2nthrt (problem 3.4.6)

Percentage Accurate: 53.7% → 85.8%

Time: 36.8s

Alternatives: 17

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% 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: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-43)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 3e-54)
       (/ (log (+ 1.0 (/ 1.0 x))) n)
       (if (<= (/ 1.0 n) 2e-17)
         (/
          (+
           (/ 0.3333333333333333 (pow x 3.0))
           (- (/ 1.0 x) (/ 0.5 (pow x 2.0))))
          n)
         (- (exp (/ (log1p x) n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = ((0.3333333333333333 / pow(x, 3.0)) + ((1.0 / x) - (0.5 / pow(x, 2.0)))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = ((0.3333333333333333 / Math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / Math.pow(x, 2.0)))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-43:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 3e-54:
		tmp = math.log((1.0 + (1.0 / x))) / n
	elif (1.0 / n) <= 2e-17:
		tmp = ((0.3333333333333333 / math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / math.pow(x, 2.0)))) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-43)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 3e-54)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	elseif (Float64(1.0 / n) <= 2e-17)
		tmp = Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0)))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-43], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e-54], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-17], N[(N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $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 4 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-43

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf 95.1%

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

      1. log-rec 95.1%

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

      2. mul-1-neg 95.1%

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

      3. associate-*r/ 95.1%

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

      4. associate-*r* 95.1%

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

      5. metadata-eval 95.1%

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

      6. *-commutative 95.1%

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

      7. associate-/l* 95.1%

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

      8. exp-to-pow 95.1%

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

      9. *-commutative 95.1%

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

   5. Simplified 95.1%

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

   if -5.00000000000000019e-43 < (/.f64 1 n) < 3.00000000000000009e-54

   1. Initial program 33.9%

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

   3. Taylor expanded in n around inf 85.7%

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

      1. +-rgt-identity 85.7%

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

      2. +-rgt-identity 85.7%

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

      3. log1p-define 85.7%

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

   5. Simplified 85.7%

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

      1. log1p-undefine 85.7%

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

      2. diff-log 85.7%

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

   7. Applied egg-rr 85.7%

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

   8. Taylor expanded in x around 0 85.7%

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

   if 3.00000000000000009e-54 < (/.f64 1 n) < 2.00000000000000014e-17

   1. Initial program 16.6%

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

   3. Taylor expanded in n around inf 46.3%

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

      1. +-rgt-identity 46.3%

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

      2. +-rgt-identity 46.3%

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

      3. log1p-define 46.3%

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

   5. Simplified 46.3%

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

   6. Taylor expanded in x around inf 74.5%

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

      1. associate--l+ 74.5%

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

      2. associate-*r/ 74.5%

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

      3. metadata-eval 74.5%

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

      4. associate-*r/ 74.5%

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

      5. metadata-eval 74.5%

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

   8. Simplified 74.5%

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

   if 2.00000000000000014e-17 < (/.f64 1 n)

   1. Initial program 53.9%

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

   3. Taylor expanded in n around 0 53.9%

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

      1. log1p-define 92.8%

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

      2. *-rgt-identity 92.8%

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

      3. associate-/l* 92.8%

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

      4. exp-to-pow 92.8%

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

   5. Simplified 92.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-43)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 3e-54)
       (/ (log (+ 1.0 (/ 1.0 x))) n)
       (if (<= (/ 1.0 n) 2e-17)
         (/
          (+
           (/ 0.3333333333333333 (pow x 3.0))
           (- (/ 1.0 x) (/ 0.5 (pow x 2.0))))
          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) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = ((0.3333333333333333 / pow(x, 3.0)) + ((1.0 / x) - (0.5 / pow(x, 2.0)))) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-43)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 3d-54) then
        tmp = log((1.0d0 + (1.0d0 / x))) / n
    else if ((1.0d0 / n) <= 2d-17) then
        tmp = ((0.3333333333333333d0 / (x ** 3.0d0)) + ((1.0d0 / x) - (0.5d0 / (x ** 2.0d0)))) / n
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = ((0.3333333333333333 / Math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / Math.pow(x, 2.0)))) / 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) <= -5e-43:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 3e-54:
		tmp = math.log((1.0 + (1.0 / x))) / n
	elif (1.0 / n) <= 2e-17:
		tmp = ((0.3333333333333333 / math.pow(x, 3.0)) + ((1.0 / x) - (0.5 / math.pow(x, 2.0)))) / 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) <= -5e-43)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 3e-54)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	elseif (Float64(1.0 / n) <= 2e-17)
		tmp = Float64(Float64(Float64(0.3333333333333333 / (x ^ 3.0)) + Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0)))) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-43)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 3e-54)
		tmp = log((1.0 + (1.0 / x))) / n;
	elseif ((1.0 / n) <= 2e-17)
		tmp = ((0.3333333333333333 / (x ^ 3.0)) + ((1.0 / x) - (0.5 / (x ^ 2.0)))) / n;
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-43], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e-54], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-17], N[(N[(N[(0.3333333333333333 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-43

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf 95.1%

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

      1. log-rec 95.1%

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

      2. mul-1-neg 95.1%

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

      3. associate-*r/ 95.1%

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

      4. associate-*r* 95.1%

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

      5. metadata-eval 95.1%

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

      6. *-commutative 95.1%

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

      7. associate-/l* 95.1%

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

      8. exp-to-pow 95.1%

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

      9. *-commutative 95.1%

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

   5. Simplified 95.1%

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

   if -5.00000000000000019e-43 < (/.f64 1 n) < 3.00000000000000009e-54

   1. Initial program 33.9%

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

   3. Taylor expanded in n around inf 85.7%

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

      1. +-rgt-identity 85.7%

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

      2. +-rgt-identity 85.7%

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

      3. log1p-define 85.7%

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

   5. Simplified 85.7%

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

      1. log1p-undefine 85.7%

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

      2. diff-log 85.7%

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

   7. Applied egg-rr 85.7%

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

   8. Taylor expanded in x around 0 85.7%

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

   if 3.00000000000000009e-54 < (/.f64 1 n) < 2.00000000000000014e-17

   1. Initial program 16.6%

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

   3. Taylor expanded in n around inf 46.3%

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

      1. +-rgt-identity 46.3%

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

      2. +-rgt-identity 46.3%

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

      3. log1p-define 46.3%

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

   5. Simplified 46.3%

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

   6. Taylor expanded in x around inf 74.5%

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

      1. associate--l+ 74.5%

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

      2. associate-*r/ 74.5%

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

      3. metadata-eval 74.5%

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

      4. associate-*r/ 74.5%

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

      5. metadata-eval 74.5%

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

   8. Simplified 74.5%

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

   if 2.00000000000000014e-17 < (/.f64 1 n)

   1. Initial program 53.9%

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

   3. Taylor expanded in n around 0 53.9%

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

      1. log1p-define 92.8%

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

      2. *-rgt-identity 92.8%

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

      3. associate-/l* 92.8%

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

      4. exp-to-pow 92.8%

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

   5. Simplified 92.8%

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

   6. Taylor expanded in x around 0 92.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{{x}^{3}} + \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-43)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 3e-54)
       (/ (log (+ 1.0 (/ 1.0 x))) n)
       (if (<= (/ 1.0 n) 2e-17)
         (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) n)
         (if (<= (/ 1.0 n) 5e+156)
           (- (+ 1.0 (/ x n)) t_0)
           (sqrt (pow (* n x) -2.0))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e+156) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = sqrt(pow((n * x), -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-43)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 3d-54) then
        tmp = log((1.0d0 + (1.0d0 / x))) / n
    else if ((1.0d0 / n) <= 2d-17) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x ** 2.0d0))) / n
    else if ((1.0d0 / n) <= 5d+156) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = sqrt(((n * x) ** (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e+156) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.sqrt(Math.pow((n * x), -2.0));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-43:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 3e-54:
		tmp = math.log((1.0 + (1.0 / x))) / n
	elif (1.0 / n) <= 2e-17:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / n
	elif (1.0 / n) <= 5e+156:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.sqrt(math.pow((n * x), -2.0))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-43)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 3e-54)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	elseif (Float64(1.0 / n) <= 2e-17)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	elseif (Float64(1.0 / n) <= 5e+156)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = sqrt((Float64(n * x) ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-43)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 3e-54)
		tmp = log((1.0 + (1.0 / x))) / n;
	elseif ((1.0 / n) <= 2e-17)
		tmp = ((1.0 / x) - (0.5 / (x ^ 2.0))) / n;
	elseif ((1.0 / n) <= 5e+156)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = sqrt(((n * x) ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-43], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e-54], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-17], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+156], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Sqrt[N[Power[N[(n * x), $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-43

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf 95.1%

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

      1. log-rec 95.1%

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

      2. mul-1-neg 95.1%

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

      3. associate-*r/ 95.1%

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

      4. associate-*r* 95.1%

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

      5. metadata-eval 95.1%

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

      6. *-commutative 95.1%

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

      7. associate-/l* 95.1%

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

      8. exp-to-pow 95.1%

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

      9. *-commutative 95.1%

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

   5. Simplified 95.1%

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

   if -5.00000000000000019e-43 < (/.f64 1 n) < 3.00000000000000009e-54

   1. Initial program 33.9%

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

   3. Taylor expanded in n around inf 85.7%

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

      1. +-rgt-identity 85.7%

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

      2. +-rgt-identity 85.7%

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

      3. log1p-define 85.7%

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

   5. Simplified 85.7%

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

      1. log1p-undefine 85.7%

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

      2. diff-log 85.7%

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

   7. Applied egg-rr 85.7%

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

   8. Taylor expanded in x around 0 85.7%

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

   if 3.00000000000000009e-54 < (/.f64 1 n) < 2.00000000000000014e-17

   1. Initial program 16.6%

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

   3. Taylor expanded in n around inf 46.3%

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

      1. +-rgt-identity 46.3%

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

      2. +-rgt-identity 46.3%

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

      3. log1p-define 46.3%

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

   5. Simplified 46.3%

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

   6. Taylor expanded in x around inf 73.5%

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

      1. associate-*r/ 73.5%

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

      2. metadata-eval 73.5%

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

   8. Simplified 73.5%

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

   if 2.00000000000000014e-17 < (/.f64 1 n) < 4.99999999999999992e156

   1. Initial program 72.0%

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

   3. Taylor expanded in x around 0 72.1%

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

   if 4.99999999999999992e156 < (/.f64 1 n)

   1. Initial program 17.8%

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

   3. Taylor expanded in n around inf 6.8%

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

      1. +-rgt-identity 6.8%

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

      2. +-rgt-identity 6.8%

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

      3. log1p-define 6.8%

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

   5. Simplified 6.8%

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

   6. Taylor expanded in x around inf 53.7%

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

      1. *-commutative 53.7%

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

   8. Simplified 53.7%

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

      1. add-sqr-sqrt 53.7%

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

      2. sqrt-unprod 86.2%

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

      3. inv-pow 86.2%

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

      4. inv-pow 86.2%

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

      5. pow-prod-up 86.2%

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

      6. metadata-eval 86.2%

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

   10. Applied egg-rr 86.2%

       \[\leadsto \color{blue}{\sqrt{{\left(x \cdot n\right)}^{-2}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(n \cdot x\right)}^{-2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-43)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 3e-54)
       (/ (log (+ 1.0 (/ 1.0 x))) n)
       (if (<= (/ 1.0 n) 2e-17)
         (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) 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) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-43)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 3d-54) then
        tmp = log((1.0d0 + (1.0d0 / x))) / n
    else if ((1.0d0 / n) <= 2d-17) then
        tmp = ((1.0d0 / x) - (0.5d0 / (x ** 2.0d0))) / n
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / 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) <= -5e-43:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 3e-54:
		tmp = math.log((1.0 + (1.0 / x))) / n
	elif (1.0 / n) <= 2e-17:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / 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) <= -5e-43)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 3e-54)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	elseif (Float64(1.0 / n) <= 2e-17)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-43)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 3e-54)
		tmp = log((1.0 + (1.0 / x))) / n;
	elseif ((1.0 / n) <= 2e-17)
		tmp = ((1.0 / x) - (0.5 / (x ^ 2.0))) / n;
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-43], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e-54], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-17], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-43

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf 95.1%

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

      1. log-rec 95.1%

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

      2. mul-1-neg 95.1%

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

      3. associate-*r/ 95.1%

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

      4. associate-*r* 95.1%

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

      5. metadata-eval 95.1%

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

      6. *-commutative 95.1%

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

      7. associate-/l* 95.1%

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

      8. exp-to-pow 95.1%

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

      9. *-commutative 95.1%

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

   5. Simplified 95.1%

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

   if -5.00000000000000019e-43 < (/.f64 1 n) < 3.00000000000000009e-54

   1. Initial program 33.9%

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

   3. Taylor expanded in n around inf 85.7%

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

      1. +-rgt-identity 85.7%

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

      2. +-rgt-identity 85.7%

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

      3. log1p-define 85.7%

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

   5. Simplified 85.7%

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

      1. log1p-undefine 85.7%

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

      2. diff-log 85.7%

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

   7. Applied egg-rr 85.7%

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

   8. Taylor expanded in x around 0 85.7%

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

   if 3.00000000000000009e-54 < (/.f64 1 n) < 2.00000000000000014e-17

   1. Initial program 16.6%

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

   3. Taylor expanded in n around inf 46.3%

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

      1. +-rgt-identity 46.3%

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

      2. +-rgt-identity 46.3%

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

      3. log1p-define 46.3%

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

   5. Simplified 46.3%

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

   6. Taylor expanded in x around inf 73.5%

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

      1. associate-*r/ 73.5%

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

      2. metadata-eval 73.5%

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

   8. Simplified 73.5%

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

   if 2.00000000000000014e-17 < (/.f64 1 n)

   1. Initial program 53.9%

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

   3. Taylor expanded in n around 0 53.9%

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

      1. log1p-define 92.8%

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

      2. *-rgt-identity 92.8%

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

      3. associate-/l* 92.8%

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

      4. exp-to-pow 92.8%

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

   5. Simplified 92.8%

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

   6. Taylor expanded in x around 0 92.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-43)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 3e-54)
       (/ (log (+ 1.0 (/ 1.0 x))) n)
       (if (<= (/ 1.0 n) 2e-17)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 5e-16)
           (/ (log x) (- n))
           (if (<= (/ 1.0 n) 5e+156)
             (- (+ 1.0 (/ x n)) t_0)
             (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-16) {
		tmp = log(x) / -n;
	} else if ((1.0 / n) <= 5e+156) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-43)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 3d-54) then
        tmp = log((1.0d0 + (1.0d0 / x))) / n
    else if ((1.0d0 / n) <= 2d-17) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 5d-16) then
        tmp = log(x) / -n
    else if ((1.0d0 / n) <= 5d+156) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-16) {
		tmp = Math.log(x) / -n;
	} else if ((1.0 / n) <= 5e+156) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-43:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 3e-54:
		tmp = math.log((1.0 + (1.0 / x))) / n
	elif (1.0 / n) <= 2e-17:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e-16:
		tmp = math.log(x) / -n
	elif (1.0 / n) <= 5e+156:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-43)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 3e-54)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	elseif (Float64(1.0 / n) <= 2e-17)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-16)
		tmp = Float64(log(x) / Float64(-n));
	elseif (Float64(1.0 / n) <= 5e+156)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-43)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 3e-54)
		tmp = log((1.0 + (1.0 / x))) / n;
	elseif ((1.0 / n) <= 2e-17)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 5e-16)
		tmp = log(x) / -n;
	elseif ((1.0 / n) <= 5e+156)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-43], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e-54], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-17], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-16], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+156], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-43

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf 95.1%

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

      1. log-rec 95.1%

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

      2. mul-1-neg 95.1%

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

      3. associate-*r/ 95.1%

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

      4. associate-*r* 95.1%

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

      5. metadata-eval 95.1%

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

      6. *-commutative 95.1%

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

      7. associate-/l* 95.1%

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

      8. exp-to-pow 95.1%

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

      9. *-commutative 95.1%

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

   5. Simplified 95.1%

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

   if -5.00000000000000019e-43 < (/.f64 1 n) < 3.00000000000000009e-54

   1. Initial program 33.9%

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

   3. Taylor expanded in n around inf 85.7%

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

      1. +-rgt-identity 85.7%

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

      2. +-rgt-identity 85.7%

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

      3. log1p-define 85.7%

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

   5. Simplified 85.7%

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

      1. log1p-undefine 85.7%

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

      2. diff-log 85.7%

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

   7. Applied egg-rr 85.7%

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

   8. Taylor expanded in x around 0 85.7%

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

   if 3.00000000000000009e-54 < (/.f64 1 n) < 2.00000000000000014e-17

   1. Initial program 16.6%

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

   3. Taylor expanded in n around inf 46.3%

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

      1. +-rgt-identity 46.3%

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

      2. +-rgt-identity 46.3%

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

      3. log1p-define 46.3%

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

   5. Simplified 46.3%

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

   6. Taylor expanded in x around inf 71.7%

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

   if 2.00000000000000014e-17 < (/.f64 1 n) < 5.0000000000000004e-16

   1. Initial program 36.8%

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

   3. Taylor expanded in n around inf 85.4%

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

      1. +-rgt-identity 85.4%

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

      2. +-rgt-identity 85.4%

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

      3. log1p-define 85.4%

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

   5. Simplified 85.4%

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

   6. Taylor expanded in x around 0 85.4%

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

      1. neg-mul-1 85.4%

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

   8. Simplified 85.4%

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

   if 5.0000000000000004e-16 < (/.f64 1 n) < 4.99999999999999992e156

   1. Initial program 73.3%

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

   3. Taylor expanded in x around 0 73.4%

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

   if 4.99999999999999992e156 < (/.f64 1 n)

   1. Initial program 17.8%

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

   3. Taylor expanded in n around inf 6.8%

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

      1. +-rgt-identity 6.8%

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

      2. +-rgt-identity 6.8%

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

      3. log1p-define 6.8%

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

   5. Simplified 6.8%

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

   6. Taylor expanded in x around inf 53.7%

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

      1. *-commutative 53.7%

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

   8. Simplified 53.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{x}{1 + x} + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-43)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 3e-54)
       (/ (log (+ 1.0 (/ 1.0 x))) n)
       (if (<= (/ 1.0 n) 2e-17)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 5e-16)
           (/ (log x) (- n))
           (if (<= (/ 1.0 n) 5e+156)
             (- (+ 1.0 (/ x n)) t_0)
             (/ (log1p (+ (/ x (+ 1.0 x)) -1.0)) (- n)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-16) {
		tmp = log(x) / -n;
	} else if ((1.0 / n) <= 5e+156) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log1p(((x / (1.0 + x)) + -1.0)) / -n;
	}
	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) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-16) {
		tmp = Math.log(x) / -n;
	} else if ((1.0 / n) <= 5e+156) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log1p(((x / (1.0 + x)) + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-43:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 3e-54:
		tmp = math.log((1.0 + (1.0 / x))) / n
	elif (1.0 / n) <= 2e-17:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e-16:
		tmp = math.log(x) / -n
	elif (1.0 / n) <= 5e+156:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log1p(((x / (1.0 + x)) + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-43)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 3e-54)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	elseif (Float64(1.0 / n) <= 2e-17)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-16)
		tmp = Float64(log(x) / Float64(-n));
	elseif (Float64(1.0 / n) <= 5e+156)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(log1p(Float64(Float64(x / Float64(1.0 + x)) + -1.0)) / Float64(-n));
	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], -5e-43], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e-54], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-17], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-16], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+156], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Log[1 + N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-43

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf 95.1%

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

      1. log-rec 95.1%

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

      2. mul-1-neg 95.1%

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

      3. associate-*r/ 95.1%

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

      4. associate-*r* 95.1%

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

      5. metadata-eval 95.1%

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

      6. *-commutative 95.1%

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

      7. associate-/l* 95.1%

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

      8. exp-to-pow 95.1%

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

      9. *-commutative 95.1%

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

   5. Simplified 95.1%

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

   if -5.00000000000000019e-43 < (/.f64 1 n) < 3.00000000000000009e-54

   1. Initial program 33.9%

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

   3. Taylor expanded in n around inf 85.7%

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

      1. +-rgt-identity 85.7%

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

      2. +-rgt-identity 85.7%

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

      3. log1p-define 85.7%

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

   5. Simplified 85.7%

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

      1. log1p-undefine 85.7%

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

      2. diff-log 85.7%

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

   7. Applied egg-rr 85.7%

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

   8. Taylor expanded in x around 0 85.7%

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

   if 3.00000000000000009e-54 < (/.f64 1 n) < 2.00000000000000014e-17

   1. Initial program 16.6%

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

   3. Taylor expanded in n around inf 46.3%

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

      1. +-rgt-identity 46.3%

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

      2. +-rgt-identity 46.3%

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

      3. log1p-define 46.3%

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

   5. Simplified 46.3%

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

   6. Taylor expanded in x around inf 71.7%

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

   if 2.00000000000000014e-17 < (/.f64 1 n) < 5.0000000000000004e-16

   1. Initial program 36.8%

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

   3. Taylor expanded in n around inf 85.4%

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

      1. +-rgt-identity 85.4%

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

      2. +-rgt-identity 85.4%

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

      3. log1p-define 85.4%

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

   5. Simplified 85.4%

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

   6. Taylor expanded in x around 0 85.4%

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

      1. neg-mul-1 85.4%

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

   8. Simplified 85.4%

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

   if 5.0000000000000004e-16 < (/.f64 1 n) < 4.99999999999999992e156

   1. Initial program 73.3%

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

   3. Taylor expanded in x around 0 73.4%

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

   if 4.99999999999999992e156 < (/.f64 1 n)

   1. Initial program 17.8%

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

   3. Taylor expanded in n around inf 6.8%

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

      1. +-rgt-identity 6.8%

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

      2. +-rgt-identity 6.8%

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

      3. log1p-define 6.8%

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

   5. Simplified 6.8%

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

      1. log1p-undefine 6.8%

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

      2. diff-log 6.8%

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

   7. Applied egg-rr 6.8%

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

      1. clear-num 6.8%

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

      2. log-rec 6.8%

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

   9. Applied egg-rr 6.8%

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

       1. log1p-expm1-u 79.6%

          \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{x}{1 + x}\right)\right)\right)}}{n} \]

       2. expm1-undefine 79.6%

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

       3. add-exp-log 79.6%

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

       4. +-commutative 79.6%

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

   11. Applied egg-rr 79.6%

       \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(\frac{x}{x + 1} - 1\right)}}{n} \]
3. Recombined 6 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{x}{1 + x} + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -5e+88)
     t_0
     (if (<= (/ 1.0 n) 3e-54)
       (/ (log (+ 1.0 (/ 1.0 x))) n)
       (if (<= (/ 1.0 n) 2e-17)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 5e-16)
           (/ (log x) (- n))
           (if (<= (/ 1.0 n) 5e+156) t_0 (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e+88) {
		tmp = t_0;
	} else if ((1.0 / n) <= 3e-54) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-16) {
		tmp = log(x) / -n;
	} else if ((1.0 / n) <= 5e+156) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-5d+88)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 3d-54) then
        tmp = log((1.0d0 + (1.0d0 / x))) / n
    else if ((1.0d0 / n) <= 2d-17) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 5d-16) then
        tmp = log(x) / -n
    else if ((1.0d0 / n) <= 5d+156) then
        tmp = t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e+88) {
		tmp = t_0;
	} else if ((1.0 / n) <= 3e-54) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-16) {
		tmp = Math.log(x) / -n;
	} else if ((1.0 / n) <= 5e+156) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e+88:
		tmp = t_0
	elif (1.0 / n) <= 3e-54:
		tmp = math.log((1.0 + (1.0 / x))) / n
	elif (1.0 / n) <= 2e-17:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e-16:
		tmp = math.log(x) / -n
	elif (1.0 / n) <= 5e+156:
		tmp = t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e+88)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 3e-54)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	elseif (Float64(1.0 / n) <= 2e-17)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-16)
		tmp = Float64(log(x) / Float64(-n));
	elseif (Float64(1.0 / n) <= 5e+156)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -5e+88)
		tmp = t_0;
	elseif ((1.0 / n) <= 3e-54)
		tmp = log((1.0 + (1.0 / x))) / n;
	elseif ((1.0 / n) <= 2e-17)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 5e-16)
		tmp = log(x) / -n;
	elseif ((1.0 / n) <= 5e+156)
		tmp = t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e+88], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e-54], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-17], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-16], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+156], t$95$0, N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -4.99999999999999997e88 or 5.0000000000000004e-16 < (/.f64 1 n) < 4.99999999999999992e156

   1. Initial program 90.3%

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

   3. Taylor expanded in x around 0 64.1%

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

      1. *-rgt-identity 64.1%

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

      2. associate-/l* 64.1%

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

      3. exp-to-pow 64.1%

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

   5. Simplified 64.1%

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

   if -4.99999999999999997e88 < (/.f64 1 n) < 3.00000000000000009e-54

   1. Initial program 40.5%

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

   3. Taylor expanded in n around inf 81.0%

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

      1. +-rgt-identity 81.0%

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

      2. +-rgt-identity 81.0%

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

      3. log1p-define 81.0%

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

   5. Simplified 81.0%

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

      1. log1p-undefine 81.0%

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

      2. diff-log 81.0%

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

   7. Applied egg-rr 81.0%

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

   8. Taylor expanded in x around 0 81.0%

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

   if 3.00000000000000009e-54 < (/.f64 1 n) < 2.00000000000000014e-17

   1. Initial program 16.6%

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

   3. Taylor expanded in n around inf 46.3%

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

      1. +-rgt-identity 46.3%

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

      2. +-rgt-identity 46.3%

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

      3. log1p-define 46.3%

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

   5. Simplified 46.3%

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

   6. Taylor expanded in x around inf 71.7%

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

   if 2.00000000000000014e-17 < (/.f64 1 n) < 5.0000000000000004e-16

   1. Initial program 36.8%

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

   3. Taylor expanded in n around inf 85.4%

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

      1. +-rgt-identity 85.4%

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

      2. +-rgt-identity 85.4%

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

      3. log1p-define 85.4%

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

   5. Simplified 85.4%

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

   6. Taylor expanded in x around 0 85.4%

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

      1. neg-mul-1 85.4%

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

   8. Simplified 85.4%

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

   if 4.99999999999999992e156 < (/.f64 1 n)

   1. Initial program 17.8%

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

   3. Taylor expanded in n around inf 6.8%

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

      1. +-rgt-identity 6.8%

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

      2. +-rgt-identity 6.8%

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

      3. log1p-define 6.8%

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

   5. Simplified 6.8%

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

   6. Taylor expanded in x around inf 53.7%

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

      1. *-commutative 53.7%

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

   8. Simplified 53.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{+88}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+156}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+156}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-43)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 3e-54)
       (/ (log (+ 1.0 (/ 1.0 x))) n)
       (if (<= (/ 1.0 n) 2e-17)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 5e-16)
           (/ (log x) (- n))
           (if (<= (/ 1.0 n) 5e+156) (- 1.0 t_0) (/ 1.0 (* n x)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-16) {
		tmp = log(x) / -n;
	} else if ((1.0 / n) <= 5e+156) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-5d-43)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 3d-54) then
        tmp = log((1.0d0 + (1.0d0 / x))) / n
    else if ((1.0d0 / n) <= 2d-17) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 5d-16) then
        tmp = log(x) / -n
    else if ((1.0d0 / n) <= 5d+156) then
        tmp = 1.0d0 - t_0
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 5e-16) {
		tmp = Math.log(x) / -n;
	} else if ((1.0 / n) <= 5e+156) {
		tmp = 1.0 - t_0;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-43:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 3e-54:
		tmp = math.log((1.0 + (1.0 / x))) / n
	elif (1.0 / n) <= 2e-17:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 5e-16:
		tmp = math.log(x) / -n
	elif (1.0 / n) <= 5e+156:
		tmp = 1.0 - t_0
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-43)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 3e-54)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	elseif (Float64(1.0 / n) <= 2e-17)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 5e-16)
		tmp = Float64(log(x) / Float64(-n));
	elseif (Float64(1.0 / n) <= 5e+156)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -5e-43)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 3e-54)
		tmp = log((1.0 + (1.0 / x))) / n;
	elseif ((1.0 / n) <= 2e-17)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 5e-16)
		tmp = log(x) / -n;
	elseif ((1.0 / n) <= 5e+156)
		tmp = 1.0 - t_0;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-43], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e-54], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-17], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-16], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+156], N[(1.0 - t$95$0), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-43

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf 95.1%

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

      1. log-rec 95.1%

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

      2. mul-1-neg 95.1%

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

      3. associate-*r/ 95.1%

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

      4. associate-*r* 95.1%

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

      5. metadata-eval 95.1%

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

      6. *-commutative 95.1%

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

      7. associate-/l* 95.1%

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

      8. exp-to-pow 95.1%

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

      9. *-commutative 95.1%

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

   5. Simplified 95.1%

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

   if -5.00000000000000019e-43 < (/.f64 1 n) < 3.00000000000000009e-54

   1. Initial program 33.9%

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

   3. Taylor expanded in n around inf 85.7%

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

      1. +-rgt-identity 85.7%

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

      2. +-rgt-identity 85.7%

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

      3. log1p-define 85.7%

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

   5. Simplified 85.7%

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

      1. log1p-undefine 85.7%

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

      2. diff-log 85.7%

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

   7. Applied egg-rr 85.7%

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

   8. Taylor expanded in x around 0 85.7%

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

   if 3.00000000000000009e-54 < (/.f64 1 n) < 2.00000000000000014e-17

   1. Initial program 16.6%

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

   3. Taylor expanded in n around inf 46.3%

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

      1. +-rgt-identity 46.3%

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

      2. +-rgt-identity 46.3%

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

      3. log1p-define 46.3%

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

   5. Simplified 46.3%

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

   6. Taylor expanded in x around inf 71.7%

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

   if 2.00000000000000014e-17 < (/.f64 1 n) < 5.0000000000000004e-16

   1. Initial program 36.8%

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

   3. Taylor expanded in n around inf 85.4%

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

      1. +-rgt-identity 85.4%

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

      2. +-rgt-identity 85.4%

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

      3. log1p-define 85.4%

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

   5. Simplified 85.4%

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

   6. Taylor expanded in x around 0 85.4%

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

      1. neg-mul-1 85.4%

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

   8. Simplified 85.4%

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

   if 5.0000000000000004e-16 < (/.f64 1 n) < 4.99999999999999992e156

   1. Initial program 73.3%

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

   3. Taylor expanded in x around 0 73.3%

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

      1. *-rgt-identity 73.3%

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

      2. associate-/l* 73.3%

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

      3. exp-to-pow 73.3%

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

   5. Simplified 73.3%

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

   if 4.99999999999999992e156 < (/.f64 1 n)

   1. Initial program 17.8%

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

   3. Taylor expanded in n around inf 6.8%

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

      1. +-rgt-identity 6.8%

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

      2. +-rgt-identity 6.8%

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

      3. log1p-define 6.8%

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

   5. Simplified 6.8%

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

   6. Taylor expanded in x around inf 53.7%

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

      1. *-commutative 53.7%

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

   8. Simplified 53.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+156}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{x}{1 + x} + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-43)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 3e-54)
       (/ (log (+ 1.0 (/ 1.0 x))) n)
       (if (<= (/ 1.0 n) 2e-17)
         (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) n)
         (if (<= (/ 1.0 n) 5e+156)
           (- (+ 1.0 (/ x n)) t_0)
           (/ (log1p (+ (/ x (+ 1.0 x)) -1.0)) (- n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e+156) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log1p(((x / (1.0 + x)) + -1.0)) / -n;
	}
	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) <= -5e-43) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 3e-54) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e-17) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e+156) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log1p(((x / (1.0 + x)) + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-43:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 3e-54:
		tmp = math.log((1.0 + (1.0 / x))) / n
	elif (1.0 / n) <= 2e-17:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / n
	elif (1.0 / n) <= 5e+156:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log1p(((x / (1.0 + x)) + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-43)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 3e-54)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	elseif (Float64(1.0 / n) <= 2e-17)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	elseif (Float64(1.0 / n) <= 5e+156)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(log1p(Float64(Float64(x / Float64(1.0 + x)) + -1.0)) / Float64(-n));
	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], -5e-43], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 3e-54], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-17], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+156], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[Log[1 + N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-43

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf 95.1%

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

      1. log-rec 95.1%

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

      2. mul-1-neg 95.1%

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

      3. associate-*r/ 95.1%

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

      4. associate-*r* 95.1%

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

      5. metadata-eval 95.1%

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

      6. *-commutative 95.1%

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

      7. associate-/l* 95.1%

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

      8. exp-to-pow 95.1%

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

      9. *-commutative 95.1%

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

   5. Simplified 95.1%

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

   if -5.00000000000000019e-43 < (/.f64 1 n) < 3.00000000000000009e-54

   1. Initial program 33.9%

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

   3. Taylor expanded in n around inf 85.7%

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

      1. +-rgt-identity 85.7%

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

      2. +-rgt-identity 85.7%

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

      3. log1p-define 85.7%

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

   5. Simplified 85.7%

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

      1. log1p-undefine 85.7%

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

      2. diff-log 85.7%

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

   7. Applied egg-rr 85.7%

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

   8. Taylor expanded in x around 0 85.7%

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

   if 3.00000000000000009e-54 < (/.f64 1 n) < 2.00000000000000014e-17

   1. Initial program 16.6%

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

   3. Taylor expanded in n around inf 46.3%

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

      1. +-rgt-identity 46.3%

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

      2. +-rgt-identity 46.3%

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

      3. log1p-define 46.3%

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

   5. Simplified 46.3%

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

   6. Taylor expanded in x around inf 73.5%

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

      1. associate-*r/ 73.5%

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

      2. metadata-eval 73.5%

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

   8. Simplified 73.5%

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

   if 2.00000000000000014e-17 < (/.f64 1 n) < 4.99999999999999992e156

   1. Initial program 72.0%

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

   3. Taylor expanded in x around 0 72.1%

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

   if 4.99999999999999992e156 < (/.f64 1 n)

   1. Initial program 17.8%

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

   3. Taylor expanded in n around inf 6.8%

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

      1. +-rgt-identity 6.8%

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

      2. +-rgt-identity 6.8%

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

      3. log1p-define 6.8%

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

   5. Simplified 6.8%

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

      1. log1p-undefine 6.8%

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

      2. diff-log 6.8%

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

   7. Applied egg-rr 6.8%

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

      1. clear-num 6.8%

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

      2. log-rec 6.8%

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

   9. Applied egg-rr 6.8%

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

       1. log1p-expm1-u 79.6%

          \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{x}{1 + x}\right)\right)\right)}}{n} \]

       2. expm1-undefine 79.6%

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

       3. add-exp-log 79.6%

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

       4. +-commutative 79.6%

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

   11. Applied egg-rr 79.6%

       \[\leadsto \frac{-\color{blue}{\mathsf{log1p}\left(\frac{x}{x + 1} - 1\right)}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{x}{1 + x} + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 2.2 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 2.2e-105)
     t_0
     (if (<= x 8e-77)
       t_1
       (if (<= x 2.8e-46)
         t_0
         (if (<= x 3.05e-37)
           t_1
           (if (<= x 1.0) (- (/ x n) (/ (log x) n)) (/ (/ 1.0 x) n))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 2.2e-105) {
		tmp = t_0;
	} else if (x <= 8e-77) {
		tmp = t_1;
	} else if (x <= 2.8e-46) {
		tmp = t_0;
	} else if (x <= 3.05e-37) {
		tmp = t_1;
	} else if (x <= 1.0) {
		tmp = (x / n) - (log(x) / n);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 2.2d-105) then
        tmp = t_0
    else if (x <= 8d-77) then
        tmp = t_1
    else if (x <= 2.8d-46) then
        tmp = t_0
    else if (x <= 3.05d-37) then
        tmp = t_1
    else if (x <= 1.0d0) then
        tmp = (x / n) - (log(x) / n)
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 2.2e-105) {
		tmp = t_0;
	} else if (x <= 8e-77) {
		tmp = t_1;
	} else if (x <= 2.8e-46) {
		tmp = t_0;
	} else if (x <= 3.05e-37) {
		tmp = t_1;
	} else if (x <= 1.0) {
		tmp = (x / n) - (Math.log(x) / n);
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 2.2e-105:
		tmp = t_0
	elif x <= 8e-77:
		tmp = t_1
	elif x <= 2.8e-46:
		tmp = t_0
	elif x <= 3.05e-37:
		tmp = t_1
	elif x <= 1.0:
		tmp = (x / n) - (math.log(x) / n)
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 2.2e-105)
		tmp = t_0;
	elseif (x <= 8e-77)
		tmp = t_1;
	elseif (x <= 2.8e-46)
		tmp = t_0;
	elseif (x <= 3.05e-37)
		tmp = t_1;
	elseif (x <= 1.0)
		tmp = Float64(Float64(x / n) - Float64(log(x) / n));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 2.2e-105)
		tmp = t_0;
	elseif (x <= 8e-77)
		tmp = t_1;
	elseif (x <= 2.8e-46)
		tmp = t_0;
	elseif (x <= 3.05e-37)
		tmp = t_1;
	elseif (x <= 1.0)
		tmp = (x / n) - (log(x) / n);
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.2e-105], t$95$0, If[LessEqual[x, 8e-77], t$95$1, If[LessEqual[x, 2.8e-46], t$95$0, If[LessEqual[x, 3.05e-37], t$95$1, If[LessEqual[x, 1.0], N[(N[(x / n), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.20000000000000004e-105 or 7.9999999999999994e-77 < x < 2.7999999999999998e-46

   1. Initial program 37.1%

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

   3. Taylor expanded in n around inf 60.9%

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

      1. +-rgt-identity 60.9%

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

      2. +-rgt-identity 60.9%

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

      3. log1p-define 60.9%

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

   5. Simplified 60.9%

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

   6. Taylor expanded in x around 0 60.9%

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

      1. neg-mul-1 60.9%

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

   8. Simplified 60.9%

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

   if 2.20000000000000004e-105 < x < 7.9999999999999994e-77 or 2.7999999999999998e-46 < x < 3.0500000000000002e-37

   1. Initial program 72.5%

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

   3. Taylor expanded in x around 0 72.5%

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

      1. *-rgt-identity 72.5%

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

      2. associate-/l* 72.5%

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

      3. exp-to-pow 72.5%

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

   5. Simplified 72.5%

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

   if 3.0500000000000002e-37 < x < 1

   1. Initial program 22.2%

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

   3. Taylor expanded in n around inf 64.1%

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

      1. +-rgt-identity 64.1%

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

      2. +-rgt-identity 64.1%

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

      3. log1p-define 64.1%

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

   5. Simplified 64.1%

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

   6. Taylor expanded in x around 0 62.4%

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

      1. neg-mul-1 62.4%

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

      2. +-commutative 62.4%

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

      3. unsub-neg 62.4%

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

   8. Simplified 62.4%

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

   if 1 < x

   1. Initial program 67.6%

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

   3. Taylor expanded in n around inf 66.3%

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

      1. +-rgt-identity 66.3%

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

      2. +-rgt-identity 66.3%

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

      3. log1p-define 66.3%

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

   5. Simplified 66.3%

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

   6. Taylor expanded in x around inf 68.0%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-77}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-37}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x}{n} - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 3.1 \cdot 10^{-104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 3.1e-104)
     t_0
     (if (<= x 2.5e-76)
       t_1
       (if (<= x 2.75e-46)
         t_0
         (if (<= x 3e-37)
           t_1
           (if (<= x 1.0) (/ (- x (log x)) n) (/ (/ 1.0 x) n))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 3.1e-104) {
		tmp = t_0;
	} else if (x <= 2.5e-76) {
		tmp = t_1;
	} else if (x <= 2.75e-46) {
		tmp = t_0;
	} else if (x <= 3e-37) {
		tmp = t_1;
	} else if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(x) / -n
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 3.1d-104) then
        tmp = t_0
    else if (x <= 2.5d-76) then
        tmp = t_1
    else if (x <= 2.75d-46) then
        tmp = t_0
    else if (x <= 3d-37) then
        tmp = t_1
    else if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 3.1e-104) {
		tmp = t_0;
	} else if (x <= 2.5e-76) {
		tmp = t_1;
	} else if (x <= 2.75e-46) {
		tmp = t_0;
	} else if (x <= 3e-37) {
		tmp = t_1;
	} else if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 3.1e-104:
		tmp = t_0
	elif x <= 2.5e-76:
		tmp = t_1
	elif x <= 2.75e-46:
		tmp = t_0
	elif x <= 3e-37:
		tmp = t_1
	elif x <= 1.0:
		tmp = (x - math.log(x)) / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 3.1e-104)
		tmp = t_0;
	elseif (x <= 2.5e-76)
		tmp = t_1;
	elseif (x <= 2.75e-46)
		tmp = t_0;
	elseif (x <= 3e-37)
		tmp = t_1;
	elseif (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 3.1e-104)
		tmp = t_0;
	elseif (x <= 2.5e-76)
		tmp = t_1;
	elseif (x <= 2.75e-46)
		tmp = t_0;
	elseif (x <= 3e-37)
		tmp = t_1;
	elseif (x <= 1.0)
		tmp = (x - log(x)) / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.1e-104], t$95$0, If[LessEqual[x, 2.5e-76], t$95$1, If[LessEqual[x, 2.75e-46], t$95$0, If[LessEqual[x, 3e-37], t$95$1, If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 3.09999999999999976e-104 or 2.4999999999999999e-76 < x < 2.74999999999999992e-46

   1. Initial program 37.1%

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

   3. Taylor expanded in n around inf 60.9%

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

      1. +-rgt-identity 60.9%

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

      2. +-rgt-identity 60.9%

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

      3. log1p-define 60.9%

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

   5. Simplified 60.9%

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

   6. Taylor expanded in x around 0 60.9%

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

      1. neg-mul-1 60.9%

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

   8. Simplified 60.9%

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

   if 3.09999999999999976e-104 < x < 2.4999999999999999e-76 or 2.74999999999999992e-46 < x < 3e-37

   1. Initial program 72.5%

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

   3. Taylor expanded in x around 0 72.5%

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

      1. *-rgt-identity 72.5%

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

      2. associate-/l* 72.5%

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

      3. exp-to-pow 72.5%

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

   5. Simplified 72.5%

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

   if 3e-37 < x < 1

   1. Initial program 22.2%

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

   3. Taylor expanded in n around inf 64.1%

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

      1. +-rgt-identity 64.1%

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

      2. +-rgt-identity 64.1%

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

      3. log1p-define 64.1%

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

   5. Simplified 64.1%

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

   6. Taylor expanded in x around 0 62.3%

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

      1. neg-mul-1 62.3%

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

      2. unsub-neg 62.3%

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

   8. Simplified 62.3%

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

   if 1 < x

   1. Initial program 67.6%

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

   3. Taylor expanded in n around inf 66.3%

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

      1. +-rgt-identity 66.3%

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

      2. +-rgt-identity 66.3%

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

      3. log1p-define 66.3%

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

   5. Simplified 66.3%

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

   6. Taylor expanded in x around inf 68.0%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-104}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-76}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-46}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-37}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 40.5%

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

   3. Taylor expanded in n around inf 54.7%

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

      1. +-rgt-identity 54.7%

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

      2. +-rgt-identity 54.7%

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

      3. log1p-define 54.7%

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

   5. Simplified 54.7%

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

   6. Taylor expanded in x around 0 54.6%

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

      1. neg-mul-1 54.6%

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

      2. unsub-neg 54.6%

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

   8. Simplified 54.6%

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

   if 1 < x

   1. Initial program 67.6%

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

   3. Taylor expanded in n around inf 66.3%

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

      1. +-rgt-identity 66.3%

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

      2. +-rgt-identity 66.3%

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

      3. log1p-define 66.3%

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

   5. Simplified 66.3%

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

   6. Taylor expanded in x around inf 68.0%

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

4. Final simplification 60.2%

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

Alternative 13: 56.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.55000000000000004

   1. Initial program 40.5%

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

   3. Taylor expanded in n around inf 54.7%

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

      1. +-rgt-identity 54.7%

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

      2. +-rgt-identity 54.7%

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

      3. log1p-define 54.7%

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

   5. Simplified 54.7%

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

   6. Taylor expanded in x around 0 54.0%

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

      1. neg-mul-1 54.0%

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

   8. Simplified 54.0%

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

   if 0.55000000000000004 < x

   1. Initial program 67.6%

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

   3. Taylor expanded in n around inf 66.3%

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

      1. +-rgt-identity 66.3%

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

      2. +-rgt-identity 66.3%

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

      3. log1p-define 66.3%

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

   5. Simplified 66.3%

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

   6. Taylor expanded in x around inf 68.0%

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

4. Final simplification 59.9%

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

Alternative 14: 47.8% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -200000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -200000.0) (/ 0.0 n) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -200000.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-200000.0d0)) then
        tmp = 0.0d0 / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -200000.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -200000.0:
		tmp = 0.0 / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -200000.0)
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -200000.0)
		tmp = 0.0 / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -200000.0], N[(0.0 / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -2e5

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 49.7%

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

      1. +-rgt-identity 49.7%

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

      2. +-rgt-identity 49.7%

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

      3. log1p-define 49.7%

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

   5. Simplified 49.7%

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

      1. log1p-undefine 49.7%

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

      2. diff-log 49.7%

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

   7. Applied egg-rr 49.7%

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

   8. Taylor expanded in x around inf 50.8%

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

   if -2e5 < (/.f64 1 n)

   1. Initial program 36.0%

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

   3. Taylor expanded in n around inf 62.9%

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

      1. +-rgt-identity 62.9%

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

      2. +-rgt-identity 62.9%

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

      3. log1p-define 62.9%

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

   5. Simplified 62.9%

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

   6. Taylor expanded in x around inf 44.2%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -200000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.1% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in n around inf 59.6%

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

   1. +-rgt-identity 59.6%

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

   2. +-rgt-identity 59.6%

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

   3. log1p-define 59.6%

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

5. Simplified 59.6%

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

6. Taylor expanded in x around inf 41.2%

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

   1. *-commutative 41.2%

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

8. Simplified 41.2%

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

9. Final simplification 41.2%

   \[\leadsto \frac{1}{n \cdot x} \]
10. Add Preprocessing

Alternative 16: 41.6% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in n around inf 59.6%

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

   1. +-rgt-identity 59.6%

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

   2. +-rgt-identity 59.6%

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

   3. log1p-define 59.6%

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

5. Simplified 59.6%

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

6. Taylor expanded in x around inf 42.1%

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

7. Final simplification 42.1%

   \[\leadsto \frac{\frac{1}{x}}{n} \]
8. Add Preprocessing

Alternative 17: 4.5% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in x around 0 31.0%

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

4. Taylor expanded in x around inf 4.4%

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

5. Final simplification 4.4%

   \[\leadsto \frac{x}{n} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))