2nthrt (problem 3.4.6)

Percentage Accurate: 53.6% → 85.5%

Time: 22.2s

Alternatives: 17

Speedup: 2.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + x\right)\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \left(0.5 \cdot \frac{{t_0}^{2}}{{n}^{2}} + \frac{t_0}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{t_1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (+ 1.0 x)))
        (t_1 (pow x (/ 1.0 n)))
        (t_2
         (-
          (+ (* 0.5 (/ (pow t_0 2.0) (pow n 2.0))) (/ t_0 n))
          (+ (* 0.5 (/ (pow (log x) 2.0) (pow n 2.0))) (/ (log x) n)))))
   (if (<= (/ 1.0 n) -5e-25)
     (/ (/ t_1 n) x)
     (if (<= (/ 1.0 n) -1e-129)
       t_2
       (if (<= (/ 1.0 n) -5e-165)
         (/ 1.0 (* n (+ x 0.5)))
         (if (<= (/ 1.0 n) 5e-14) t_2 (- (exp (/ (log1p x) n)) t_1)))))))

C

double code(double x, double n) {
	double t_0 = log((1.0 + x));
	double t_1 = pow(x, (1.0 / n));
	double t_2 = ((0.5 * (pow(t_0, 2.0) / pow(n, 2.0))) + (t_0 / n)) - ((0.5 * (pow(log(x), 2.0) / pow(n, 2.0))) + (log(x) / n));
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_2;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 5e-14) {
		tmp = t_2;
	} else {
		tmp = exp((log1p(x) / n)) - t_1;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.log((1.0 + x));
	double t_1 = Math.pow(x, (1.0 / n));
	double t_2 = ((0.5 * (Math.pow(t_0, 2.0) / Math.pow(n, 2.0))) + (t_0 / n)) - ((0.5 * (Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0))) + (Math.log(x) / n));
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_2;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 5e-14) {
		tmp = t_2;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_1;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log((1.0 + x))
	t_1 = math.pow(x, (1.0 / n))
	t_2 = ((0.5 * (math.pow(t_0, 2.0) / math.pow(n, 2.0))) + (t_0 / n)) - ((0.5 * (math.pow(math.log(x), 2.0) / math.pow(n, 2.0))) + (math.log(x) / n))
	tmp = 0
	if (1.0 / n) <= -5e-25:
		tmp = (t_1 / n) / x
	elif (1.0 / n) <= -1e-129:
		tmp = t_2
	elif (1.0 / n) <= -5e-165:
		tmp = 1.0 / (n * (x + 0.5))
	elif (1.0 / n) <= 5e-14:
		tmp = t_2
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_1
	return tmp

Julia

function code(x, n)
	t_0 = log(Float64(1.0 + x))
	t_1 = x ^ Float64(1.0 / n)
	t_2 = Float64(Float64(Float64(0.5 * Float64((t_0 ^ 2.0) / (n ^ 2.0))) + Float64(t_0 / n)) - Float64(Float64(0.5 * Float64((log(x) ^ 2.0) / (n ^ 2.0))) + Float64(log(x) / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-25)
		tmp = Float64(Float64(t_1 / n) / x);
	elseif (Float64(1.0 / n) <= -1e-129)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= -5e-165)
		tmp = Float64(1.0 / Float64(n * Float64(x + 0.5)));
	elseif (Float64(1.0 / n) <= 5e-14)
		tmp = t_2;
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_1);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.5 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-25], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-129], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-165], N[(1.0 / N[(n * N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-14], t$95$2, N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -4.99999999999999962e-25

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf 95.9%

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

      1. mul-1-neg 95.9%

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

      2. log-rec 95.9%

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

      3. mul-1-neg 95.9%

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

      4. distribute-neg-frac 95.9%

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

      5. mul-1-neg 95.9%

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

      6. remove-double-neg 95.9%

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

      7. *-commutative 95.9%

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

   4. Simplified 95.9%

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

      1. div-inv 95.9%

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

      2. pow-to-exp 95.9%

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

      3. *-un-lft-identity 95.9%

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

      4. times-frac 95.9%

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

   6. Applied egg-rr 95.9%

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

      1. associate-*l/ 96.0%

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

      2. *-un-lft-identity 96.0%

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

   8. Applied egg-rr 96.0%

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

   if -4.99999999999999962e-25 < (/.f64 1 n) < -9.9999999999999993e-130 or -4.99999999999999981e-165 < (/.f64 1 n) < 5.0000000000000002e-14

   1. Initial program 26.8%

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

   2. Taylor expanded in n around inf 81.5%

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

   if -9.9999999999999993e-130 < (/.f64 1 n) < -4.99999999999999981e-165

   1. Initial program 25.6%

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

   2. Taylor expanded in n around inf 34.3%

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

      1. +-rgt-identity 34.3%

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

      2. +-rgt-identity 34.3%

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

      3. log1p-def 34.3%

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

   4. Simplified 34.3%

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

      1. clear-num 34.3%

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

      2. inv-pow 34.3%

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

   6. Applied egg-rr 34.3%

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

      1. unpow-1 34.3%

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

   8. Simplified 34.3%

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

   9. Taylor expanded in x around inf 91.7%

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

       1. *-commutative 91.7%

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

       2. distribute-lft-out 91.7%

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

   11. Simplified 91.7%

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

   if 5.0000000000000002e-14 < (/.f64 1 n)

   1. Initial program 57.5%

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

   2. Taylor expanded in n around 0 57.5%

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

      1. log1p-def 96.0%

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

   4. Simplified 96.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \frac{\log \left(1 + x\right)}{n}\right) - \left(0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} + \frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\mathsf{log1p}\left(x\right) - \log x}{n} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n)))
        (t_1
         (+
          (/ (- (log1p x) (log x)) n)
          (*
           0.5
           (-
            (/ (pow (log1p x) 2.0) (pow n 2.0))
            (/ (pow (log x) 2.0) (pow n 2.0)))))))
   (if (<= (/ 1.0 n) -5e-25)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) -1e-129)
       t_1
       (if (<= (/ 1.0 n) -5e-165)
         (/ 1.0 (* n (+ x 0.5)))
         (if (<= (/ 1.0 n) 5e-14) t_1 (- (exp (/ (log1p x) n)) t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = ((log1p(x) - log(x)) / n) + (0.5 * ((pow(log1p(x), 2.0) / pow(n, 2.0)) - (pow(log(x), 2.0) / pow(n, 2.0))));
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 5e-14) {
		tmp = t_1;
	} 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 t_1 = ((Math.log1p(x) - Math.log(x)) / n) + (0.5 * ((Math.pow(Math.log1p(x), 2.0) / Math.pow(n, 2.0)) - (Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0))));
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 5e-14) {
		tmp = t_1;
	} 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))
	t_1 = ((math.log1p(x) - math.log(x)) / n) + (0.5 * ((math.pow(math.log1p(x), 2.0) / math.pow(n, 2.0)) - (math.pow(math.log(x), 2.0) / math.pow(n, 2.0))))
	tmp = 0
	if (1.0 / n) <= -5e-25:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= -1e-129:
		tmp = t_1
	elif (1.0 / n) <= -5e-165:
		tmp = 1.0 / (n * (x + 0.5))
	elif (1.0 / n) <= 5e-14:
		tmp = t_1
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(Float64(log1p(x) - log(x)) / n) + Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) / (n ^ 2.0)) - Float64((log(x) ^ 2.0) / (n ^ 2.0)))))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-25)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= -1e-129)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-165)
		tmp = Float64(1.0 / Float64(n * Float64(x + 0.5)));
	elseif (Float64(1.0 / n) <= 5e-14)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-25], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-129], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-165], N[(1.0 / N[(n * N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-14], t$95$1, 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 1 n) < -4.99999999999999962e-25

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf 95.9%

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

      1. mul-1-neg 95.9%

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

      2. log-rec 95.9%

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

      3. mul-1-neg 95.9%

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

      4. distribute-neg-frac 95.9%

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

      5. mul-1-neg 95.9%

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

      6. remove-double-neg 95.9%

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

      7. *-commutative 95.9%

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

   4. Simplified 95.9%

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

      1. div-inv 95.9%

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

      2. pow-to-exp 95.9%

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

      3. *-un-lft-identity 95.9%

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

      4. times-frac 95.9%

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

   6. Applied egg-rr 95.9%

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

      1. associate-*l/ 96.0%

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

      2. *-un-lft-identity 96.0%

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

   8. Applied egg-rr 96.0%

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

   if -4.99999999999999962e-25 < (/.f64 1 n) < -9.9999999999999993e-130 or -4.99999999999999981e-165 < (/.f64 1 n) < 5.0000000000000002e-14

   1. Initial program 26.8%

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

   2. Taylor expanded in n around inf 81.5%

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

      1. associate--l+ 73.6%

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

      2. +-commutative 73.6%

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

      3. associate--r+ 81.5%

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

      4. div-sub 81.4%

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

      5. remove-double-neg 81.4%

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

      6. mul-1-neg 81.4%

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

      7. distribute-lft-out-- 81.4%

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

      8. distribute-neg-frac 81.4%

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

      9. mul-1-neg 81.4%

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

   4. Simplified 81.4%

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

   if -9.9999999999999993e-130 < (/.f64 1 n) < -4.99999999999999981e-165

   1. Initial program 25.6%

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

   2. Taylor expanded in n around inf 34.3%

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

      1. +-rgt-identity 34.3%

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

      2. +-rgt-identity 34.3%

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

      3. log1p-def 34.3%

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

   4. Simplified 34.3%

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

      1. clear-num 34.3%

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

      2. inv-pow 34.3%

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

   6. Applied egg-rr 34.3%

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

      1. unpow-1 34.3%

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

   8. Simplified 34.3%

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

   9. Taylor expanded in x around inf 91.7%

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

       1. *-commutative 91.7%

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

       2. distribute-lft-out 91.7%

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

   11. Simplified 91.7%

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

   if 5.0000000000000002e-14 < (/.f64 1 n)

   1. Initial program 57.5%

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

   2. Taylor expanded in n around 0 57.5%

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

      1. log1p-def 96.0%

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

   4. Simplified 96.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 3: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -5e-25)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) -1e-129)
       t_1
       (if (<= (/ 1.0 n) -5e-165)
         (/ 1.0 (* n (+ x 0.5)))
         (if (<= (/ 1.0 n) 5e-14) t_1 (- (exp (/ (log1p x) n)) t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 5e-14) {
		tmp = t_1;
	} 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 t_1 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 5e-14) {
		tmp = t_1;
	} 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))
	t_1 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -5e-25:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= -1e-129:
		tmp = t_1
	elif (1.0 / n) <= -5e-165:
		tmp = 1.0 / (n * (x + 0.5))
	elif (1.0 / n) <= 5e-14:
		tmp = t_1
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-25)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= -1e-129)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-165)
		tmp = Float64(1.0 / Float64(n * Float64(x + 0.5)));
	elseif (Float64(1.0 / n) <= 5e-14)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-25], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-129], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-165], N[(1.0 / N[(n * N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-14], t$95$1, 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 1 n) < -4.99999999999999962e-25

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf 95.9%

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

      1. mul-1-neg 95.9%

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

      2. log-rec 95.9%

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

      3. mul-1-neg 95.9%

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

      4. distribute-neg-frac 95.9%

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

      5. mul-1-neg 95.9%

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

      6. remove-double-neg 95.9%

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

      7. *-commutative 95.9%

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

   4. Simplified 95.9%

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

      1. div-inv 95.9%

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

      2. pow-to-exp 95.9%

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

      3. *-un-lft-identity 95.9%

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

      4. times-frac 95.9%

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

   6. Applied egg-rr 95.9%

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

      1. associate-*l/ 96.0%

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

      2. *-un-lft-identity 96.0%

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

   8. Applied egg-rr 96.0%

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

   if -4.99999999999999962e-25 < (/.f64 1 n) < -9.9999999999999993e-130 or -4.99999999999999981e-165 < (/.f64 1 n) < 5.0000000000000002e-14

   1. Initial program 26.8%

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

   2. Taylor expanded in n around inf 81.2%

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

      1. +-rgt-identity 81.2%

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

      2. +-rgt-identity 81.2%

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

      3. log1p-def 81.2%

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

   4. Simplified 81.2%

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

      1. log1p-udef 81.2%

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

      2. diff-log 81.3%

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

   6. Applied egg-rr 81.3%

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

   if -9.9999999999999993e-130 < (/.f64 1 n) < -4.99999999999999981e-165

   1. Initial program 25.6%

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

   2. Taylor expanded in n around inf 34.3%

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

      1. +-rgt-identity 34.3%

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

      2. +-rgt-identity 34.3%

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

      3. log1p-def 34.3%

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

   4. Simplified 34.3%

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

      1. clear-num 34.3%

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

      2. inv-pow 34.3%

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

   6. Applied egg-rr 34.3%

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

      1. unpow-1 34.3%

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

   8. Simplified 34.3%

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

   9. Taylor expanded in x around inf 91.7%

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

       1. *-commutative 91.7%

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

       2. distribute-lft-out 91.7%

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

   11. Simplified 91.7%

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

   if 5.0000000000000002e-14 < (/.f64 1 n)

   1. Initial program 57.5%

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

   2. Taylor expanded in n around 0 57.5%

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

      1. log1p-def 96.0%

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

   4. Simplified 96.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 4: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+125}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot x\right)}^{-3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -5e-25)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) -1e-129)
       t_1
       (if (<= (/ 1.0 n) -5e-165)
         (/ 1.0 (* n (+ x 0.5)))
         (if (<= (/ 1.0 n) 1e-5)
           t_1
           (if (<= (/ 1.0 n) 1e+125)
             (- (+ 1.0 (/ x n)) t_0)
             (cbrt (pow (* n x) -3.0)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 1e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+125) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = cbrt(pow((n * x), -3.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 1e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+125) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.cbrt(Math.pow((n * x), -3.0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-25)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= -1e-129)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-165)
		tmp = Float64(1.0 / Float64(n * Float64(x + 0.5)));
	elseif (Float64(1.0 / n) <= 1e-5)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+125)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = cbrt((Float64(n * x) ^ -3.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-25], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-129], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-165], N[(1.0 / N[(n * N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-5], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+125], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Power[N[Power[N[(n * x), $MachinePrecision], -3.0], $MachinePrecision], 1/3], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.99999999999999962e-25

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf 95.9%

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

      1. mul-1-neg 95.9%

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

      2. log-rec 95.9%

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

      3. mul-1-neg 95.9%

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

      4. distribute-neg-frac 95.9%

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

      5. mul-1-neg 95.9%

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

      6. remove-double-neg 95.9%

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

      7. *-commutative 95.9%

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

   4. Simplified 95.9%

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

      1. div-inv 95.9%

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

      2. pow-to-exp 95.9%

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

      3. *-un-lft-identity 95.9%

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

      4. times-frac 95.9%

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

   6. Applied egg-rr 95.9%

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

      1. associate-*l/ 96.0%

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

      2. *-un-lft-identity 96.0%

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

   8. Applied egg-rr 96.0%

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

   if -4.99999999999999962e-25 < (/.f64 1 n) < -9.9999999999999993e-130 or -4.99999999999999981e-165 < (/.f64 1 n) < 1.00000000000000008e-5

   1. Initial program 26.4%

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

   2. Taylor expanded in n around inf 80.0%

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

      1. +-rgt-identity 80.0%

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

      2. +-rgt-identity 80.0%

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

      3. log1p-def 80.0%

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

   4. Simplified 80.0%

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

      1. log1p-udef 80.0%

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

      2. diff-log 80.0%

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

   6. Applied egg-rr 80.0%

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

   if -9.9999999999999993e-130 < (/.f64 1 n) < -4.99999999999999981e-165

   1. Initial program 25.6%

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

   2. Taylor expanded in n around inf 34.3%

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

      1. +-rgt-identity 34.3%

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

      2. +-rgt-identity 34.3%

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

      3. log1p-def 34.3%

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

   4. Simplified 34.3%

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

      1. clear-num 34.3%

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

      2. inv-pow 34.3%

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

   6. Applied egg-rr 34.3%

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

      1. unpow-1 34.3%

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

   8. Simplified 34.3%

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

   9. Taylor expanded in x around inf 91.7%

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

       1. *-commutative 91.7%

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

       2. distribute-lft-out 91.7%

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

   11. Simplified 91.7%

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

   if 1.00000000000000008e-5 < (/.f64 1 n) < 9.9999999999999992e124

   1. Initial program 90.8%

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

   2. Taylor expanded in x around 0 86.2%

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

   if 9.9999999999999992e124 < (/.f64 1 n)

   1. Initial program 31.4%

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

   2. Taylor expanded in n around inf 6.6%

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

      1. +-rgt-identity 6.6%

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

      2. +-rgt-identity 6.6%

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

      3. log1p-def 6.6%

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

   4. Simplified 6.6%

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

   5. Taylor expanded in x around inf 42.2%

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

      1. *-commutative 42.2%

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

   7. Simplified 42.2%

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

      1. add-sqr-sqrt 42.2%

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

      2. sqrt-unprod 74.8%

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

      3. inv-pow 74.8%

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

      4. inv-pow 74.8%

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

      5. pow-prod-up 74.8%

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

      6. metadata-eval 74.8%

         \[\leadsto \sqrt{{\left(x \cdot n\right)}^{\color{blue}{-2}}} \]

   9. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
   10. Step-by-step derivation

       1. add-cbrt-cube 78.9%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{{\left(x \cdot n\right)}^{-2}} \cdot \sqrt{{\left(x \cdot n\right)}^{-2}}\right) \cdot \sqrt{{\left(x \cdot n\right)}^{-2}}}} \]

       2. pow1/3 78.9%

          \[\leadsto \color{blue}{{\left(\left(\sqrt{{\left(x \cdot n\right)}^{-2}} \cdot \sqrt{{\left(x \cdot n\right)}^{-2}}\right) \cdot \sqrt{{\left(x \cdot n\right)}^{-2}}\right)}^{0.3333333333333333}} \]

       3. add-sqr-sqrt 78.9%

          \[\leadsto {\left(\color{blue}{{\left(x \cdot n\right)}^{-2}} \cdot \sqrt{{\left(x \cdot n\right)}^{-2}}\right)}^{0.3333333333333333} \]

       4. sqrt-pow1 78.9%

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

       5. metadata-eval 78.9%

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

       6. pow-prod-up 78.9%

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

       7. metadata-eval 78.9%

          \[\leadsto {\left({\left(x \cdot n\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} \]

   11. Applied egg-rr 78.9%

       \[\leadsto \color{blue}{{\left({\left(x \cdot n\right)}^{-3}\right)}^{0.3333333333333333}} \]
   12. Step-by-step derivation

       1. unpow1/3 78.9%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(x \cdot n\right)}^{-3}}} \]

   13. Simplified 78.9%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+125}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot x\right)}^{-3}}\\ \end{array} \]

Alternative 5: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -5e-25)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) -1e-129)
       t_1
       (if (<= (/ 1.0 n) -5e-165)
         (/ 1.0 (* n (+ x 0.5)))
         (if (<= (/ 1.0 n) 1e-5) t_1 (- (exp (/ x n)) t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 1e-5) {
		tmp = t_1;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-5d-25)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= (-1d-129)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-165)) then
        tmp = 1.0d0 / (n * (x + 0.5d0))
    else if ((1.0d0 / n) <= 1d-5) then
        tmp = t_1
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 1e-5) {
		tmp = t_1;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -5e-25:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= -1e-129:
		tmp = t_1
	elif (1.0 / n) <= -5e-165:
		tmp = 1.0 / (n * (x + 0.5))
	elif (1.0 / n) <= 1e-5:
		tmp = t_1
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-25)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= -1e-129)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-165)
		tmp = Float64(1.0 / Float64(n * Float64(x + 0.5)));
	elseif (Float64(1.0 / n) <= 1e-5)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-25)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= -1e-129)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-165)
		tmp = 1.0 / (n * (x + 0.5));
	elseif ((1.0 / n) <= 1e-5)
		tmp = t_1;
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-25], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-129], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-165], N[(1.0 / N[(n * N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-5], t$95$1, N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -4.99999999999999962e-25

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf 95.9%

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

      1. mul-1-neg 95.9%

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

      2. log-rec 95.9%

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

      3. mul-1-neg 95.9%

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

      4. distribute-neg-frac 95.9%

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

      5. mul-1-neg 95.9%

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

      6. remove-double-neg 95.9%

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

      7. *-commutative 95.9%

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

   4. Simplified 95.9%

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

      1. div-inv 95.9%

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

      2. pow-to-exp 95.9%

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

      3. *-un-lft-identity 95.9%

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

      4. times-frac 95.9%

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

   6. Applied egg-rr 95.9%

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

      1. associate-*l/ 96.0%

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

      2. *-un-lft-identity 96.0%

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

   8. Applied egg-rr 96.0%

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

   if -4.99999999999999962e-25 < (/.f64 1 n) < -9.9999999999999993e-130 or -4.99999999999999981e-165 < (/.f64 1 n) < 1.00000000000000008e-5

   1. Initial program 26.4%

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

   2. Taylor expanded in n around inf 80.0%

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

      1. +-rgt-identity 80.0%

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

      2. +-rgt-identity 80.0%

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

      3. log1p-def 80.0%

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

   4. Simplified 80.0%

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

      1. log1p-udef 80.0%

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

      2. diff-log 80.0%

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

   6. Applied egg-rr 80.0%

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

   if -9.9999999999999993e-130 < (/.f64 1 n) < -4.99999999999999981e-165

   1. Initial program 25.6%

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

   2. Taylor expanded in n around inf 34.3%

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

      1. +-rgt-identity 34.3%

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

      2. +-rgt-identity 34.3%

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

      3. log1p-def 34.3%

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

   4. Simplified 34.3%

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

      1. clear-num 34.3%

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

      2. inv-pow 34.3%

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

   6. Applied egg-rr 34.3%

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

      1. unpow-1 34.3%

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

   8. Simplified 34.3%

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

   9. Taylor expanded in x around inf 91.7%

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

       1. *-commutative 91.7%

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

       2. distribute-lft-out 91.7%

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

   11. Simplified 91.7%

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

   if 1.00000000000000008e-5 < (/.f64 1 n)

   1. Initial program 59.7%

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

   2. Taylor expanded in n around 0 59.7%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 6: 81.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{t_1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+125}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_0 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x)))
        (t_1 (pow x (/ 1.0 n)))
        (t_2 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -5e-25)
     (/ (/ t_1 n) x)
     (if (<= (/ 1.0 n) -1e-129)
       t_2
       (if (<= (/ 1.0 n) -5e-165)
         (/ 1.0 (* n (+ x 0.5)))
         (if (<= (/ 1.0 n) 1e-5)
           t_2
           (if (<= (/ 1.0 n) 1e+125)
             (- (+ 1.0 (/ x n)) t_1)
             (sqrt (* t_0 t_0)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = pow(x, (1.0 / n));
	double t_2 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_2;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 1e-5) {
		tmp = t_2;
	} else if ((1.0 / n) <= 1e+125) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = sqrt((t_0 * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / (n * x)
    t_1 = x ** (1.0d0 / n)
    t_2 = log(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-5d-25)) then
        tmp = (t_1 / n) / x
    else if ((1.0d0 / n) <= (-1d-129)) then
        tmp = t_2
    else if ((1.0d0 / n) <= (-5d-165)) then
        tmp = 1.0d0 / (n * (x + 0.5d0))
    else if ((1.0d0 / n) <= 1d-5) then
        tmp = t_2
    else if ((1.0d0 / n) <= 1d+125) then
        tmp = (1.0d0 + (x / n)) - t_1
    else
        tmp = sqrt((t_0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = Math.pow(x, (1.0 / n));
	double t_2 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_1 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_2;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 1e-5) {
		tmp = t_2;
	} else if ((1.0 / n) <= 1e+125) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = Math.sqrt((t_0 * t_0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	t_1 = math.pow(x, (1.0 / n))
	t_2 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -5e-25:
		tmp = (t_1 / n) / x
	elif (1.0 / n) <= -1e-129:
		tmp = t_2
	elif (1.0 / n) <= -5e-165:
		tmp = 1.0 / (n * (x + 0.5))
	elif (1.0 / n) <= 1e-5:
		tmp = t_2
	elif (1.0 / n) <= 1e+125:
		tmp = (1.0 + (x / n)) - t_1
	else:
		tmp = math.sqrt((t_0 * t_0))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	t_1 = x ^ Float64(1.0 / n)
	t_2 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-25)
		tmp = Float64(Float64(t_1 / n) / x);
	elseif (Float64(1.0 / n) <= -1e-129)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= -5e-165)
		tmp = Float64(1.0 / Float64(n * Float64(x + 0.5)));
	elseif (Float64(1.0 / n) <= 1e-5)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 1e+125)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	else
		tmp = sqrt(Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (n * x);
	t_1 = x ^ (1.0 / n);
	t_2 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-25)
		tmp = (t_1 / n) / x;
	elseif ((1.0 / n) <= -1e-129)
		tmp = t_2;
	elseif ((1.0 / n) <= -5e-165)
		tmp = 1.0 / (n * (x + 0.5));
	elseif ((1.0 / n) <= 1e-5)
		tmp = t_2;
	elseif ((1.0 / n) <= 1e+125)
		tmp = (1.0 + (x / n)) - t_1;
	else
		tmp = sqrt((t_0 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-25], N[(N[(t$95$1 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-129], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-165], N[(1.0 / N[(n * N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-5], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+125], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.99999999999999962e-25

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf 95.9%

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

      1. mul-1-neg 95.9%

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

      2. log-rec 95.9%

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

      3. mul-1-neg 95.9%

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

      4. distribute-neg-frac 95.9%

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

      5. mul-1-neg 95.9%

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

      6. remove-double-neg 95.9%

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

      7. *-commutative 95.9%

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

   4. Simplified 95.9%

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

      1. div-inv 95.9%

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

      2. pow-to-exp 95.9%

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

      3. *-un-lft-identity 95.9%

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

      4. times-frac 95.9%

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

   6. Applied egg-rr 95.9%

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

      1. associate-*l/ 96.0%

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

      2. *-un-lft-identity 96.0%

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

   8. Applied egg-rr 96.0%

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

   if -4.99999999999999962e-25 < (/.f64 1 n) < -9.9999999999999993e-130 or -4.99999999999999981e-165 < (/.f64 1 n) < 1.00000000000000008e-5

   1. Initial program 26.4%

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

   2. Taylor expanded in n around inf 80.0%

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

      1. +-rgt-identity 80.0%

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

      2. +-rgt-identity 80.0%

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

      3. log1p-def 80.0%

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

   4. Simplified 80.0%

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

      1. log1p-udef 80.0%

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

      2. diff-log 80.0%

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

   6. Applied egg-rr 80.0%

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

   if -9.9999999999999993e-130 < (/.f64 1 n) < -4.99999999999999981e-165

   1. Initial program 25.6%

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

   2. Taylor expanded in n around inf 34.3%

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

      1. +-rgt-identity 34.3%

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

      2. +-rgt-identity 34.3%

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

      3. log1p-def 34.3%

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

   4. Simplified 34.3%

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

      1. clear-num 34.3%

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

      2. inv-pow 34.3%

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

   6. Applied egg-rr 34.3%

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

      1. unpow-1 34.3%

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

   8. Simplified 34.3%

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

   9. Taylor expanded in x around inf 91.7%

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

       1. *-commutative 91.7%

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

       2. distribute-lft-out 91.7%

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

   11. Simplified 91.7%

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

   if 1.00000000000000008e-5 < (/.f64 1 n) < 9.9999999999999992e124

   1. Initial program 90.8%

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

   2. Taylor expanded in x around 0 86.2%

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

   if 9.9999999999999992e124 < (/.f64 1 n)

   1. Initial program 31.4%

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

   2. Taylor expanded in n around inf 6.6%

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

      1. +-rgt-identity 6.6%

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

      2. +-rgt-identity 6.6%

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

      3. log1p-def 6.6%

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

   4. Simplified 6.6%

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

   5. Taylor expanded in x around inf 42.2%

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

      1. *-commutative 42.2%

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

   7. Simplified 42.2%

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

      1. add-sqr-sqrt 42.2%

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

      2. sqrt-unprod 74.8%

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

      3. inv-pow 74.8%

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

      4. inv-pow 74.8%

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

      5. pow-prod-up 74.8%

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

      6. metadata-eval 74.8%

         \[\leadsto \sqrt{{\left(x \cdot n\right)}^{\color{blue}{-2}}} \]

   9. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
   10. Step-by-step derivation

       1. metadata-eval 74.8%

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

       2. pow-prod-up 74.8%

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

       3. unpow-1 74.8%

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

       4. unpow-1 74.8%

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

   11. Applied egg-rr 74.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+125}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{n \cdot x} \cdot \frac{1}{n \cdot x}}\\ \end{array} \]

Alternative 7: 80.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+238}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -5e-25)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) -1e-129)
       t_1
       (if (<= (/ 1.0 n) -5e-165)
         (/ 1.0 (* n (+ x 0.5)))
         (if (<= (/ 1.0 n) 1e-5)
           t_1
           (if (<= (/ 1.0 n) 1e+238)
             (- (+ 1.0 (/ x n)) t_0)
             (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 1e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+238) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-5d-25)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= (-1d-129)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-165)) then
        tmp = 1.0d0 / (n * (x + 0.5d0))
    else if ((1.0d0 / n) <= 1d-5) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+238) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 1e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+238) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -5e-25:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= -1e-129:
		tmp = t_1
	elif (1.0 / n) <= -5e-165:
		tmp = 1.0 / (n * (x + 0.5))
	elif (1.0 / n) <= 1e-5:
		tmp = t_1
	elif (1.0 / n) <= 1e+238:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-25)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= -1e-129)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-165)
		tmp = Float64(1.0 / Float64(n * Float64(x + 0.5)));
	elseif (Float64(1.0 / n) <= 1e-5)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+238)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-25)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= -1e-129)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-165)
		tmp = 1.0 / (n * (x + 0.5));
	elseif ((1.0 / n) <= 1e-5)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+238)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-25], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-129], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-165], N[(1.0 / N[(n * N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-5], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+238], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.99999999999999962e-25

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf 95.9%

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

      1. mul-1-neg 95.9%

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

      2. log-rec 95.9%

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

      3. mul-1-neg 95.9%

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

      4. distribute-neg-frac 95.9%

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

      5. mul-1-neg 95.9%

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

      6. remove-double-neg 95.9%

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

      7. *-commutative 95.9%

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

   4. Simplified 95.9%

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

      1. div-inv 95.9%

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

      2. pow-to-exp 95.9%

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

      3. *-un-lft-identity 95.9%

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

      4. times-frac 95.9%

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

   6. Applied egg-rr 95.9%

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

      1. associate-*l/ 96.0%

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

      2. *-un-lft-identity 96.0%

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

   8. Applied egg-rr 96.0%

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

   if -4.99999999999999962e-25 < (/.f64 1 n) < -9.9999999999999993e-130 or -4.99999999999999981e-165 < (/.f64 1 n) < 1.00000000000000008e-5

   1. Initial program 26.4%

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

   2. Taylor expanded in n around inf 80.0%

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

      1. +-rgt-identity 80.0%

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

      2. +-rgt-identity 80.0%

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

      3. log1p-def 80.0%

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

   4. Simplified 80.0%

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

      1. log1p-udef 80.0%

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

      2. diff-log 80.0%

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

   6. Applied egg-rr 80.0%

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

   if -9.9999999999999993e-130 < (/.f64 1 n) < -4.99999999999999981e-165

   1. Initial program 25.6%

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

   2. Taylor expanded in n around inf 34.3%

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

      1. +-rgt-identity 34.3%

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

      2. +-rgt-identity 34.3%

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

      3. log1p-def 34.3%

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

   4. Simplified 34.3%

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

      1. clear-num 34.3%

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

      2. inv-pow 34.3%

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

   6. Applied egg-rr 34.3%

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

      1. unpow-1 34.3%

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

   8. Simplified 34.3%

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

   9. Taylor expanded in x around inf 91.7%

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

       1. *-commutative 91.7%

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

       2. distribute-lft-out 91.7%

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

   11. Simplified 91.7%

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

   if 1.00000000000000008e-5 < (/.f64 1 n) < 1e238

   1. Initial program 72.3%

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

   2. Taylor expanded in x around 0 65.3%

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

   if 1e238 < (/.f64 1 n)

   1. Initial program 3.1%

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

   2. Taylor expanded in n around inf 10.1%

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

      1. +-rgt-identity 10.1%

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

      2. +-rgt-identity 10.1%

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

      3. log1p-def 10.1%

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

   4. Simplified 10.1%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+238}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 8: 66.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+238}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -2e-7)
     t_0
     (if (<= (/ 1.0 n) -1e-129)
       t_1
       (if (<= (/ 1.0 n) -5e-165)
         (/ 1.0 (* n (+ x 0.5)))
         (if (<= (/ 1.0 n) 1e-5)
           t_1
           (if (<= (/ 1.0 n) 1e+238) t_0 (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -2e-7) {
		tmp = t_0;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 1e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+238) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = log(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-2d-7)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-1d-129)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-165)) then
        tmp = 1.0d0 / (n * (x + 0.5d0))
    else if ((1.0d0 / n) <= 1d-5) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+238) then
        tmp = t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -2e-7) {
		tmp = t_0;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 1e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+238) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -2e-7:
		tmp = t_0
	elif (1.0 / n) <= -1e-129:
		tmp = t_1
	elif (1.0 / n) <= -5e-165:
		tmp = 1.0 / (n * (x + 0.5))
	elif (1.0 / n) <= 1e-5:
		tmp = t_1
	elif (1.0 / n) <= 1e+238:
		tmp = t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-7)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -1e-129)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-165)
		tmp = Float64(1.0 / Float64(n * Float64(x + 0.5)));
	elseif (Float64(1.0 / n) <= 1e-5)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+238)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -2e-7)
		tmp = t_0;
	elseif ((1.0 / n) <= -1e-129)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-165)
		tmp = 1.0 / (n * (x + 0.5));
	elseif ((1.0 / n) <= 1e-5)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+238)
		tmp = t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-7], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-129], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-165], N[(1.0 / N[(n * N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-5], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+238], t$95$0, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -1.9999999999999999e-7 or 1.00000000000000008e-5 < (/.f64 1 n) < 1e238

   1. Initial program 89.1%

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

   2. Taylor expanded in x around 0 61.1%

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

   if -1.9999999999999999e-7 < (/.f64 1 n) < -9.9999999999999993e-130 or -4.99999999999999981e-165 < (/.f64 1 n) < 1.00000000000000008e-5

   1. Initial program 25.8%

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

   2. Taylor expanded in n around inf 78.0%

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

      1. +-rgt-identity 78.0%

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

      2. +-rgt-identity 78.0%

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

      3. log1p-def 78.0%

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

   4. Simplified 78.0%

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

      1. log1p-udef 78.0%

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

      2. diff-log 78.1%

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

   6. Applied egg-rr 78.1%

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

   if -9.9999999999999993e-130 < (/.f64 1 n) < -4.99999999999999981e-165

   1. Initial program 25.6%

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

   2. Taylor expanded in n around inf 34.3%

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

      1. +-rgt-identity 34.3%

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

      2. +-rgt-identity 34.3%

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

      3. log1p-def 34.3%

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

   4. Simplified 34.3%

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

      1. clear-num 34.3%

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

      2. inv-pow 34.3%

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

   6. Applied egg-rr 34.3%

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

      1. unpow-1 34.3%

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

   8. Simplified 34.3%

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

   9. Taylor expanded in x around inf 91.7%

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

       1. *-commutative 91.7%

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

       2. distribute-lft-out 91.7%

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

   11. Simplified 91.7%

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

   if 1e238 < (/.f64 1 n)

   1. Initial program 3.1%

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

   2. Taylor expanded in n around inf 10.1%

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

      1. +-rgt-identity 10.1%

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

      2. +-rgt-identity 10.1%

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

      3. log1p-def 10.1%

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

   4. Simplified 10.1%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-7}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+238}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 9: 80.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{t_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+238}:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ 1.0 x) x)) n)))
   (if (<= (/ 1.0 n) -5e-25)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) -1e-129)
       t_1
       (if (<= (/ 1.0 n) -5e-165)
         (/ 1.0 (* n (+ x 0.5)))
         (if (<= (/ 1.0 n) 1e-5)
           t_1
           (if (<= (/ 1.0 n) 1e+238) (- 1.0 t_0) (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 1e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+238) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(((1.0d0 + x) / x)) / n
    if ((1.0d0 / n) <= (-5d-25)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= (-1d-129)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-165)) then
        tmp = 1.0d0 / (n * (x + 0.5d0))
    else if ((1.0d0 / n) <= 1d-5) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+238) then
        tmp = 1.0d0 - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((1.0 + x) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -5e-25) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= -1e-129) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-165) {
		tmp = 1.0 / (n * (x + 0.5));
	} else if ((1.0 / n) <= 1e-5) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+238) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((1.0 + x) / x)) / n
	tmp = 0
	if (1.0 / n) <= -5e-25:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= -1e-129:
		tmp = t_1
	elif (1.0 / n) <= -5e-165:
		tmp = 1.0 / (n * (x + 0.5))
	elif (1.0 / n) <= 1e-5:
		tmp = t_1
	elif (1.0 / n) <= 1e+238:
		tmp = 1.0 - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-25)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= -1e-129)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-165)
		tmp = Float64(1.0 / Float64(n * Float64(x + 0.5)));
	elseif (Float64(1.0 / n) <= 1e-5)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+238)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(((1.0 + x) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -5e-25)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= -1e-129)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-165)
		tmp = 1.0 / (n * (x + 0.5));
	elseif ((1.0 / n) <= 1e-5)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+238)
		tmp = 1.0 - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-25], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-129], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-165], N[(1.0 / N[(n * N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-5], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+238], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.99999999999999962e-25

   1. Initial program 91.8%

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

   2. Taylor expanded in x around inf 95.9%

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

      1. mul-1-neg 95.9%

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

      2. log-rec 95.9%

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

      3. mul-1-neg 95.9%

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

      4. distribute-neg-frac 95.9%

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

      5. mul-1-neg 95.9%

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

      6. remove-double-neg 95.9%

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

      7. *-commutative 95.9%

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

   4. Simplified 95.9%

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

      1. div-inv 95.9%

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

      2. pow-to-exp 95.9%

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

      3. *-un-lft-identity 95.9%

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

      4. times-frac 95.9%

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

   6. Applied egg-rr 95.9%

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

      1. associate-*l/ 96.0%

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

      2. *-un-lft-identity 96.0%

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

   8. Applied egg-rr 96.0%

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

   if -4.99999999999999962e-25 < (/.f64 1 n) < -9.9999999999999993e-130 or -4.99999999999999981e-165 < (/.f64 1 n) < 1.00000000000000008e-5

   1. Initial program 26.4%

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

   2. Taylor expanded in n around inf 80.0%

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

      1. +-rgt-identity 80.0%

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

      2. +-rgt-identity 80.0%

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

      3. log1p-def 80.0%

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

   4. Simplified 80.0%

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

      1. log1p-udef 80.0%

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

      2. diff-log 80.0%

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

   6. Applied egg-rr 80.0%

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

   if -9.9999999999999993e-130 < (/.f64 1 n) < -4.99999999999999981e-165

   1. Initial program 25.6%

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

   2. Taylor expanded in n around inf 34.3%

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

      1. +-rgt-identity 34.3%

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

      2. +-rgt-identity 34.3%

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

      3. log1p-def 34.3%

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

   4. Simplified 34.3%

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

      1. clear-num 34.3%

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

      2. inv-pow 34.3%

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

   6. Applied egg-rr 34.3%

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

      1. unpow-1 34.3%

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

   8. Simplified 34.3%

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

   9. Taylor expanded in x around inf 91.7%

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

       1. *-commutative 91.7%

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

       2. distribute-lft-out 91.7%

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

   11. Simplified 91.7%

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

   if 1.00000000000000008e-5 < (/.f64 1 n) < 1e238

   1. Initial program 72.3%

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

   2. Taylor expanded in x around 0 64.3%

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

   if 1e238 < (/.f64 1 n)

   1. Initial program 3.1%

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

   2. Taylor expanded in n around inf 10.1%

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

      1. +-rgt-identity 10.1%

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

      2. +-rgt-identity 10.1%

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

      3. log1p-def 10.1%

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

   4. Simplified 10.1%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-129}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{n \cdot \left(x + 0.5\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-5}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+238}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]

Alternative 10: 55.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.2 \cdot 10^{-199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{-158}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 1.2e-199)
     t_0
     (if (<= x 1.46e-158)
       (/ (- (log x)) n)
       (if (<= x 1.6e-110)
         t_0
         (if (<= x 1.0) (- (/ x n) (/ (log x) n)) (/ (/ 1.0 x) n)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.2e-199) {
		tmp = t_0;
	} else if (x <= 1.46e-158) {
		tmp = -log(x) / n;
	} else if (x <= 1.6e-110) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (x / n) - (log(x) / n);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 1.2d-199) then
        tmp = t_0
    else if (x <= 1.46d-158) then
        tmp = -log(x) / n
    else if (x <= 1.6d-110) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = (x / n) - (log(x) / n)
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.2e-199) {
		tmp = t_0;
	} else if (x <= 1.46e-158) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.6e-110) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (x / n) - (Math.log(x) / n);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.2e-199:
		tmp = t_0
	elif x <= 1.46e-158:
		tmp = -math.log(x) / n
	elif x <= 1.6e-110:
		tmp = t_0
	elif x <= 1.0:
		tmp = (x / n) - (math.log(x) / n)
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 1.2e-199)
		tmp = t_0;
	elseif (x <= 1.46e-158)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.6e-110)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 1.2e-199)
		tmp = t_0;
	elseif (x <= 1.46e-158)
		tmp = -log(x) / n;
	elseif (x <= 1.6e-110)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = (x / n) - (log(x) / n);
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.2e-199], t$95$0, If[LessEqual[x, 1.46e-158], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.6e-110], t$95$0, If[LessEqual[x, 1.0], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.19999999999999998e-199 or 1.4599999999999999e-158 < x < 1.60000000000000014e-110

   1. Initial program 60.5%

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

   2. Taylor expanded in x around 0 60.5%

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

   if 1.19999999999999998e-199 < x < 1.4599999999999999e-158

   1. Initial program 39.1%

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

   2. Taylor expanded in x around 0 39.1%

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

   3. Taylor expanded in n around inf 58.9%

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

      1. neg-mul-1 58.9%

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

      2. distribute-neg-frac 58.9%

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

   5. Simplified 58.9%

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

   if 1.60000000000000014e-110 < x < 1

   1. Initial program 32.1%

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

   2. Taylor expanded in n around inf 54.2%

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

      1. +-rgt-identity 54.2%

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

      2. +-rgt-identity 54.2%

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

      3. log1p-def 54.2%

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

   4. Simplified 54.2%

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

   5. Taylor expanded in x around 0 51.6%

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

      1. neg-mul-1 51.6%

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

      2. +-commutative 51.6%

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

      3. unsub-neg 51.6%

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

   7. Simplified 51.6%

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

   if 1 < x

   1. Initial program 61.6%

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

   2. Taylor expanded in n around inf 62.2%

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

      1. +-rgt-identity 62.2%

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

      2. +-rgt-identity 62.2%

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

      3. log1p-def 62.2%

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

   4. Simplified 62.2%

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

   5. Taylor expanded in x around inf 66.8%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-199}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{-158}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-110}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 11: 55.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.45 \cdot 10^{-199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-158}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 1.45e-199)
     t_0
     (if (<= x 2e-158)
       (/ (- (log x)) n)
       (if (<= x 2.3e-110)
         t_0
         (if (<= x 1.0) (/ (- x (log x)) n) (/ (/ 1.0 x) n)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.45e-199) {
		tmp = t_0;
	} else if (x <= 2e-158) {
		tmp = -log(x) / n;
	} else if (x <= 2.3e-110) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 1.45d-199) then
        tmp = t_0
    else if (x <= 2d-158) then
        tmp = -log(x) / n
    else if (x <= 2.3d-110) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.45e-199) {
		tmp = t_0;
	} else if (x <= 2e-158) {
		tmp = -Math.log(x) / n;
	} else if (x <= 2.3e-110) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.45e-199:
		tmp = t_0
	elif x <= 2e-158:
		tmp = -math.log(x) / n
	elif x <= 2.3e-110:
		tmp = t_0
	elif x <= 1.0:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 1.45e-199)
		tmp = t_0;
	elseif (x <= 2e-158)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 2.3e-110)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 1.45e-199)
		tmp = t_0;
	elseif (x <= 2e-158)
		tmp = -log(x) / n;
	elseif (x <= 2.3e-110)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = (x - log(x)) / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.45e-199], t$95$0, If[LessEqual[x, 2e-158], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 2.3e-110], t$95$0, If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.45e-199 or 2.00000000000000013e-158 < x < 2.3000000000000001e-110

   1. Initial program 60.5%

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

   2. Taylor expanded in x around 0 60.5%

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

   if 1.45e-199 < x < 2.00000000000000013e-158

   1. Initial program 39.1%

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

   2. Taylor expanded in x around 0 39.1%

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

   3. Taylor expanded in n around inf 58.9%

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

      1. neg-mul-1 58.9%

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

      2. distribute-neg-frac 58.9%

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

   5. Simplified 58.9%

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

   if 2.3000000000000001e-110 < x < 1

   1. Initial program 32.1%

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

   2. Taylor expanded in n around inf 54.2%

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

      1. +-rgt-identity 54.2%

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

      2. +-rgt-identity 54.2%

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

      3. log1p-def 54.2%

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

   4. Simplified 54.2%

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

   5. Taylor expanded in x around 0 51.5%

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

      1. neg-mul-1 51.5%

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

      2. unsub-neg 51.5%

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

   7. Simplified 51.5%

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

   if 1 < x

   1. Initial program 61.6%

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

   2. Taylor expanded in n around inf 62.2%

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

      1. +-rgt-identity 62.2%

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

      2. +-rgt-identity 62.2%

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

      3. log1p-def 62.2%

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

   4. Simplified 62.2%

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

   5. Taylor expanded in x around inf 66.8%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-199}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-158}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-110}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 12: 56.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0) (/ (- x (log x)) n) (/ (/ 1.0 x) n)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 46.9%

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

   2. Taylor expanded in n around inf 48.2%

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

      1. +-rgt-identity 48.2%

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

      2. +-rgt-identity 48.2%

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

      3. log1p-def 48.2%

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

   4. Simplified 48.2%

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

   5. Taylor expanded in x around 0 47.2%

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

      1. neg-mul-1 47.2%

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

      2. unsub-neg 47.2%

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

   7. Simplified 47.2%

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

   if 1 < x

   1. Initial program 61.6%

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

   2. Taylor expanded in n around inf 62.2%

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

      1. +-rgt-identity 62.2%

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

      2. +-rgt-identity 62.2%

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

      3. log1p-def 62.2%

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

   4. Simplified 62.2%

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

   5. Taylor expanded in x around inf 66.8%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 13: 56.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.55) (/ (- (log x)) n) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -log(x) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.55d0) then
        tmp = -log(x) / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.55:
		tmp = -math.log(x) / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.55)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.55)
		tmp = -log(x) / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.55], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.55000000000000004

   1. Initial program 46.9%

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

   2. Taylor expanded in x around 0 45.1%

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

   3. Taylor expanded in n around inf 46.4%

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

      1. neg-mul-1 46.4%

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

      2. distribute-neg-frac 46.4%

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

   5. Simplified 46.4%

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

   if 0.55000000000000004 < x

   1. Initial program 61.6%

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

   2. Taylor expanded in n around inf 62.2%

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

      1. +-rgt-identity 62.2%

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

      2. +-rgt-identity 62.2%

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

      3. log1p-def 62.2%

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

   4. Simplified 62.2%

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

   5. Taylor expanded in x around inf 66.8%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 14: 39.8% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{n \cdot x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* n x)))

C

double code(double x, double n) {
	return 1.0 / (n * x);
}

Fortran

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

Java

public static double code(double x, double n) {
	return 1.0 / (n * x);
}

Python

def code(x, n):
	return 1.0 / (n * x)

Julia

function code(x, n)
	return Float64(1.0 / Float64(n * x))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (n * x);
end

Wolfram

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

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in n around inf 53.5%

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

   1. +-rgt-identity 53.5%

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

   2. +-rgt-identity 53.5%

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

   3. log1p-def 53.5%

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

4. Simplified 53.5%

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

5. Taylor expanded in x around inf 41.0%

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

   1. *-commutative 41.0%

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

7. Simplified 41.0%

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

8. Final simplification 41.0%

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

Alternative 15: 40.3% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n}}{x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 n) x))

C

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

Fortran

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

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in x around inf 55.5%

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

   1. mul-1-neg 55.5%

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

   2. log-rec 55.5%

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

   3. mul-1-neg 55.5%

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

   4. distribute-neg-frac 55.5%

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

   5. mul-1-neg 55.5%

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

   6. remove-double-neg 55.5%

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

   7. *-commutative 55.5%

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

4. Simplified 55.5%

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

5. Taylor expanded in n around inf 41.0%

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

   1. associate-/r* 41.5%

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

7. Simplified 41.5%

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

8. Final simplification 41.5%

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

Alternative 16: 40.3% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 x) n))

C

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

Fortran

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

Java

public static double code(double x, double n) {
	return (1.0 / x) / n;
}

Python

def code(x, n):
	return (1.0 / x) / n

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in n around inf 53.5%

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

   1. +-rgt-identity 53.5%

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

   2. +-rgt-identity 53.5%

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

   3. log1p-def 53.5%

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

4. Simplified 53.5%

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

5. Taylor expanded in x around inf 41.5%

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

6. Final simplification 41.5%

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

Alternative 17: 4.5% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ x n))

C

double code(double x, double n) {
	return x / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = x / n
end function

Java

public static double code(double x, double n) {
	return x / n;
}

Python

def code(x, n):
	return x / n

Julia

function code(x, n)
	return Float64(x / n)
end

MATLAB

function tmp = code(x, n)
	tmp = x / n;
end

Wolfram

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

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in x around 0 32.6%

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

3. Taylor expanded in x around inf 4.5%

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

4. Final simplification 4.5%

   \[\leadsto \frac{x}{n} \]

Reproduce

herbie shell --seed 2023311 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))