + (-0.5 * (pow(log(x), 2.0) / n))) - log(x)) / n;
	} else {
		tmp = t_0 * (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 (x <= (-2000.0d0)) then
        tmp = (t_0 / x) / n
    else if (x <= 1.6d-308) then
        tmp = exp(((x * (1.0d0 + (x * (-0.5d0)))) / n)) - t_0
    else if (x <= 1.95d-28) then
        tmp = ((x + ((-0.5d0) * ((log(x) ** 2.0d0) / n))) - log(x)) / n
    else
        tmp = t_0 * (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 (x <= -2000.0) {
		tmp = (t_0 / x) / n;
	} else if (x <= 1.6e-308) {
		tmp = Math.exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	} else if (x <= 1.95e-28) {
		tmp = ((x + (-0.5 * (Math.pow(Math.log(x), 2.0) / n))) - Math.log(x)) / n;
	} else {
		tmp = t_0 * (1.0 / (x * n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -2000.0:
		tmp = (t_0 / x) / n
	elif x <= 1.6e-308:
		tmp = math.exp(((x * (1.0 + (x * -0.5))) / n)) - t_0
	elif x <= 1.95e-28:
		tmp = ((x + (-0.5 * (math.pow(math.log(x), 2.0) / n))) - math.log(x)) / n
	else:
		tmp = t_0 * (1.0 / (x * n))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -2000.0)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (x <= 1.6e-308)
		tmp = Float64(exp(Float64(Float64(x * Float64(1.0 + Float64(x * -0.5))) / n)) - t_0);
	elseif (x <= 1.95e-28)
		tmp = Float64(Float64(Float64(x + Float64(-0.5 * Float64((log(x) ^ 2.0) / n))) - log(x)) / n);
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(x * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= -2000.0)
		tmp = (t_0 / x) / n;
	elseif (x <= 1.6e-308)
		tmp = exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	elseif (x <= 1.95e-28)
		tmp = ((x + (-0.5 * ((log(x) ^ 2.0) / n))) - log(x)) / n;
	else
		tmp = t_0 * (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[x, -2000.0], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.6e-308], N[(N[Exp[N[(N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 1.95e-28], N[(N[(N[(x + N[(-0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. log-rec 0.0%

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

      3. mul-1-neg 0.0%

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

      4. associate-*r/ 0.0%

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

      5. associate-*r* 0.0%

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

      6. metadata-eval 0.0%

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

      7. *-commutative 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 0.0%

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

      10. *-commutative 0.0%

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

      11. log-rec 0.0%

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

      12. mul-1-neg 0.0%

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

      13. associate-*r/ 0.0%

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

      14. associate-*r* 0.0%

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

      15. metadata-eval 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-/l* 0.0%

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

   5. Simplified 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-inv 100.0%

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

      3. fma-define 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. fma-undefine 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. +-inverses 100.0%

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

   9. Simplified 100.0%

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

       1. +-rgt-identity 100.0%

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

       2. associate-/r* 100.0%

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

       3. pow1 100.0%

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

       4. pow-div 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. pow-sub 100.0%

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

       2. pow1 100.0%

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

       3. div-inv 100.0%

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

   13. Applied egg-rr 100.0%

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

       1. associate-*r/ 100.0%

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

       2. *-rgt-identity 100.0%

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

   15. Simplified 100.0%

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

   if -2e3 < x < 1.6000000000000001e-308

   1. Initial program 46.2%

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

   3. Taylor expanded in n around 0 0.0%

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

      1. log1p-define 0.0%

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

      2. *-rgt-identity 0.0%

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

      3. associate-*l/ 0.0%

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

      4. associate-/l* 0.0%

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

      5. exp-to-pow 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in x around 0 93.9%

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

      1. *-commutative 93.9%

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

   8. Simplified 93.9%

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

   if 1.6000000000000001e-308 < x < 1.94999999999999999e-28

   1. Initial program 40.2%

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

   3. Taylor expanded in x around 0 40.4%

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

   4. Taylor expanded in n around inf 67.5%

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

   if 1.94999999999999999e-28 < x

   1. Initial program 65.2%

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

   3. Taylor expanded in x around inf 56.4%

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

      1. +-commutative 56.4%

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

      2. log-rec 56.4%

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

      3. mul-1-neg 56.4%

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

      4. associate-*r/ 56.4%

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

      5. associate-*r* 56.4%

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

      6. metadata-eval 56.4%

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

      7. *-commutative 56.4%

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

      8. associate-/l* 56.4%

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

      9. exp-to-pow 56.4%

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

      10. *-commutative 56.4%

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

      11. log-rec 56.4%

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

      12. mul-1-neg 56.4%

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

      13. associate-*r/ 56.4%

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

      14. associate-*r* 56.4%

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

      15. metadata-eval 56.4%

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

      16. *-commutative 56.4%

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

      17. associate-/l* 56.4%

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

   5. Simplified 56.4%

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

      1. associate--l+ 85.3%

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

      2. div-inv 85.3%

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

      3. fma-define 85.3%

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

   7. Applied egg-rr 85.3%

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

      1. fma-undefine 85.3%

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

      2. associate-*r/ 85.3%

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

      3. *-rgt-identity 85.3%

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

      4. +-inverses 93.4%

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

   9. Simplified 93.4%

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

       1. +-rgt-identity 93.4%

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

       2. div-inv 93.4%

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

   11. Applied egg-rr 93.4%

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

Alternative 7: 76.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -2000:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-12}:\\ \;\;\;\;e^{\frac{x \cdot \left(1 + x \cdot -0.5\right)}{n}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -2000.0)
     (/ (/ t_0 x) n)
     (if (<= x 3.4e-12)
       (- (exp (/ (* x (+ 1.0 (* x -0.5))) n)) t_0)
       (* t_0 (/ 1.0 (* x n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -2000.0) {
		tmp = (t_0 / x) / n;
	} else if (x <= 3.4e-12) {
		tmp = exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	} else {
		tmp = t_0 * (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 (x <= (-2000.0d0)) then
        tmp = (t_0 / x) / n
    else if (x <= 3.4d-12) then
        tmp = exp(((x * (1.0d0 + (x * (-0.5d0)))) / n)) - t_0
    else
        tmp = t_0 * (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 (x <= -2000.0) {
		tmp = (t_0 / x) / n;
	} else if (x <= 3.4e-12) {
		tmp = Math.exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	} else {
		tmp = t_0 * (1.0 / (x * n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -2000.0:
		tmp = (t_0 / x) / n
	elif x <= 3.4e-12:
		tmp = math.exp(((x * (1.0 + (x * -0.5))) / n)) - t_0
	else:
		tmp = t_0 * (1.0 / (x * n))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -2000.0)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (x <= 3.4e-12)
		tmp = Float64(exp(Float64(Float64(x * Float64(1.0 + Float64(x * -0.5))) / n)) - t_0);
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(x * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= -2000.0)
		tmp = (t_0 / x) / n;
	elseif (x <= 3.4e-12)
		tmp = exp(((x * (1.0 + (x * -0.5))) / n)) - t_0;
	else
		tmp = t_0 * (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[x, -2000.0], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 3.4e-12], N[(N[Exp[N[(N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. log-rec 0.0%

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

      3. mul-1-neg 0.0%

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

      4. associate-*r/ 0.0%

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

      5. associate-*r* 0.0%

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

      6. metadata-eval 0.0%

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

      7. *-commutative 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 0.0%

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

      10. *-commutative 0.0%

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

      11. log-rec 0.0%

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

      12. mul-1-neg 0.0%

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

      13. associate-*r/ 0.0%

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

      14. associate-*r* 0.0%

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

      15. metadata-eval 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-/l* 0.0%

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

   5. Simplified 100.0%

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

      1. associate--l+ 100.0%

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

      2. div-inv 100.0%

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

      3. fma-define 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. fma-undefine 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. +-inverses 100.0%

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

   9. Simplified 100.0%

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

       1. +-rgt-identity 100.0%

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

       2. associate-/r* 100.0%

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

       3. pow1 100.0%

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

       4. pow-div 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. pow-sub 100.0%

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

       2. pow1 100.0%

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

       3. div-inv 100.0%

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

   13. Applied egg-rr 100.0%

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

       1. associate-*r/ 100.0%

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

       2. *-rgt-identity 100.0%

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

   15. Simplified 100.0%

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

   if -2e3 < x < 3.4000000000000001e-12

   1. Initial program 43.5%

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

   3. Taylor expanded in n around 0 32.9%

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

      1. log1p-define 42.4%

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

      2. *-rgt-identity 42.4%

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

      3. associate-*l/ 42.4%

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

      4. associate-/l* 42.4%

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

      5. exp-to-pow 64.1%

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

   5. Simplified 64.1%

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

   6. Taylor expanded in x around 0 64.1%

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

      1. *-commutative 64.1%

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

   8. Simplified 64.1%

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

   if 3.4000000000000001e-12 < x

   1. Initial program 64.8%

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

   3. Taylor expanded in x around inf 62.6%

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

      1. +-commutative 62.6%

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

      2. log-rec 62.6%

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

      3. mul-1-neg 62.6%

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

      4. associate-*r/ 62.6%

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

      5. associate-*r* 62.6%

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

      6. metadata-eval 62.6%

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

      7. *-commutative 62.6%

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

      8. associate-/l* 62.6%

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

      9. exp-to-pow 62.6%

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

      10. *-commutative 62.6%

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

      11. log-rec 62.6%

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

      12. mul-1-neg 62.6%

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

      13. associate-*r/ 62.6%

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

      14. associate-*r* 62.6%

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

      15. metadata-eval 62.6%

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

      16. *-commutative 62.6%

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

      17. associate-/l* 62.6%

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

   5. Simplified 62.6%

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

      1. associate--l+ 94.6%

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

      2. div-inv 94.6%

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

      3. fma-define 94.6%

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

   7. Applied egg-rr 94.6%

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

      1. fma-undefine 94.6%

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

      2. associate-*r/ 94.6%

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

      3. *-rgt-identity 94.6%

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

      4. +-inverses 98.0%

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

   9. Simplified 98.0%

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

       1. +-rgt-identity 98.0%

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

       2. div-inv 98.0%

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

   11. Applied egg-rr 98.0%

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

Alternative 8: 71.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -3.05 \cdot 10^{-167}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-247}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{\frac{0.5 + \left(x \cdot -0.5 + 0.16666666666666666 \cdot \frac{x}{n}\right)}{n} - x \cdot -0.3333333333333333}{n} + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -3.05e-167)
     (/ (/ t_0 x) n)
     (if (<= x 2.1e-247)
       (- (+ 1.0 (/ x n)) t_0)
       (if (<= x 3.6e-13)
         (-
          (+
           1.0
           (*
            x
            (+
             (/ 1.0 n)
             (*
              x
              (+
               (/
                (-
                 (/ (+ 0.5 (+ (* x -0.5) (* 0.16666666666666666 (/ x n)))) n)
                 (* x -0.3333333333333333))
                n)
               (* 0.5 (/ -1.0 n)))))))
          t_0)
         (* t_0 (/ 1.0 (* x n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -3.05e-167) {
		tmp = (t_0 / x) / n;
	} else if (x <= 2.1e-247) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 3.6e-13) {
		tmp = (1.0 + (x * ((1.0 / n) + (x * (((((0.5 + ((x * -0.5) + (0.16666666666666666 * (x / n)))) / n) - (x * -0.3333333333333333)) / n) + (0.5 * (-1.0 / n))))))) - t_0;
	} else {
		tmp = t_0 * (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 (x <= (-3.05d-167)) then
        tmp = (t_0 / x) / n
    else if (x <= 2.1d-247) then
        tmp = (1.0d0 + (x / n)) - t_0
    else if (x <= 3.6d-13) then
        tmp = (1.0d0 + (x * ((1.0d0 / n) + (x * (((((0.5d0 + ((x * (-0.5d0)) + (0.16666666666666666d0 * (x / n)))) / n) - (x * (-0.3333333333333333d0))) / n) + (0.5d0 * ((-1.0d0) / n))))))) - t_0
    else
        tmp = t_0 * (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 (x <= -3.05e-167) {
		tmp = (t_0 / x) / n;
	} else if (x <= 2.1e-247) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 3.6e-13) {
		tmp = (1.0 + (x * ((1.0 / n) + (x * (((((0.5 + ((x * -0.5) + (0.16666666666666666 * (x / n)))) / n) - (x * -0.3333333333333333)) / n) + (0.5 * (-1.0 / n))))))) - t_0;
	} else {
		tmp = t_0 * (1.0 / (x * n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -3.05e-167:
		tmp = (t_0 / x) / n
	elif x <= 2.1e-247:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 3.6e-13:
		tmp = (1.0 + (x * ((1.0 / n) + (x * (((((0.5 + ((x * -0.5) + (0.16666666666666666 * (x / n)))) / n) - (x * -0.3333333333333333)) / n) + (0.5 * (-1.0 / n))))))) - t_0
	else:
		tmp = t_0 * (1.0 / (x * n))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -3.05e-167)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (x <= 2.1e-247)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 3.6e-13)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(x * Float64(Float64(Float64(Float64(Float64(0.5 + Float64(Float64(x * -0.5) + Float64(0.16666666666666666 * Float64(x / n)))) / n) - Float64(x * -0.3333333333333333)) / n) + Float64(0.5 * Float64(-1.0 / n))))))) - t_0);
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(x * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= -3.05e-167)
		tmp = (t_0 / x) / n;
	elseif (x <= 2.1e-247)
		tmp = (1.0 + (x / n)) - t_0;
	elseif (x <= 3.6e-13)
		tmp = (1.0 + (x * ((1.0 / n) + (x * (((((0.5 + ((x * -0.5) + (0.16666666666666666 * (x / n)))) / n) - (x * -0.3333333333333333)) / n) + (0.5 * (-1.0 / n))))))) - t_0;
	else
		tmp = t_0 * (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[x, -3.05e-167], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.1e-247], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 3.6e-13], N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(x * N[(N[(N[(N[(N[(0.5 + N[(N[(x * -0.5), $MachinePrecision] + N[(0.16666666666666666 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(0.5 * N[(-1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.0499999999999999e-167

   1. Initial program 71.8%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. log-rec 0.0%

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

      3. mul-1-neg 0.0%

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

      4. associate-*r/ 0.0%

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

      5. associate-*r* 0.0%

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

      6. metadata-eval 0.0%

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

      7. *-commutative 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 0.0%

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

      10. *-commutative 0.0%

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

      11. log-rec 0.0%

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

      12. mul-1-neg 0.0%

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

      13. associate-*r/ 0.0%

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

      14. associate-*r* 0.0%

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

      15. metadata-eval 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-/l* 0.0%

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

   5. Simplified 89.6%

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

      1. associate--l+ 89.6%

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

      2. div-inv 87.0%

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

      3. fma-define 87.0%

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

   7. Applied egg-rr 87.0%

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

      1. fma-undefine 87.0%

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

      2. associate-*r/ 89.6%

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

      3. *-rgt-identity 89.6%

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

      4. +-inverses 89.6%

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

   9. Simplified 89.6%

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

       1. +-rgt-identity 89.6%

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

       2. associate-/r* 94.9%

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

       3. pow1 94.9%

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

       4. pow-div 94.9%

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

   11. Applied egg-rr 94.9%

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

       1. pow-sub 94.9%

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

       2. pow1 94.9%

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

       3. div-inv 94.9%

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

   13. Applied egg-rr 94.9%

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

       1. associate-*r/ 94.9%

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

       2. *-rgt-identity 94.9%

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

   15. Simplified 94.9%

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

   if -3.0499999999999999e-167 < x < 2.10000000000000014e-247

   1. Initial program 48.6%

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

   3. Taylor expanded in x around 0 49.1%

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

   if 2.10000000000000014e-247 < x < 3.5999999999999998e-13

   1. Initial program 45.2%

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

   3. Taylor expanded in x around 0 25.7%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\left(0.5 \cdot \frac{1}{{n}^{2}} + x \cdot \left(\left(0.16666666666666666 \cdot \frac{1}{{n}^{3}} + 0.3333333333333333 \cdot \frac{1}{n}\right) - 0.5 \cdot \frac{1}{{n}^{2}}\right)\right) - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in n around -inf 55.8%

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

   if 3.5999999999999998e-13 < x

   1. Initial program 64.8%

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

   3. Taylor expanded in x around inf 62.6%

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

      1. +-commutative 62.6%

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

      2. log-rec 62.6%

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

      3. mul-1-neg 62.6%

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

      4. associate-*r/ 62.6%

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

      5. associate-*r* 62.6%

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

      6. metadata-eval 62.6%

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

      7. *-commutative 62.6%

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

      8. associate-/l* 62.6%

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

      9. exp-to-pow 62.6%

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

      10. *-commutative 62.6%

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

      11. log-rec 62.6%

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

      12. mul-1-neg 62.6%

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

      13. associate-*r/ 62.6%

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

      14. associate-*r* 62.6%

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

      15. metadata-eval 62.6%

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

      16. *-commutative 62.6%

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

      17. associate-/l* 62.6%

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

   5. Simplified 62.6%

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

      1. associate--l+ 94.6%

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

      2. div-inv 94.6%

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

      3. fma-define 94.6%

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

   7. Applied egg-rr 94.6%

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

      1. fma-undefine 94.6%

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

      2. associate-*r/ 94.6%

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

      3. *-rgt-identity 94.6%

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

      4. +-inverses 98.0%

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

   9. Simplified 98.0%

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

       1. +-rgt-identity 98.0%

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

       2. div-inv 98.0%

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

   11. Applied egg-rr 98.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-167}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-247}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + x \cdot \left(\frac{\frac{0.5 + \left(x \cdot -0.5 + 0.16666666666666666 \cdot \frac{x}{n}\right)}{n} - x \cdot -0.3333333333333333}{n} + 0.5 \cdot \frac{-1}{n}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{t\_0}{x}}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-13}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -1.45e-166)
     (/ (/ t_0 x) n)
     (if (<= x 2.4e-13) (- (+ 1.0 (/ x n)) t_0) (* t_0 (/ 1.0 (* x n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -1.45e-166) {
		tmp = (t_0 / x) / n;
	} else if (x <= 2.4e-13) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = t_0 * (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 (x <= (-1.45d-166)) then
        tmp = (t_0 / x) / n
    else if (x <= 2.4d-13) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = t_0 * (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 (x <= -1.45e-166) {
		tmp = (t_0 / x) / n;
	} else if (x <= 2.4e-13) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = t_0 * (1.0 / (x * n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -1.45e-166:
		tmp = (t_0 / x) / n
	elif x <= 2.4e-13:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = t_0 * (1.0 / (x * n))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -1.45e-166)
		tmp = Float64(Float64(t_0 / x) / n);
	elseif (x <= 2.4e-13)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(x * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= -1.45e-166)
		tmp = (t_0 / x) / n;
	elseif (x <= 2.4e-13)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = t_0 * (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[x, -1.45e-166], N[(N[(t$95$0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.4e-13], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e-166

   1. Initial program 71.8%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. log-rec 0.0%

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

      3. mul-1-neg 0.0%

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

      4. associate-*r/ 0.0%

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

      5. associate-*r* 0.0%

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

      6. metadata-eval 0.0%

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

      7. *-commutative 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 0.0%

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

      10. *-commutative 0.0%

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

      11. log-rec 0.0%

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

      12. mul-1-neg 0.0%

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

      13. associate-*r/ 0.0%

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

      14. associate-*r* 0.0%

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

      15. metadata-eval 0.0%

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

      16. *-commutative 0.0%

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

      17. associate-/l* 0.0%

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

   5. Simplified 89.6%

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

      1. associate--l+ 89.6%

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

      2. div-inv 87.0%

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

      3. fma-define 87.0%

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

   7. Applied egg-rr 87.0%

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

      1. fma-undefine 87.0%

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

      2. associate-*r/ 89.6%

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

      3. *-rgt-identity 89.6%

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

      4. +-inverses 89.6%

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

   9. Simplified 89.6%

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

       1. +-rgt-identity 89.6%

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

       2. associate-/r* 94.9%

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

       3. pow1 94.9%

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

       4. pow-div 94.9%

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

   11. Applied egg-rr 94.9%

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

       1. pow-sub 94.9%

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

       2. pow1 94.9%

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

       3. div-inv 94.9%

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

   13. Applied egg-rr 94.9%

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

       1. associate-*r/ 94.9%

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

       2. *-rgt-identity 94.9%

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

   15. Simplified 94.9%

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

   if -1.45e-166 < x < 2.3999999999999999e-13

   1. Initial program 46.1%

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

   3. Taylor expanded in x around 0 46.6%

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

   if 2.3999999999999999e-13 < x

   1. Initial program 64.8%

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

   3. Taylor expanded in x around inf 62.6%

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

      1. +-commutative 62.6%

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

      2. log-rec 62.6%

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

      3. mul-1-neg 62.6%

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

      4. associate-*r/ 62.6%

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

      5. associate-*r* 62.6%

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

      6. metadata-eval 62.6%

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

      7. *-commutative 62.6%

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

      8. associate-/l* 62.6%

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

      9. exp-to-pow 62.6%

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

      10. *-commutative 62.6%

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

      11. log-rec 62.6%

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

      12. mul-1-neg 62.6%

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

      13. associate-*r/ 62.6%

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

      14. associate-*r* 62.6%

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

      15. metadata-eval 62.6%

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

      16. *-commutative 62.6%

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

      17. associate-/l* 62.6%

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

   5. Simplified 62.6%

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

      1. associate--l+ 94.6%

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

      2. div-inv 94.6%

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

      3. fma-define 94.6%

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

   7. Applied egg-rr 94.6%

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

      1. fma-undefine 94.6%

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

      2. associate-*r/ 94.6%

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

      3. *-rgt-identity 94.6%

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

      4. +-inverses 98.0%

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

   9. Simplified 98.0%

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

       1. +-rgt-identity 98.0%

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

       2. div-inv 98.0%

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

   11. Applied egg-rr 98.0%

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

Alternative 10: 61.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in x around inf 22.7%

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

   1. +-commutative 22.7%

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

   2. log-rec 22.7%

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

   3. mul-1-neg 22.7%

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

   4. associate-*r/ 22.7%

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

   5. associate-*r* 22.7%

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

   6. metadata-eval 22.7%

      \[\leadsto \left(\frac{e^{\frac{\color{blue}{1} \cdot \log x}{n}}}{n \cdot x} + e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   7. *-commutative 22.7%

      \[\leadsto \left(\frac{e^{\frac{\color{blue}{\log x \cdot 1}}{n}}}{n \cdot x} + e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   8. associate-/l* 22.7%

      \[\leadsto \left(\frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{n \cdot x} + e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   9. exp-to-pow 22.7%

      \[\leadsto \left(\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{n \cdot x} + e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   10. *-commutative 22.7%

       \[\leadsto \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x \cdot n}} + e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   11. log-rec 22.7%

       \[\leadsto \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + e^{-1 \cdot \frac{\color{blue}{-\log x}}{n}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   12. mul-1-neg 22.7%

       \[\leadsto \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + e^{-1 \cdot \frac{\color{blue}{-1 \cdot \log x}}{n}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   13. associate-*r/ 22.7%

       \[\leadsto \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + e^{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \log x\right)}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   14. associate-*r* 22.7%

       \[\leadsto \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + e^{\frac{\color{blue}{\left(-1 \cdot -1\right) \cdot \log x}}{n}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   15. metadata-eval 22.7%

       \[\leadsto \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + e^{\frac{\color{blue}{1} \cdot \log x}{n}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   16. *-commutative 22.7%

       \[\leadsto \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + e^{\frac{\color{blue}{\log x \cdot 1}}{n}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

   17. associate-/l* 22.7%

       \[\leadsto \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + e^{\color{blue}{\log x \cdot \frac{1}{n}}}\right) - {x}^{\left(\frac{1}{n}\right)} \]

5. Simplified 36.1%

   \[\leadsto \color{blue}{\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + {x}^{\left(\frac{1}{n}\right)}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
6. Step-by-step derivation

   1. associate--l+ 47.1%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \left({x}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \]

   2. div-inv 46.7%

      \[\leadsto \color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x \cdot n}} + \left({x}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \]

   3. fma-define 46.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{1}{x \cdot n}, {x}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \]

7. Applied egg-rr 46.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{\left(\frac{1}{n}\right)}, \frac{1}{x \cdot n}, {x}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \]
8. Step-by-step derivation

   1. fma-undefine 46.7%

      \[\leadsto \color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x \cdot n} + \left({x}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)} \]

   2. associate-*r/ 47.1%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)} \cdot 1}{x \cdot n}} + \left({x}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \]

   3. *-rgt-identity 47.1%

      \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x \cdot n} + \left({x}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \]

   4. +-inverses 60.0%

      \[\leadsto \frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + \color{blue}{0} \]

9. Simplified 60.0%

   \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n} + 0} \]
10. Step-by-step derivation

    1. +-rgt-identity 60.0%

       \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]

    2. associate-/r* 62.9%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} \]

    3. pow1 62.9%

       \[\leadsto \frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{{x}^{1}}}}{n} \]

    4. pow-div 62.7%

       \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n} - 1\right)}}}{n} \]

11. Applied egg-rr 62.7%

    \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n} - 1\right)}}{n}} \]
12. Step-by-step derivation

    1. pow-sub 62.9%

       \[\leadsto \frac{\color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{{x}^{1}}}}{n} \]

    2. pow1 62.9%

       \[\leadsto \frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{\color{blue}{x}}}{n} \]

    3. div-inv 62.9%

       \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x}}}{n} \]

13. Applied egg-rr 62.9%

    \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)} \cdot \frac{1}{x}}}{n} \]
14. Step-by-step derivation

    1. associate-*r/ 62.9%

       \[\leadsto \frac{\color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)} \cdot 1}{x}}}{n} \]

    2. *-rgt-identity 62.9%

       \[\leadsto \frac{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x}}{n} \]

15. Simplified 62.9%

    \[\leadsto \frac{\color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}}{n} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024179 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))