2nthrt (problem 3.4.6)

Percentage Accurate: 52.4% → 85.9%

Time: 20.9s

Alternatives: 19

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: 52.4% 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: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-14)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 4e-18)
       (fma
        0.5
        (/ (pow (log1p x) 2.0) (* n n))
        (- (/ (- (log1p x) (log x)) n) (* 0.5 (/ (pow (log x) 2.0) (* n n)))))
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-14) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 4e-18) {
		tmp = fma(0.5, (pow(log1p(x), 2.0) / (n * n)), (((log1p(x) - log(x)) / n) - (0.5 * (pow(log(x), 2.0) / (n * n)))));
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-14)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 4e-18)
		tmp = fma(0.5, Float64((log1p(x) ^ 2.0) / Float64(n * n)), Float64(Float64(Float64(log1p(x) - log(x)) / n) - Float64(0.5 * Float64((log(x) ^ 2.0) / Float64(n * n)))));
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-14], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-18], N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] - N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2e-14

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -2e-14 < (/.f64 1 n) < 4.0000000000000003e-18

   1. Initial program 30.7%

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

   2. Taylor expanded in n around inf 77.7%

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

      1. associate--l+ 71.7%

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

      2. fma-def 71.7%

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

      3. log1p-def 71.7%

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

      4. unpow2 71.7%

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

      5. associate--r+ 77.7%

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

      6. +-rgt-identity 77.7%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 77.7%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 77.7%

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

      9. log1p-def 77.7%

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

      10. unpow2 77.7%

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

   4. Simplified 77.7%

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

   if 4.0000000000000003e-18 < (/.f64 1 n)

   1. Initial program 53.0%

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

   2. Taylor expanded in n around 0 53.0%

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

      1. log1p-def 94.6%

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

   4. Simplified 94.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\mathsf{log1p}\left(x\right) - \log x}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-14)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 4e-18)
       (/ (- (log1p x) (log x)) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-14) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 4e-18) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-14) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 4e-18) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-14:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 4e-18:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-14)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 4e-18)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-14], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-18], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2e-14

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -2e-14 < (/.f64 1 n) < 4.0000000000000003e-18

   1. Initial program 30.7%

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

   2. Taylor expanded in n around inf 77.6%

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

      1. +-rgt-identity 77.6%

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

      2. +-rgt-identity 77.6%

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

      3. log1p-def 77.6%

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

   4. Simplified 77.6%

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

   if 4.0000000000000003e-18 < (/.f64 1 n)

   1. Initial program 53.0%

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

   2. Taylor expanded in n around 0 53.0%

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

      1. log1p-def 94.6%

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

   4. Simplified 94.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 3: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-14)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e-7)
       (/ (- (log1p x) (log x)) n)
       (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-14) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-7) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-14) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-7) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-14:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e-7:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-14)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-7)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-14], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-7], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2e-14

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -2e-14 < (/.f64 1 n) < 9.9999999999999995e-8

   1. Initial program 30.3%

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

   2. Taylor expanded in n around inf 76.5%

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

      1. +-rgt-identity 76.5%

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

      2. +-rgt-identity 76.5%

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

      3. log1p-def 76.5%

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

   4. Simplified 76.5%

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

   if 9.9999999999999995e-8 < (/.f64 1 n)

   1. Initial program 55.9%

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

   2. Taylor expanded in n around 0 55.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 4: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right) + \left(1 + \frac{x}{n}\right)\right) - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-14)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 1e-7)
       (/ (- (log1p x) (log x)) n)
       (-
        (+ (* (- (/ 0.5 (* n n)) (/ 0.5 n)) (* x x)) (+ 1.0 (/ x n)))
        t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-14) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-7) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (x / n))) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e-14) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 1e-7) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (x / n))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e-14:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 1e-7:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (x / n))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-14)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-7)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)) * Float64(x * x)) + Float64(1.0 + Float64(x / n))) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-14], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-7], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 n) < -2e-14

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -2e-14 < (/.f64 1 n) < 9.9999999999999995e-8

   1. Initial program 30.3%

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

   2. Taylor expanded in n around inf 76.5%

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

      1. +-rgt-identity 76.5%

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

      2. +-rgt-identity 76.5%

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

      3. log1p-def 76.5%

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

   4. Simplified 76.5%

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

   if 9.9999999999999995e-8 < (/.f64 1 n)

   1. Initial program 55.9%

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

   2. Taylor expanded in x around 0 67.5%

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

      1. associate-+r+ 67.5%

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

      2. +-commutative 67.5%

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

      3. associate-*r/ 67.5%

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

      4. metadata-eval 67.5%

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

      5. unpow2 67.5%

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

      6. associate-*r/ 67.5%

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

      7. metadata-eval 67.5%

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

      8. unpow2 67.5%

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

   4. Simplified 67.5%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-7}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right) + \left(1 + \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 5: 65.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -4.2 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -98000000000000:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;n \leq 8800000:\\ \;\;\;\;\left(\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right) + \left(1 + \frac{x}{n}\right)\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= n -4.2e+112)
     (/
      (+ (/ 1.0 x) (+ (/ 0.3333333333333333 (pow x 3.0)) (/ -0.5 (* x x))))
      n)
     (if (<= n -98000000000000.0)
       (/ (- x (log x)) n)
       (if (<= n -5e-309)
         (/ t_0 (* n x))
         (if (<= n 8800000.0)
           (-
            (+ (* (- (/ 0.5 (* n n)) (/ 0.5 n)) (* x x)) (+ 1.0 (/ x n)))
            t_0)
           (/ (+ (/ 1.0 n) (/ (log x) (* n n))) x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (n <= -4.2e+112) {
		tmp = ((1.0 / x) + ((0.3333333333333333 / pow(x, 3.0)) + (-0.5 / (x * x)))) / n;
	} else if (n <= -98000000000000.0) {
		tmp = (x - log(x)) / n;
	} else if (n <= -5e-309) {
		tmp = t_0 / (n * x);
	} else if (n <= 8800000.0) {
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (x / n))) - t_0;
	} else {
		tmp = ((1.0 / n) + (log(x) / (n * n))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (n <= (-4.2d+112)) then
        tmp = ((1.0d0 / x) + ((0.3333333333333333d0 / (x ** 3.0d0)) + ((-0.5d0) / (x * x)))) / n
    else if (n <= (-98000000000000.0d0)) then
        tmp = (x - log(x)) / n
    else if (n <= (-5d-309)) then
        tmp = t_0 / (n * x)
    else if (n <= 8800000.0d0) then
        tmp = ((((0.5d0 / (n * n)) - (0.5d0 / n)) * (x * x)) + (1.0d0 + (x / n))) - t_0
    else
        tmp = ((1.0d0 / n) + (log(x) / (n * 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 tmp;
	if (n <= -4.2e+112) {
		tmp = ((1.0 / x) + ((0.3333333333333333 / Math.pow(x, 3.0)) + (-0.5 / (x * x)))) / n;
	} else if (n <= -98000000000000.0) {
		tmp = (x - Math.log(x)) / n;
	} else if (n <= -5e-309) {
		tmp = t_0 / (n * x);
	} else if (n <= 8800000.0) {
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (x / n))) - t_0;
	} else {
		tmp = ((1.0 / n) + (Math.log(x) / (n * n))) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if n <= -4.2e+112:
		tmp = ((1.0 / x) + ((0.3333333333333333 / math.pow(x, 3.0)) + (-0.5 / (x * x)))) / n
	elif n <= -98000000000000.0:
		tmp = (x - math.log(x)) / n
	elif n <= -5e-309:
		tmp = t_0 / (n * x)
	elif n <= 8800000.0:
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (x / n))) - t_0
	else:
		tmp = ((1.0 / n) + (math.log(x) / (n * n))) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (n <= -4.2e+112)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(-0.5 / Float64(x * x)))) / n);
	elseif (n <= -98000000000000.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (n <= -5e-309)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (n <= 8800000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 / Float64(n * n)) - Float64(0.5 / n)) * Float64(x * x)) + Float64(1.0 + Float64(x / n))) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(log(x) / Float64(n * n))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (n <= -4.2e+112)
		tmp = ((1.0 / x) + ((0.3333333333333333 / (x ^ 3.0)) + (-0.5 / (x * x)))) / n;
	elseif (n <= -98000000000000.0)
		tmp = (x - log(x)) / n;
	elseif (n <= -5e-309)
		tmp = t_0 / (n * x);
	elseif (n <= 8800000.0)
		tmp = ((((0.5 / (n * n)) - (0.5 / n)) * (x * x)) + (1.0 + (x / n))) - t_0;
	else
		tmp = ((1.0 / n) + (log(x) / (n * n))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -4.2e+112], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -98000000000000.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -5e-309], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8800000.0], N[(N[(N[(N[(N[(0.5 / N[(n * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -4.1999999999999998e112

   1. Initial program 42.1%

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

   2. Taylor expanded in n around inf 77.9%

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

      1. +-rgt-identity 77.9%

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

      2. +-rgt-identity 77.9%

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

      3. log1p-def 77.9%

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

   4. Simplified 77.9%

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

   5. Taylor expanded in x around inf 67.9%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. sub-neg 67.9%

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

      2. +-commutative 67.9%

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

      3. associate-+l+ 67.9%

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

      4. associate-*r/ 67.9%

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

      5. metadata-eval 67.9%

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

      6. associate-*r/ 67.9%

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

      7. metadata-eval 67.9%

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

      8. distribute-neg-frac 67.9%

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

      9. metadata-eval 67.9%

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

      10. unpow2 67.9%

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

   7. Simplified 67.9%

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

   if -4.1999999999999998e112 < n < -9.8e13

   1. Initial program 9.5%

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

   2. Taylor expanded in n around inf 71.3%

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

      1. +-rgt-identity 71.3%

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

      2. +-rgt-identity 71.3%

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

      3. log1p-def 71.3%

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

   4. Simplified 71.3%

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

   5. Taylor expanded in x around 0 64.5%

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

      1. mul-1-neg 64.5%

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

      2. sub-neg 64.5%

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

      3. div-sub 64.5%

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

   7. Simplified 64.5%

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

   if -9.8e13 < n < -4.9999999999999995e-309

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -4.9999999999999995e-309 < n < 8.8e6

   1. Initial program 55.9%

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

   2. Taylor expanded in x around 0 67.5%

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

      1. associate-+r+ 67.5%

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

      2. +-commutative 67.5%

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

      3. associate-*r/ 67.5%

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

      4. metadata-eval 67.5%

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

      5. unpow2 67.5%

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

      6. associate-*r/ 67.5%

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

      7. metadata-eval 67.5%

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

      8. unpow2 67.5%

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

   4. Simplified 67.5%

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

   if 8.8e6 < n

   1. Initial program 34.0%

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

   2. Taylor expanded in n around inf 78.0%

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

      1. associate--l+ 70.5%

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

      2. fma-def 70.5%

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

      3. log1p-def 70.5%

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

      4. unpow2 70.5%

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

      5. associate--r+ 78.0%

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

      6. +-rgt-identity 78.0%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 78.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 78.1%

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

      9. log1p-def 78.1%

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

      10. unpow2 78.1%

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

   4. Simplified 78.1%

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

   5. Taylor expanded in x around inf 55.6%

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

      1. mul-1-neg 55.6%

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

      2. distribute-neg-frac 55.6%

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

      3. log-rec 55.6%

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

      4. remove-double-neg 55.6%

         \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

      5. unpow2 55.6%

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

   7. Simplified 55.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -98000000000000:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;n \leq 8800000:\\ \;\;\;\;\left(\left(\frac{0.5}{n \cdot n} - \frac{0.5}{n}\right) \cdot \left(x \cdot x\right) + \left(1 + \frac{x}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}\\ \end{array} \]

Alternative 6: 66.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -2.9 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -4.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_1}{n \cdot x}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-100}:\\ \;\;\;\;\sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{elif}\;n \leq 13500000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x))) (t_1 (pow x (/ 1.0 n))))
   (if (<= n -2.9e+112)
     (/
      (+ (/ 1.0 x) (+ (/ 0.3333333333333333 (pow x 3.0)) (/ -0.5 (* x x))))
      n)
     (if (<= n -4.4e+14)
       (/ (- x (log x)) n)
       (if (<= n -1e-310)
         (/ t_1 (* n x))
         (if (<= n 2.4e-100)
           (cbrt (* t_0 (* t_0 t_0)))
           (if (<= n 13500000.0)
             (- (+ 1.0 (/ x n)) t_1)
             (/ (+ (/ 1.0 n) (/ (log x) (* n n))) x))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if (n <= -2.9e+112) {
		tmp = ((1.0 / x) + ((0.3333333333333333 / pow(x, 3.0)) + (-0.5 / (x * x)))) / n;
	} else if (n <= -4.4e+14) {
		tmp = (x - log(x)) / n;
	} else if (n <= -1e-310) {
		tmp = t_1 / (n * x);
	} else if (n <= 2.4e-100) {
		tmp = cbrt((t_0 * (t_0 * t_0)));
	} else if (n <= 13500000.0) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = ((1.0 / n) + (log(x) / (n * n))) / x;
	}
	return tmp;
}

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 tmp;
	if (n <= -2.9e+112) {
		tmp = ((1.0 / x) + ((0.3333333333333333 / Math.pow(x, 3.0)) + (-0.5 / (x * x)))) / n;
	} else if (n <= -4.4e+14) {
		tmp = (x - Math.log(x)) / n;
	} else if (n <= -1e-310) {
		tmp = t_1 / (n * x);
	} else if (n <= 2.4e-100) {
		tmp = Math.cbrt((t_0 * (t_0 * t_0)));
	} else if (n <= 13500000.0) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = ((1.0 / n) + (Math.log(x) / (n * n))) / x;
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (n <= -2.9e+112)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(-0.5 / Float64(x * x)))) / n);
	elseif (n <= -4.4e+14)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (n <= -1e-310)
		tmp = Float64(t_1 / Float64(n * x));
	elseif (n <= 2.4e-100)
		tmp = cbrt(Float64(t_0 * Float64(t_0 * t_0)));
	elseif (n <= 13500000.0)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(log(x) / Float64(n * n))) / x);
	end
	return 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]}, If[LessEqual[n, -2.9e+112], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -4.4e+14], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -1e-310], N[(t$95$1 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.4e-100], N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[n, 13500000.0], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if n < -2.9000000000000002e112

   1. Initial program 42.1%

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

   2. Taylor expanded in n around inf 77.9%

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

      1. +-rgt-identity 77.9%

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

      2. +-rgt-identity 77.9%

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

      3. log1p-def 77.9%

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

   4. Simplified 77.9%

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

   5. Taylor expanded in x around inf 67.9%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. sub-neg 67.9%

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

      2. +-commutative 67.9%

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

      3. associate-+l+ 67.9%

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

      4. associate-*r/ 67.9%

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

      5. metadata-eval 67.9%

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

      6. associate-*r/ 67.9%

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

      7. metadata-eval 67.9%

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

      8. distribute-neg-frac 67.9%

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

      9. metadata-eval 67.9%

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

      10. unpow2 67.9%

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

   7. Simplified 67.9%

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

   if -2.9000000000000002e112 < n < -4.4e14

   1. Initial program 9.5%

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

   2. Taylor expanded in n around inf 71.3%

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

      1. +-rgt-identity 71.3%

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

      2. +-rgt-identity 71.3%

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

      3. log1p-def 71.3%

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

   4. Simplified 71.3%

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

   5. Taylor expanded in x around 0 64.5%

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

      1. mul-1-neg 64.5%

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

      2. sub-neg 64.5%

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

      3. div-sub 64.5%

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

   7. Simplified 64.5%

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

   if -4.4e14 < n < -9.999999999999969e-311

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -9.999999999999969e-311 < n < 2.4000000000000003e-100

   1. Initial program 36.8%

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

   2. Taylor expanded in n around inf 0.1%

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

      1. associate--l+ 0.1%

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

      2. fma-def 0.1%

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

      3. log1p-def 0.1%

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

      4. unpow2 0.1%

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

      5. associate--r+ 0.1%

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

      6. +-rgt-identity 0.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 0.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 0.1%

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

      9. log1p-def 0.1%

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

      10. unpow2 0.1%

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

   4. Simplified 0.1%

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

   5. Taylor expanded in x around inf 0.2%

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

      1. mul-1-neg 0.2%

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

      2. distribute-neg-frac 0.2%

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

      3. log-rec 0.2%

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

      4. remove-double-neg 0.2%

         \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

      5. unpow2 0.2%

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

   7. Simplified 0.2%

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

   8. Taylor expanded in n around inf 37.9%

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

      1. associate-/l/ 37.9%

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

      2. add-cbrt-cube 70.5%

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

   10. Applied egg-rr 70.5%

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

   if 2.4000000000000003e-100 < n < 1.35e7

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 90.3%

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

   if 1.35e7 < n

   1. Initial program 34.0%

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

   2. Taylor expanded in n around inf 78.0%

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

      1. associate--l+ 70.5%

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

      2. fma-def 70.5%

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

      3. log1p-def 70.5%

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

      4. unpow2 70.5%

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

      5. associate--r+ 78.0%

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

      6. +-rgt-identity 78.0%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 78.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 78.1%

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

      9. log1p-def 78.1%

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

      10. unpow2 78.1%

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

   4. Simplified 78.1%

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

   5. Taylor expanded in x around inf 55.6%

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

      1. mul-1-neg 55.6%

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

      2. distribute-neg-frac 55.6%

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

      3. log-rec 55.6%

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

      4. remove-double-neg 55.6%

         \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

      5. unpow2 55.6%

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

   7. Simplified 55.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.9 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -4.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{-100}:\\ \;\;\;\;\sqrt[3]{\frac{1}{n \cdot x} \cdot \left(\frac{1}{n \cdot x} \cdot \frac{1}{n \cdot x}\right)}\\ \mathbf{elif}\;n \leq 13500000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}\\ \end{array} \]

Alternative 7: 65.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -8 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \mathbf{elif}\;n \leq -68000000000000:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-227}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq 1850000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= n -8e+150)
     (/ (- (/ 1.0 x) (/ (/ 0.5 x) x)) n)
     (if (<= n -68000000000000.0)
       (/ (- x (log x)) n)
       (if (<= n -1e-310)
         (/ t_0 (* n x))
         (if (<= n 3.1e-227)
           (/ 1.0 (* n x))
           (if (<= n 1850000.0)
             (- (+ 1.0 (/ x n)) t_0)
             (/ (+ (/ 1.0 n) (/ (log x) (* n n))) x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (n <= -8e+150) {
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	} else if (n <= -68000000000000.0) {
		tmp = (x - log(x)) / n;
	} else if (n <= -1e-310) {
		tmp = t_0 / (n * x);
	} else if (n <= 3.1e-227) {
		tmp = 1.0 / (n * x);
	} else if (n <= 1850000.0) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((1.0 / n) + (log(x) / (n * n))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (n <= (-8d+150)) then
        tmp = ((1.0d0 / x) - ((0.5d0 / x) / x)) / n
    else if (n <= (-68000000000000.0d0)) then
        tmp = (x - log(x)) / n
    else if (n <= (-1d-310)) then
        tmp = t_0 / (n * x)
    else if (n <= 3.1d-227) then
        tmp = 1.0d0 / (n * x)
    else if (n <= 1850000.0d0) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = ((1.0d0 / n) + (log(x) / (n * 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 tmp;
	if (n <= -8e+150) {
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	} else if (n <= -68000000000000.0) {
		tmp = (x - Math.log(x)) / n;
	} else if (n <= -1e-310) {
		tmp = t_0 / (n * x);
	} else if (n <= 3.1e-227) {
		tmp = 1.0 / (n * x);
	} else if (n <= 1850000.0) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((1.0 / n) + (Math.log(x) / (n * n))) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if n <= -8e+150:
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n
	elif n <= -68000000000000.0:
		tmp = (x - math.log(x)) / n
	elif n <= -1e-310:
		tmp = t_0 / (n * x)
	elif n <= 3.1e-227:
		tmp = 1.0 / (n * x)
	elif n <= 1850000.0:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = ((1.0 / n) + (math.log(x) / (n * n))) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (n <= -8e+150)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(Float64(0.5 / x) / x)) / n);
	elseif (n <= -68000000000000.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (n <= -1e-310)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (n <= 3.1e-227)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (n <= 1850000.0)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(log(x) / Float64(n * n))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (n <= -8e+150)
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	elseif (n <= -68000000000000.0)
		tmp = (x - log(x)) / n;
	elseif (n <= -1e-310)
		tmp = t_0 / (n * x);
	elseif (n <= 3.1e-227)
		tmp = 1.0 / (n * x);
	elseif (n <= 1850000.0)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = ((1.0 / n) + (log(x) / (n * n))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -8e+150], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -68000000000000.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -1e-310], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.1e-227], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1850000.0], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if n < -7.99999999999999985e150

   1. Initial program 47.7%

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

   2. Taylor expanded in n around inf 80.5%

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

      1. +-rgt-identity 80.5%

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

      2. +-rgt-identity 80.5%

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

      3. log1p-def 80.5%

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

   4. Simplified 80.5%

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

   5. Taylor expanded in x around inf 70.4%

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

      1. associate-/l/ 70.5%

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

      2. associate-*r/ 70.5%

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

      3. metadata-eval 70.5%

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

      4. associate-/l/ 70.5%

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

      5. metadata-eval 70.5%

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

      6. associate-*r/ 70.5%

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

      7. div-sub 70.5%

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

      8. associate-*r/ 70.5%

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

      9. metadata-eval 70.5%

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

      10. unpow2 70.5%

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

      11. associate-/r* 70.5%

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

   7. Simplified 70.5%

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

   if -7.99999999999999985e150 < n < -6.8e13

   1. Initial program 11.3%

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

   2. Taylor expanded in n around inf 70.8%

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

      1. +-rgt-identity 70.8%

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

      2. +-rgt-identity 70.8%

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

      3. log1p-def 70.8%

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

   4. Simplified 70.8%

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

   5. Taylor expanded in x around 0 62.4%

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

      1. mul-1-neg 62.4%

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

      2. sub-neg 62.4%

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

      3. div-sub 62.4%

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

   7. Simplified 62.4%

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

   if -6.8e13 < n < -9.999999999999969e-311

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -9.999999999999969e-311 < n < 3.09999999999999979e-227

   1. Initial program 3.1%

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

   2. Taylor expanded in x around inf 0.2%

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

      1. log-rec 0.2%

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

      2. mul-1-neg 0.2%

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

      3. associate-*r/ 0.2%

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

      4. neg-mul-1 0.2%

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

      5. mul-1-neg 0.2%

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

      6. remove-double-neg 0.2%

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

      7. *-commutative 0.2%

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

   4. Simplified 0.2%

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

   5. Taylor expanded in n 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%

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

   if 3.09999999999999979e-227 < n < 1.85e6

   1. Initial program 70.1%

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

   2. Taylor expanded in x around 0 63.0%

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

   if 1.85e6 < n

   1. Initial program 34.0%

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

   2. Taylor expanded in n around inf 78.0%

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

      1. associate--l+ 70.5%

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

      2. fma-def 70.5%

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

      3. log1p-def 70.5%

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

      4. unpow2 70.5%

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

      5. associate--r+ 78.0%

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

      6. +-rgt-identity 78.0%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 78.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 78.1%

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

      9. log1p-def 78.1%

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

      10. unpow2 78.1%

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

   4. Simplified 78.1%

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

   5. Taylor expanded in x around inf 55.6%

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

      1. mul-1-neg 55.6%

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

      2. distribute-neg-frac 55.6%

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

      3. log-rec 55.6%

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

      4. remove-double-neg 55.6%

         \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

      5. unpow2 55.6%

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

   7. Simplified 55.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \mathbf{elif}\;n \leq -68000000000000:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{-227}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq 1850000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}\\ \end{array} \]

Alternative 8: 65.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -68000000000000:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_0}{n \cdot x}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-228}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq 13500000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= n -9.5e+106)
     (/
      (+ (/ 1.0 x) (+ (/ 0.3333333333333333 (pow x 3.0)) (/ -0.5 (* x x))))
      n)
     (if (<= n -68000000000000.0)
       (/ (- x (log x)) n)
       (if (<= n -1e-310)
         (/ t_0 (* n x))
         (if (<= n 1.15e-228)
           (/ 1.0 (* n x))
           (if (<= n 13500000.0)
             (- (+ 1.0 (/ x n)) t_0)
             (/ (+ (/ 1.0 n) (/ (log x) (* n n))) x))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (n <= -9.5e+106) {
		tmp = ((1.0 / x) + ((0.3333333333333333 / pow(x, 3.0)) + (-0.5 / (x * x)))) / n;
	} else if (n <= -68000000000000.0) {
		tmp = (x - log(x)) / n;
	} else if (n <= -1e-310) {
		tmp = t_0 / (n * x);
	} else if (n <= 1.15e-228) {
		tmp = 1.0 / (n * x);
	} else if (n <= 13500000.0) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((1.0 / n) + (log(x) / (n * n))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (n <= (-9.5d+106)) then
        tmp = ((1.0d0 / x) + ((0.3333333333333333d0 / (x ** 3.0d0)) + ((-0.5d0) / (x * x)))) / n
    else if (n <= (-68000000000000.0d0)) then
        tmp = (x - log(x)) / n
    else if (n <= (-1d-310)) then
        tmp = t_0 / (n * x)
    else if (n <= 1.15d-228) then
        tmp = 1.0d0 / (n * x)
    else if (n <= 13500000.0d0) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = ((1.0d0 / n) + (log(x) / (n * 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 tmp;
	if (n <= -9.5e+106) {
		tmp = ((1.0 / x) + ((0.3333333333333333 / Math.pow(x, 3.0)) + (-0.5 / (x * x)))) / n;
	} else if (n <= -68000000000000.0) {
		tmp = (x - Math.log(x)) / n;
	} else if (n <= -1e-310) {
		tmp = t_0 / (n * x);
	} else if (n <= 1.15e-228) {
		tmp = 1.0 / (n * x);
	} else if (n <= 13500000.0) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((1.0 / n) + (Math.log(x) / (n * n))) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if n <= -9.5e+106:
		tmp = ((1.0 / x) + ((0.3333333333333333 / math.pow(x, 3.0)) + (-0.5 / (x * x)))) / n
	elif n <= -68000000000000.0:
		tmp = (x - math.log(x)) / n
	elif n <= -1e-310:
		tmp = t_0 / (n * x)
	elif n <= 1.15e-228:
		tmp = 1.0 / (n * x)
	elif n <= 13500000.0:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = ((1.0 / n) + (math.log(x) / (n * n))) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (n <= -9.5e+106)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(-0.5 / Float64(x * x)))) / n);
	elseif (n <= -68000000000000.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (n <= -1e-310)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (n <= 1.15e-228)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (n <= 13500000.0)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(log(x) / Float64(n * n))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (n <= -9.5e+106)
		tmp = ((1.0 / x) + ((0.3333333333333333 / (x ^ 3.0)) + (-0.5 / (x * x)))) / n;
	elseif (n <= -68000000000000.0)
		tmp = (x - log(x)) / n;
	elseif (n <= -1e-310)
		tmp = t_0 / (n * x);
	elseif (n <= 1.15e-228)
		tmp = 1.0 / (n * x);
	elseif (n <= 13500000.0)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = ((1.0 / n) + (log(x) / (n * n))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -9.5e+106], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -68000000000000.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -1e-310], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.15e-228], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 13500000.0], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if n < -9.4999999999999995e106

   1. Initial program 42.1%

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

   2. Taylor expanded in n around inf 77.9%

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

      1. +-rgt-identity 77.9%

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

      2. +-rgt-identity 77.9%

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

      3. log1p-def 77.9%

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

   4. Simplified 77.9%

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

   5. Taylor expanded in x around inf 67.9%

      \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - 0.5 \cdot \frac{1}{{x}^{2}}}}{n} \]
   6. Step-by-step derivation

      1. sub-neg 67.9%

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

      2. +-commutative 67.9%

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

      3. associate-+l+ 67.9%

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

      4. associate-*r/ 67.9%

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

      5. metadata-eval 67.9%

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

      6. associate-*r/ 67.9%

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

      7. metadata-eval 67.9%

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

      8. distribute-neg-frac 67.9%

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

      9. metadata-eval 67.9%

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

      10. unpow2 67.9%

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

   7. Simplified 67.9%

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

   if -9.4999999999999995e106 < n < -6.8e13

   1. Initial program 9.5%

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

   2. Taylor expanded in n around inf 71.3%

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

      1. +-rgt-identity 71.3%

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

      2. +-rgt-identity 71.3%

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

      3. log1p-def 71.3%

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

   4. Simplified 71.3%

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

   5. Taylor expanded in x around 0 64.5%

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

      1. mul-1-neg 64.5%

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

      2. sub-neg 64.5%

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

      3. div-sub 64.5%

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

   7. Simplified 64.5%

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

   if -6.8e13 < n < -9.999999999999969e-311

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -9.999999999999969e-311 < n < 1.1499999999999999e-228

   1. Initial program 3.1%

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

   2. Taylor expanded in x around inf 0.2%

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

      1. log-rec 0.2%

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

      2. mul-1-neg 0.2%

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

      3. associate-*r/ 0.2%

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

      4. neg-mul-1 0.2%

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

      5. mul-1-neg 0.2%

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

      6. remove-double-neg 0.2%

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

      7. *-commutative 0.2%

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

   4. Simplified 0.2%

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

   5. Taylor expanded in n 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%

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

   if 1.1499999999999999e-228 < n < 1.35e7

   1. Initial program 70.1%

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

   2. Taylor expanded in x around 0 63.0%

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

   if 1.35e7 < n

   1. Initial program 34.0%

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

   2. Taylor expanded in n around inf 78.0%

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

      1. associate--l+ 70.5%

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

      2. fma-def 70.5%

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

      3. log1p-def 70.5%

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

      4. unpow2 70.5%

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

      5. associate--r+ 78.0%

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

      6. +-rgt-identity 78.0%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 78.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 78.1%

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

      9. log1p-def 78.1%

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

      10. unpow2 78.1%

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

   4. Simplified 78.1%

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

   5. Taylor expanded in x around inf 55.6%

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

      1. mul-1-neg 55.6%

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

      2. distribute-neg-frac 55.6%

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

      3. log-rec 55.6%

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

      4. remove-double-neg 55.6%

         \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

      5. unpow2 55.6%

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

   7. Simplified 55.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{1}{x} + \left(\frac{0.3333333333333333}{{x}^{3}} + \frac{-0.5}{x \cdot x}\right)}{n}\\ \mathbf{elif}\;n \leq -68000000000000:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-228}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq 13500000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\log x}{n \cdot n}}{x}\\ \end{array} \]

Alternative 9: 60.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 6 \cdot 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-248}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-233}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25}{n}}{{x}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (- (log x)) n)))
   (if (<= x 6e-296)
     t_0
     (if (<= x 3.35e-248)
       t_1
       (if (<= x 5.1e-233)
         t_0
         (if (<= x 1.8e-190)
           t_1
           (if (<= x 4.9e-156)
             t_0
             (if (<= x 0.98)
               (/ (- x (log x)) n)
               (if (<= x 2.7e+136)
                 (/ (- (/ 1.0 x) (/ (/ 0.5 x) x)) n)
                 (/ (/ -0.25 n) (pow x 4.0)))))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = -log(x) / n;
	double tmp;
	if (x <= 6e-296) {
		tmp = t_0;
	} else if (x <= 3.35e-248) {
		tmp = t_1;
	} else if (x <= 5.1e-233) {
		tmp = t_0;
	} else if (x <= 1.8e-190) {
		tmp = t_1;
	} else if (x <= 4.9e-156) {
		tmp = t_0;
	} else if (x <= 0.98) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.7e+136) {
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	} else {
		tmp = (-0.25 / n) / pow(x, 4.0);
	}
	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 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = -log(x) / n
    if (x <= 6d-296) then
        tmp = t_0
    else if (x <= 3.35d-248) then
        tmp = t_1
    else if (x <= 5.1d-233) then
        tmp = t_0
    else if (x <= 1.8d-190) then
        tmp = t_1
    else if (x <= 4.9d-156) then
        tmp = t_0
    else if (x <= 0.98d0) then
        tmp = (x - log(x)) / n
    else if (x <= 2.7d+136) then
        tmp = ((1.0d0 / x) - ((0.5d0 / x) / x)) / n
    else
        tmp = ((-0.25d0) / n) / (x ** 4.0d0)
    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(x) / n;
	double tmp;
	if (x <= 6e-296) {
		tmp = t_0;
	} else if (x <= 3.35e-248) {
		tmp = t_1;
	} else if (x <= 5.1e-233) {
		tmp = t_0;
	} else if (x <= 1.8e-190) {
		tmp = t_1;
	} else if (x <= 4.9e-156) {
		tmp = t_0;
	} else if (x <= 0.98) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.7e+136) {
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	} else {
		tmp = (-0.25 / n) / Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = -math.log(x) / n
	tmp = 0
	if x <= 6e-296:
		tmp = t_0
	elif x <= 3.35e-248:
		tmp = t_1
	elif x <= 5.1e-233:
		tmp = t_0
	elif x <= 1.8e-190:
		tmp = t_1
	elif x <= 4.9e-156:
		tmp = t_0
	elif x <= 0.98:
		tmp = (x - math.log(x)) / n
	elif x <= 2.7e+136:
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n
	else:
		tmp = (-0.25 / n) / math.pow(x, 4.0)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 6e-296)
		tmp = t_0;
	elseif (x <= 3.35e-248)
		tmp = t_1;
	elseif (x <= 5.1e-233)
		tmp = t_0;
	elseif (x <= 1.8e-190)
		tmp = t_1;
	elseif (x <= 4.9e-156)
		tmp = t_0;
	elseif (x <= 0.98)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.7e+136)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(Float64(0.5 / x) / x)) / n);
	else
		tmp = Float64(Float64(-0.25 / n) / (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = -log(x) / n;
	tmp = 0.0;
	if (x <= 6e-296)
		tmp = t_0;
	elseif (x <= 3.35e-248)
		tmp = t_1;
	elseif (x <= 5.1e-233)
		tmp = t_0;
	elseif (x <= 1.8e-190)
		tmp = t_1;
	elseif (x <= 4.9e-156)
		tmp = t_0;
	elseif (x <= 0.98)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.7e+136)
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	else
		tmp = (-0.25 / n) / (x ^ 4.0);
	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[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 6e-296], t$95$0, If[LessEqual[x, 3.35e-248], t$95$1, If[LessEqual[x, 5.1e-233], t$95$0, If[LessEqual[x, 1.8e-190], t$95$1, If[LessEqual[x, 4.9e-156], t$95$0, If[LessEqual[x, 0.98], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.7e+136], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.25 / n), $MachinePrecision] / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 5.9999999999999995e-296 or 3.35e-248 < x < 5.1000000000000003e-233 or 1.80000000000000003e-190 < x < 4.89999999999999951e-156

   1. Initial program 81.0%

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

   2. Taylor expanded in x around 0 81.0%

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

   if 5.9999999999999995e-296 < x < 3.35e-248 or 5.1000000000000003e-233 < x < 1.80000000000000003e-190

   1. Initial program 36.3%

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

   2. Taylor expanded in x around 0 36.3%

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

   3. Taylor expanded in n around inf 66.8%

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

      1. associate-*r/ 66.8%

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

      2. neg-mul-1 66.8%

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

   5. Simplified 66.8%

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

   if 4.89999999999999951e-156 < x < 0.97999999999999998

   1. Initial program 37.5%

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

   2. Taylor expanded in n around inf 49.1%

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

      1. +-rgt-identity 49.1%

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

      2. +-rgt-identity 49.1%

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

      3. log1p-def 49.1%

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

   4. Simplified 49.1%

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

   5. Taylor expanded in x around 0 47.9%

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

      1. mul-1-neg 47.9%

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

      2. sub-neg 47.9%

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

      3. div-sub 47.9%

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

   7. Simplified 47.9%

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

   if 0.97999999999999998 < x < 2.7000000000000002e136

   1. Initial program 40.1%

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

   2. Taylor expanded in n around inf 43.4%

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

      1. +-rgt-identity 43.4%

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

      2. +-rgt-identity 43.4%

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

      3. log1p-def 43.4%

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

   4. Simplified 43.4%

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

   5. Taylor expanded in x around inf 69.8%

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

      1. associate-/l/ 70.4%

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

      2. associate-*r/ 70.4%

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

      3. metadata-eval 70.4%

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

      4. associate-/l/ 70.4%

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

      5. metadata-eval 70.4%

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

      6. associate-*r/ 70.4%

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

      7. div-sub 70.5%

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

      8. associate-*r/ 70.5%

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

      9. metadata-eval 70.5%

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

      10. unpow2 70.5%

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

      11. associate-/r* 70.5%

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

   7. Simplified 70.5%

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

   if 2.7000000000000002e136 < x

   1. Initial program 90.8%

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

   2. Taylor expanded in n around inf 66.1%

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

      1. associate--l+ 53.6%

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

      2. fma-def 53.6%

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

      3. log1p-def 53.6%

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

      4. unpow2 53.6%

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

      5. associate--r+ 66.1%

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

      6. +-rgt-identity 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 66.1%

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

      9. log1p-def 66.1%

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

      10. unpow2 66.1%

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

   4. Simplified 66.1%

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

   5. Taylor expanded in x around inf 66.1%

      \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - \left(0.25 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]
   6. Step-by-step derivation

      1. +-commutative 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} - \left(0.25 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      2. associate-*r/ 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}}\right) - \left(0.25 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      3. metadata-eval 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \frac{\color{blue}{0.3333333333333333}}{{x}^{3}}\right) - \left(0.25 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      4. +-commutative 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + 0.25 \cdot \frac{1}{{x}^{4}}\right)}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      5. associate-*r/ 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + 0.25 \cdot \frac{1}{{x}^{4}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      6. metadata-eval 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{\color{blue}{0.5}}{{x}^{2}} + 0.25 \cdot \frac{1}{{x}^{4}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      7. unpow2 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{0.5}{\color{blue}{x \cdot x}} + 0.25 \cdot \frac{1}{{x}^{4}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      8. associate-*r/ 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{0.5}{x \cdot x} + \color{blue}{\frac{0.25 \cdot 1}{{x}^{4}}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      9. metadata-eval 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{0.5}{x \cdot x} + \frac{\color{blue}{0.25}}{{x}^{4}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

   7. Simplified 66.1%

      \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\color{blue}{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{0.5}{x \cdot x} + \frac{0.25}{{x}^{4}}\right)}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

   8. Taylor expanded in x around 0 90.8%

      \[\leadsto \color{blue}{\frac{-0.25}{n \cdot {x}^{4}}} \]
   9. Step-by-step derivation

      1. associate-/r* 90.8%

         \[\leadsto \color{blue}{\frac{\frac{-0.25}{n}}{{x}^{4}}} \]

   10. Simplified 90.8%

       \[\leadsto \color{blue}{\frac{\frac{-0.25}{n}}{{x}^{4}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-296}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-248}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-190}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-156}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25}{n}}{{x}^{4}}\\ \end{array} \]

Alternative 10: 59.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 4.7 \cdot 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-250}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-233}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.94:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25}{n}}{{x}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (- (log x)) n)))
   (if (<= x 4.7e-296)
     t_0
     (if (<= x 8.2e-250)
       t_1
       (if (<= x 3.05e-233)
         t_0
         (if (<= x 1.6e-190)
           t_1
           (if (<= x 6.6e-156)
             t_0
             (if (<= x 0.94)
               (- (/ x n) (/ (log x) n))
               (if (<= x 2.7e+136)
                 (/ (- (/ 1.0 x) (/ (/ 0.5 x) x)) n)
                 (/ (/ -0.25 n) (pow x 4.0)))))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = -log(x) / n;
	double tmp;
	if (x <= 4.7e-296) {
		tmp = t_0;
	} else if (x <= 8.2e-250) {
		tmp = t_1;
	} else if (x <= 3.05e-233) {
		tmp = t_0;
	} else if (x <= 1.6e-190) {
		tmp = t_1;
	} else if (x <= 6.6e-156) {
		tmp = t_0;
	} else if (x <= 0.94) {
		tmp = (x / n) - (log(x) / n);
	} else if (x <= 2.7e+136) {
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	} else {
		tmp = (-0.25 / n) / pow(x, 4.0);
	}
	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 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = -log(x) / n
    if (x <= 4.7d-296) then
        tmp = t_0
    else if (x <= 8.2d-250) then
        tmp = t_1
    else if (x <= 3.05d-233) then
        tmp = t_0
    else if (x <= 1.6d-190) then
        tmp = t_1
    else if (x <= 6.6d-156) then
        tmp = t_0
    else if (x <= 0.94d0) then
        tmp = (x / n) - (log(x) / n)
    else if (x <= 2.7d+136) then
        tmp = ((1.0d0 / x) - ((0.5d0 / x) / x)) / n
    else
        tmp = ((-0.25d0) / n) / (x ** 4.0d0)
    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(x) / n;
	double tmp;
	if (x <= 4.7e-296) {
		tmp = t_0;
	} else if (x <= 8.2e-250) {
		tmp = t_1;
	} else if (x <= 3.05e-233) {
		tmp = t_0;
	} else if (x <= 1.6e-190) {
		tmp = t_1;
	} else if (x <= 6.6e-156) {
		tmp = t_0;
	} else if (x <= 0.94) {
		tmp = (x / n) - (Math.log(x) / n);
	} else if (x <= 2.7e+136) {
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	} else {
		tmp = (-0.25 / n) / Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = -math.log(x) / n
	tmp = 0
	if x <= 4.7e-296:
		tmp = t_0
	elif x <= 8.2e-250:
		tmp = t_1
	elif x <= 3.05e-233:
		tmp = t_0
	elif x <= 1.6e-190:
		tmp = t_1
	elif x <= 6.6e-156:
		tmp = t_0
	elif x <= 0.94:
		tmp = (x / n) - (math.log(x) / n)
	elif x <= 2.7e+136:
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n
	else:
		tmp = (-0.25 / n) / math.pow(x, 4.0)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 4.7e-296)
		tmp = t_0;
	elseif (x <= 8.2e-250)
		tmp = t_1;
	elseif (x <= 3.05e-233)
		tmp = t_0;
	elseif (x <= 1.6e-190)
		tmp = t_1;
	elseif (x <= 6.6e-156)
		tmp = t_0;
	elseif (x <= 0.94)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	elseif (x <= 2.7e+136)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(Float64(0.5 / x) / x)) / n);
	else
		tmp = Float64(Float64(-0.25 / n) / (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = -log(x) / n;
	tmp = 0.0;
	if (x <= 4.7e-296)
		tmp = t_0;
	elseif (x <= 8.2e-250)
		tmp = t_1;
	elseif (x <= 3.05e-233)
		tmp = t_0;
	elseif (x <= 1.6e-190)
		tmp = t_1;
	elseif (x <= 6.6e-156)
		tmp = t_0;
	elseif (x <= 0.94)
		tmp = (x / n) - (log(x) / n);
	elseif (x <= 2.7e+136)
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	else
		tmp = (-0.25 / n) / (x ^ 4.0);
	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[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 4.7e-296], t$95$0, If[LessEqual[x, 8.2e-250], t$95$1, If[LessEqual[x, 3.05e-233], t$95$0, If[LessEqual[x, 1.6e-190], t$95$1, If[LessEqual[x, 6.6e-156], t$95$0, If[LessEqual[x, 0.94], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e+136], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(-0.25 / n), $MachinePrecision] / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 4.7e-296 or 8.20000000000000032e-250 < x < 3.0500000000000001e-233 or 1.6e-190 < x < 6.5999999999999997e-156

   1. Initial program 81.0%

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

   2. Taylor expanded in x around 0 81.0%

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

   if 4.7e-296 < x < 8.20000000000000032e-250 or 3.0500000000000001e-233 < x < 1.6e-190

   1. Initial program 36.3%

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

   2. Taylor expanded in x around 0 36.3%

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

   3. Taylor expanded in n around inf 66.8%

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

      1. associate-*r/ 66.8%

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

      2. neg-mul-1 66.8%

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

   5. Simplified 66.8%

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

   if 6.5999999999999997e-156 < x < 0.93999999999999995

   1. Initial program 37.5%

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

   2. Taylor expanded in n around inf 49.1%

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

      1. +-rgt-identity 49.1%

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

      2. +-rgt-identity 49.1%

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

      3. log1p-def 49.1%

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

   4. Simplified 49.1%

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

   5. Taylor expanded in x around 0 47.9%

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

      1. associate-*r/ 47.9%

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

      2. mul-1-neg 47.9%

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

   7. Simplified 47.9%

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

   if 0.93999999999999995 < x < 2.7000000000000002e136

   1. Initial program 40.1%

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

   2. Taylor expanded in n around inf 43.4%

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

      1. +-rgt-identity 43.4%

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

      2. +-rgt-identity 43.4%

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

      3. log1p-def 43.4%

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

   4. Simplified 43.4%

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

   5. Taylor expanded in x around inf 69.8%

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

      1. associate-/l/ 70.4%

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

      2. associate-*r/ 70.4%

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

      3. metadata-eval 70.4%

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

      4. associate-/l/ 70.4%

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

      5. metadata-eval 70.4%

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

      6. associate-*r/ 70.4%

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

      7. div-sub 70.5%

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

      8. associate-*r/ 70.5%

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

      9. metadata-eval 70.5%

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

      10. unpow2 70.5%

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

      11. associate-/r* 70.5%

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

   7. Simplified 70.5%

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

   if 2.7000000000000002e136 < x

   1. Initial program 90.8%

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

   2. Taylor expanded in n around inf 66.1%

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

      1. associate--l+ 53.6%

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

      2. fma-def 53.6%

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

      3. log1p-def 53.6%

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

      4. unpow2 53.6%

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

      5. associate--r+ 66.1%

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

      6. +-rgt-identity 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      7. div-sub 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. +-rgt-identity 66.1%

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

      9. log1p-def 66.1%

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

      10. unpow2 66.1%

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

   4. Simplified 66.1%

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

   5. Taylor expanded in x around inf 66.1%

      \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) - \left(0.25 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]
   6. Step-by-step derivation

      1. +-commutative 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\color{blue}{\left(\frac{1}{x} + 0.3333333333333333 \cdot \frac{1}{{x}^{3}}\right)} - \left(0.25 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      2. associate-*r/ 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{x}^{3}}}\right) - \left(0.25 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      3. metadata-eval 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \frac{\color{blue}{0.3333333333333333}}{{x}^{3}}\right) - \left(0.25 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      4. +-commutative 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + 0.25 \cdot \frac{1}{{x}^{4}}\right)}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      5. associate-*r/ 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + 0.25 \cdot \frac{1}{{x}^{4}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      6. metadata-eval 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{\color{blue}{0.5}}{{x}^{2}} + 0.25 \cdot \frac{1}{{x}^{4}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      7. unpow2 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{0.5}{\color{blue}{x \cdot x}} + 0.25 \cdot \frac{1}{{x}^{4}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      8. associate-*r/ 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{0.5}{x \cdot x} + \color{blue}{\frac{0.25 \cdot 1}{{x}^{4}}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

      9. metadata-eval 66.1%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{0.5}{x \cdot x} + \frac{\color{blue}{0.25}}{{x}^{4}}\right)}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

   7. Simplified 66.1%

      \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \frac{\color{blue}{\left(\frac{1}{x} + \frac{0.3333333333333333}{{x}^{3}}\right) - \left(\frac{0.5}{x \cdot x} + \frac{0.25}{{x}^{4}}\right)}}{n} - 0.5 \cdot \frac{{\log x}^{2}}{n \cdot n}\right) \]

   8. Taylor expanded in x around 0 90.8%

      \[\leadsto \color{blue}{\frac{-0.25}{n \cdot {x}^{4}}} \]
   9. Step-by-step derivation

      1. associate-/r* 90.8%

         \[\leadsto \color{blue}{\frac{\frac{-0.25}{n}}{{x}^{4}}} \]

   10. Simplified 90.8%

       \[\leadsto \color{blue}{\frac{\frac{-0.25}{n}}{{x}^{4}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{-296}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-190}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-156}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.94:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25}{n}}{{x}^{4}}\\ \end{array} \]

Alternative 11: 64.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -4.7 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -7.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_1}{n \cdot x}\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{-228}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq 13500000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (/ 1.0 x) (/ (/ 0.5 x) x)) n)) (t_1 (pow x (/ 1.0 n))))
   (if (<= n -4.7e+150)
     t_0
     (if (<= n -7.5e+14)
       (/ (- x (log x)) n)
       (if (<= n -1e-310)
         (/ t_1 (* n x))
         (if (<= n 4.8e-228)
           (/ 1.0 (* n x))
           (if (<= n 13500000.0) (- (+ 1.0 (/ x n)) t_1) t_0)))))))

C

double code(double x, double n) {
	double t_0 = ((1.0 / x) - ((0.5 / x) / x)) / n;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if (n <= -4.7e+150) {
		tmp = t_0;
	} else if (n <= -7.5e+14) {
		tmp = (x - log(x)) / n;
	} else if (n <= -1e-310) {
		tmp = t_1 / (n * x);
	} else if (n <= 4.8e-228) {
		tmp = 1.0 / (n * x);
	} else if (n <= 13500000.0) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((1.0d0 / x) - ((0.5d0 / x) / x)) / n
    t_1 = x ** (1.0d0 / n)
    if (n <= (-4.7d+150)) then
        tmp = t_0
    else if (n <= (-7.5d+14)) then
        tmp = (x - log(x)) / n
    else if (n <= (-1d-310)) then
        tmp = t_1 / (n * x)
    else if (n <= 4.8d-228) then
        tmp = 1.0d0 / (n * x)
    else if (n <= 13500000.0d0) then
        tmp = (1.0d0 + (x / n)) - t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = ((1.0 / x) - ((0.5 / x) / x)) / n;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if (n <= -4.7e+150) {
		tmp = t_0;
	} else if (n <= -7.5e+14) {
		tmp = (x - Math.log(x)) / n;
	} else if (n <= -1e-310) {
		tmp = t_1 / (n * x);
	} else if (n <= 4.8e-228) {
		tmp = 1.0 / (n * x);
	} else if (n <= 13500000.0) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = ((1.0 / x) - ((0.5 / x) / x)) / n
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if n <= -4.7e+150:
		tmp = t_0
	elif n <= -7.5e+14:
		tmp = (x - math.log(x)) / n
	elif n <= -1e-310:
		tmp = t_1 / (n * x)
	elif n <= 4.8e-228:
		tmp = 1.0 / (n * x)
	elif n <= 13500000.0:
		tmp = (1.0 + (x / n)) - t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(Float64(1.0 / x) - Float64(Float64(0.5 / x) / x)) / n)
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (n <= -4.7e+150)
		tmp = t_0;
	elseif (n <= -7.5e+14)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (n <= -1e-310)
		tmp = Float64(t_1 / Float64(n * x));
	elseif (n <= 4.8e-228)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (n <= 13500000.0)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((1.0 / x) - ((0.5 / x) / x)) / n;
	t_1 = x ^ (1.0 / n);
	tmp = 0.0;
	if (n <= -4.7e+150)
		tmp = t_0;
	elseif (n <= -7.5e+14)
		tmp = (x - log(x)) / n;
	elseif (n <= -1e-310)
		tmp = t_1 / (n * x);
	elseif (n <= 4.8e-228)
		tmp = 1.0 / (n * x);
	elseif (n <= 13500000.0)
		tmp = (1.0 + (x / n)) - t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(1.0 / x), $MachinePrecision] - N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -4.7e+150], t$95$0, If[LessEqual[n, -7.5e+14], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -1e-310], N[(t$95$1 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.8e-228], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 13500000.0], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -4.70000000000000004e150 or 1.35e7 < n

   1. Initial program 37.1%

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

   2. Taylor expanded in n around inf 78.6%

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

      1. +-rgt-identity 78.6%

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

      2. +-rgt-identity 78.6%

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

      3. log1p-def 78.6%

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

   4. Simplified 78.6%

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

   5. Taylor expanded in x around inf 56.7%

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

      1. associate-/l/ 58.5%

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

      2. associate-*r/ 58.5%

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

      3. metadata-eval 58.5%

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

      4. associate-/l/ 58.5%

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

      5. metadata-eval 58.5%

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

      6. associate-*r/ 58.5%

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

      7. div-sub 58.5%

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

      8. associate-*r/ 58.5%

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

      9. metadata-eval 58.5%

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

      10. unpow2 58.5%

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

      11. associate-/r* 58.5%

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

   7. Simplified 58.5%

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

   if -4.70000000000000004e150 < n < -7.5e14

   1. Initial program 11.3%

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

   2. Taylor expanded in n around inf 70.8%

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

      1. +-rgt-identity 70.8%

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

      2. +-rgt-identity 70.8%

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

      3. log1p-def 70.8%

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

   4. Simplified 70.8%

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

   5. Taylor expanded in x around 0 62.4%

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

      1. mul-1-neg 62.4%

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

      2. sub-neg 62.4%

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

      3. div-sub 62.4%

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

   7. Simplified 62.4%

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

   if -7.5e14 < n < -9.999999999999969e-311

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -9.999999999999969e-311 < n < 4.80000000000000004e-228

   1. Initial program 3.1%

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

   2. Taylor expanded in x around inf 0.2%

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

      1. log-rec 0.2%

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

      2. mul-1-neg 0.2%

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

      3. associate-*r/ 0.2%

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

      4. neg-mul-1 0.2%

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

      5. mul-1-neg 0.2%

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

      6. remove-double-neg 0.2%

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

      7. *-commutative 0.2%

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

   4. Simplified 0.2%

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

   5. Taylor expanded in n 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%

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

   if 4.80000000000000004e-228 < n < 1.35e7

   1. Initial program 70.1%

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

   2. Taylor expanded in x around 0 63.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.7 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \mathbf{elif}\;n \leq -7.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{-228}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq 13500000:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \end{array} \]

Alternative 12: 56.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-\log x}{n}\\ \mathbf{if}\;x \leq 5 \cdot 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.3 \cdot 10^{-249}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-191}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (- (log x)) n)))
   (if (<= x 5e-296)
     t_0
     (if (<= x 7.3e-249)
       t_1
       (if (<= x 2.35e-232)
         t_0
         (if (<= x 9.5e-191)
           t_1
           (if (<= x 3.6e-152)
             t_0
             (if (<= x 0.98)
               (/ (- x (log x)) n)
               (/ (- (/ 1.0 x) (/ (/ 0.5 x) x)) n)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = -log(x) / n;
	double tmp;
	if (x <= 5e-296) {
		tmp = t_0;
	} else if (x <= 7.3e-249) {
		tmp = t_1;
	} else if (x <= 2.35e-232) {
		tmp = t_0;
	} else if (x <= 9.5e-191) {
		tmp = t_1;
	} else if (x <= 3.6e-152) {
		tmp = t_0;
	} else if (x <= 0.98) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = -log(x) / n
    if (x <= 5d-296) then
        tmp = t_0
    else if (x <= 7.3d-249) then
        tmp = t_1
    else if (x <= 2.35d-232) then
        tmp = t_0
    else if (x <= 9.5d-191) then
        tmp = t_1
    else if (x <= 3.6d-152) then
        tmp = t_0
    else if (x <= 0.98d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / x) - ((0.5d0 / x) / x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = -Math.log(x) / n;
	double tmp;
	if (x <= 5e-296) {
		tmp = t_0;
	} else if (x <= 7.3e-249) {
		tmp = t_1;
	} else if (x <= 2.35e-232) {
		tmp = t_0;
	} else if (x <= 9.5e-191) {
		tmp = t_1;
	} else if (x <= 3.6e-152) {
		tmp = t_0;
	} else if (x <= 0.98) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = -math.log(x) / n
	tmp = 0
	if x <= 5e-296:
		tmp = t_0
	elif x <= 7.3e-249:
		tmp = t_1
	elif x <= 2.35e-232:
		tmp = t_0
	elif x <= 9.5e-191:
		tmp = t_1
	elif x <= 3.6e-152:
		tmp = t_0
	elif x <= 0.98:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(-log(x)) / n)
	tmp = 0.0
	if (x <= 5e-296)
		tmp = t_0;
	elseif (x <= 7.3e-249)
		tmp = t_1;
	elseif (x <= 2.35e-232)
		tmp = t_0;
	elseif (x <= 9.5e-191)
		tmp = t_1;
	elseif (x <= 3.6e-152)
		tmp = t_0;
	elseif (x <= 0.98)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(Float64(0.5 / x) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = -log(x) / n;
	tmp = 0.0;
	if (x <= 5e-296)
		tmp = t_0;
	elseif (x <= 7.3e-249)
		tmp = t_1;
	elseif (x <= 2.35e-232)
		tmp = t_0;
	elseif (x <= 9.5e-191)
		tmp = t_1;
	elseif (x <= 3.6e-152)
		tmp = t_0;
	elseif (x <= 0.98)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / x) - ((0.5 / x) / 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]}, Block[{t$95$1 = N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[x, 5e-296], t$95$0, If[LessEqual[x, 7.3e-249], t$95$1, If[LessEqual[x, 2.35e-232], t$95$0, If[LessEqual[x, 9.5e-191], t$95$1, If[LessEqual[x, 3.6e-152], t$95$0, If[LessEqual[x, 0.98], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 5.0000000000000003e-296 or 7.2999999999999997e-249 < x < 2.35000000000000017e-232 or 9.4999999999999996e-191 < x < 3.6e-152

   1. Initial program 81.0%

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

   2. Taylor expanded in x around 0 81.0%

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

   if 5.0000000000000003e-296 < x < 7.2999999999999997e-249 or 2.35000000000000017e-232 < x < 9.4999999999999996e-191

   1. Initial program 36.3%

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

   2. Taylor expanded in x around 0 36.3%

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

   3. Taylor expanded in n around inf 66.8%

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

      1. associate-*r/ 66.8%

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

      2. neg-mul-1 66.8%

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

   5. Simplified 66.8%

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

   if 3.6e-152 < x < 0.97999999999999998

   1. Initial program 37.5%

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

   2. Taylor expanded in n around inf 49.1%

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

      1. +-rgt-identity 49.1%

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

      2. +-rgt-identity 49.1%

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

      3. log1p-def 49.1%

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

   4. Simplified 49.1%

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

   5. Taylor expanded in x around 0 47.9%

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

      1. mul-1-neg 47.9%

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

      2. sub-neg 47.9%

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

      3. div-sub 47.9%

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

   7. Simplified 47.9%

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

   if 0.97999999999999998 < x

   1. Initial program 67.8%

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

   2. Taylor expanded in n around inf 69.3%

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

      1. +-rgt-identity 69.3%

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

      2. +-rgt-identity 69.3%

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

      3. log1p-def 69.3%

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

   4. Simplified 69.3%

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

   5. Taylor expanded in x around inf 60.7%

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

      1. associate-/l/ 62.2%

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

      2. associate-*r/ 62.2%

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

      3. metadata-eval 62.2%

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

      4. associate-/l/ 62.2%

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

      5. metadata-eval 62.2%

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

      6. associate-*r/ 62.2%

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

      7. div-sub 62.3%

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

      8. associate-*r/ 62.3%

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

      9. metadata-eval 62.3%

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

      10. unpow2 62.3%

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

      11. associate-/r* 62.3%

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

   7. Simplified 62.3%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-296}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 7.3 \cdot 10^{-249}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-232}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-191}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-152}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \end{array} \]

Alternative 13: 64.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;n \leq -8 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -6.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{t_1}{n \cdot x}\\ \mathbf{elif}\;n \leq 2.05 \cdot 10^{-226}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq 13500000:\\ \;\;\;\;1 - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (/ 1.0 x) (/ (/ 0.5 x) x)) n)) (t_1 (pow x (/ 1.0 n))))
   (if (<= n -8e+150)
     t_0
     (if (<= n -6.8e+15)
       (/ (- x (log x)) n)
       (if (<= n -1e-310)
         (/ t_1 (* n x))
         (if (<= n 2.05e-226)
           (/ 1.0 (* n x))
           (if (<= n 13500000.0) (- 1.0 t_1) t_0)))))))

C

double code(double x, double n) {
	double t_0 = ((1.0 / x) - ((0.5 / x) / x)) / n;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if (n <= -8e+150) {
		tmp = t_0;
	} else if (n <= -6.8e+15) {
		tmp = (x - log(x)) / n;
	} else if (n <= -1e-310) {
		tmp = t_1 / (n * x);
	} else if (n <= 2.05e-226) {
		tmp = 1.0 / (n * x);
	} else if (n <= 13500000.0) {
		tmp = 1.0 - t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((1.0d0 / x) - ((0.5d0 / x) / x)) / n
    t_1 = x ** (1.0d0 / n)
    if (n <= (-8d+150)) then
        tmp = t_0
    else if (n <= (-6.8d+15)) then
        tmp = (x - log(x)) / n
    else if (n <= (-1d-310)) then
        tmp = t_1 / (n * x)
    else if (n <= 2.05d-226) then
        tmp = 1.0d0 / (n * x)
    else if (n <= 13500000.0d0) then
        tmp = 1.0d0 - t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = ((1.0 / x) - ((0.5 / x) / x)) / n;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if (n <= -8e+150) {
		tmp = t_0;
	} else if (n <= -6.8e+15) {
		tmp = (x - Math.log(x)) / n;
	} else if (n <= -1e-310) {
		tmp = t_1 / (n * x);
	} else if (n <= 2.05e-226) {
		tmp = 1.0 / (n * x);
	} else if (n <= 13500000.0) {
		tmp = 1.0 - t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = ((1.0 / x) - ((0.5 / x) / x)) / n
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if n <= -8e+150:
		tmp = t_0
	elif n <= -6.8e+15:
		tmp = (x - math.log(x)) / n
	elif n <= -1e-310:
		tmp = t_1 / (n * x)
	elif n <= 2.05e-226:
		tmp = 1.0 / (n * x)
	elif n <= 13500000.0:
		tmp = 1.0 - t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(Float64(1.0 / x) - Float64(Float64(0.5 / x) / x)) / n)
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (n <= -8e+150)
		tmp = t_0;
	elseif (n <= -6.8e+15)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (n <= -1e-310)
		tmp = Float64(t_1 / Float64(n * x));
	elseif (n <= 2.05e-226)
		tmp = Float64(1.0 / Float64(n * x));
	elseif (n <= 13500000.0)
		tmp = Float64(1.0 - t_1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((1.0 / x) - ((0.5 / x) / x)) / n;
	t_1 = x ^ (1.0 / n);
	tmp = 0.0;
	if (n <= -8e+150)
		tmp = t_0;
	elseif (n <= -6.8e+15)
		tmp = (x - log(x)) / n;
	elseif (n <= -1e-310)
		tmp = t_1 / (n * x);
	elseif (n <= 2.05e-226)
		tmp = 1.0 / (n * x);
	elseif (n <= 13500000.0)
		tmp = 1.0 - t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(1.0 / x), $MachinePrecision] - N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -8e+150], t$95$0, If[LessEqual[n, -6.8e+15], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -1e-310], N[(t$95$1 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.05e-226], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 13500000.0], N[(1.0 - t$95$1), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -7.99999999999999985e150 or 1.35e7 < n

   1. Initial program 37.1%

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

   2. Taylor expanded in n around inf 78.6%

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

      1. +-rgt-identity 78.6%

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

      2. +-rgt-identity 78.6%

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

      3. log1p-def 78.6%

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

   4. Simplified 78.6%

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

   5. Taylor expanded in x around inf 56.7%

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

      1. associate-/l/ 58.5%

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

      2. associate-*r/ 58.5%

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

      3. metadata-eval 58.5%

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

      4. associate-/l/ 58.5%

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

      5. metadata-eval 58.5%

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

      6. associate-*r/ 58.5%

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

      7. div-sub 58.5%

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

      8. associate-*r/ 58.5%

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

      9. metadata-eval 58.5%

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

      10. unpow2 58.5%

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

      11. associate-/r* 58.5%

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

   7. Simplified 58.5%

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

   if -7.99999999999999985e150 < n < -6.8e15

   1. Initial program 11.3%

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

   2. Taylor expanded in n around inf 70.8%

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

      1. +-rgt-identity 70.8%

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

      2. +-rgt-identity 70.8%

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

      3. log1p-def 70.8%

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

   4. Simplified 70.8%

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

   5. Taylor expanded in x around 0 62.4%

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

      1. mul-1-neg 62.4%

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

      2. sub-neg 62.4%

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

      3. div-sub 62.4%

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

   7. Simplified 62.4%

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

   if -6.8e15 < n < -9.999999999999969e-311

   1. Initial program 97.8%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. div-inv 100.0%

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

      3. pow-to-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if -9.999999999999969e-311 < n < 2.05000000000000019e-226

   1. Initial program 3.1%

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

   2. Taylor expanded in x around inf 0.2%

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

      1. log-rec 0.2%

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

      2. mul-1-neg 0.2%

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

      3. associate-*r/ 0.2%

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

      4. neg-mul-1 0.2%

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

      5. mul-1-neg 0.2%

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

      6. remove-double-neg 0.2%

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

      7. *-commutative 0.2%

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

   4. Simplified 0.2%

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

   5. Taylor expanded in n 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%

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

   if 2.05000000000000019e-226 < n < 1.35e7

   1. Initial program 70.1%

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

   2. Taylor expanded in x around 0 62.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \mathbf{elif}\;n \leq -6.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;n \leq 2.05 \cdot 10^{-226}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq 13500000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \end{array} \]

Alternative 14: 57.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.98) (/ (- x (log x)) n) (/ (- (/ 1.0 x) (/ (/ 0.5 x) x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.98) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.98d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 / x) - ((0.5d0 / x) / x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.98) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.98:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.98)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(Float64(0.5 / x) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.98)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.98], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.97999999999999998

   1. Initial program 46.3%

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

   2. Taylor expanded in n around inf 48.7%

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

      1. +-rgt-identity 48.7%

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

      2. +-rgt-identity 48.7%

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

      3. log1p-def 48.7%

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

   4. Simplified 48.7%

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

   5. Taylor expanded in x around 0 48.1%

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

      1. mul-1-neg 48.1%

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

      2. sub-neg 48.1%

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

      3. div-sub 48.1%

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

   7. Simplified 48.1%

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

   if 0.97999999999999998 < x

   1. Initial program 67.8%

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

   2. Taylor expanded in n around inf 69.3%

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

      1. +-rgt-identity 69.3%

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

      2. +-rgt-identity 69.3%

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

      3. log1p-def 69.3%

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

   4. Simplified 69.3%

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

   5. Taylor expanded in x around inf 60.7%

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

      1. associate-/l/ 62.2%

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

      2. associate-*r/ 62.2%

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

      3. metadata-eval 62.2%

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

      4. associate-/l/ 62.2%

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

      5. metadata-eval 62.2%

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

      6. associate-*r/ 62.2%

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

      7. div-sub 62.3%

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

      8. associate-*r/ 62.3%

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

      9. metadata-eval 62.3%

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

      10. unpow2 62.3%

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

      11. associate-/r* 62.3%

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

   7. Simplified 62.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.98:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \end{array} \]

Alternative 15: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.66) (/ (- (log x)) n) (/ (- (/ 1.0 x) (/ (/ 0.5 x) x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.66) {
		tmp = -log(x) / n;
	} else {
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.66d0) then
        tmp = -log(x) / n
    else
        tmp = ((1.0d0 / x) - ((0.5d0 / x) / x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.66) {
		tmp = -Math.log(x) / n;
	} else {
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.66:
		tmp = -math.log(x) / n
	else:
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.66)
		tmp = Float64(Float64(-log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(Float64(0.5 / x) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.66)
		tmp = -log(x) / n;
	else
		tmp = ((1.0 / x) - ((0.5 / x) / x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.66], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.660000000000000031

   1. Initial program 46.3%

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

   2. Taylor expanded in x around 0 44.9%

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

   3. Taylor expanded in n around inf 47.5%

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

      1. associate-*r/ 47.5%

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

      2. neg-mul-1 47.5%

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

   5. Simplified 47.5%

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

   if 0.660000000000000031 < x

   1. Initial program 67.8%

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

   2. Taylor expanded in n around inf 69.3%

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

      1. +-rgt-identity 69.3%

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

      2. +-rgt-identity 69.3%

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

      3. log1p-def 69.3%

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

   4. Simplified 69.3%

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

   5. Taylor expanded in x around inf 60.7%

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

      1. associate-/l/ 62.2%

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

      2. associate-*r/ 62.2%

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

      3. metadata-eval 62.2%

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

      4. associate-/l/ 62.2%

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

      5. metadata-eval 62.2%

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

      6. associate-*r/ 62.2%

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

      7. div-sub 62.3%

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

      8. associate-*r/ 62.3%

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

      9. metadata-eval 62.3%

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

      10. unpow2 62.3%

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

      11. associate-/r* 62.3%

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

   7. Simplified 62.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\frac{0.5}{x}}{x}}{n}\\ \end{array} \]

Alternative 16: 39.9% 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 56.3%

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

2. Taylor expanded in x around inf 61.8%

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

   1. log-rec 61.8%

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

   2. mul-1-neg 61.8%

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

   3. associate-*r/ 61.8%

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

   4. neg-mul-1 61.8%

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

   5. mul-1-neg 61.8%

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

   6. remove-double-neg 61.8%

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

   7. *-commutative 61.8%

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

4. Simplified 61.8%

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

5. Taylor expanded in n around inf 40.4%

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

   1. *-commutative 40.4%

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

7. Simplified 40.4%

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

8. Final simplification 40.4%

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

Alternative 17: 40.4% 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 56.3%

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

2. Taylor expanded in n around inf 49.1%

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

   1. associate--l+ 45.9%

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

   2. fma-def 45.9%

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

   3. log1p-def 45.9%

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

   4. unpow2 45.9%

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

   5. associate--r+ 49.1%

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

   6. +-rgt-identity 49.1%

      \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \left(\frac{\color{blue}{\log \left(1 + x\right) + 0}}{n} - \frac{\log x}{n}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

   7. div-sub 49.1%

      \[\leadsto \mathsf{fma}\left(0.5, \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{n \cdot n}, \color{blue}{\frac{\left(\log \left(1 + x\right) + 0\right) - \log x}{n}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

   8. +-rgt-identity 49.1%

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

   9. log1p-def 49.1%

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

   10. unpow2 49.1%

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

4. Simplified 49.1%

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

5. Taylor expanded in x around inf 42.8%

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

   1. mul-1-neg 42.8%

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

   2. distribute-neg-frac 42.8%

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

   3. log-rec 42.8%

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

   4. remove-double-neg 42.8%

      \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{\log x}}{{n}^{2}}}{x} \]

   5. unpow2 42.8%

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

7. Simplified 42.8%

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

8. Taylor expanded in n around inf 41.1%

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

9. Final simplification 41.1%

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

Alternative 18: 40.4% 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 56.3%

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

2. Taylor expanded in n around inf 58.3%

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

   1. +-rgt-identity 58.3%

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

   2. +-rgt-identity 58.3%

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

   3. log1p-def 58.3%

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

4. Simplified 58.3%

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

5. Taylor expanded in x around inf 41.1%

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

6. Final simplification 41.1%

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

Alternative 19: 4.5% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.3%

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

2. Taylor expanded in x around 0 32.7%

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

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 2023195 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))