2nthrt (problem 3.4.6)

Percentage Accurate: 53.6% → 84.9%

Time: 45.4s

Alternatives: 15

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: 53.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-26)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 5e-134)
     (/ (log (/ x (+ 1.0 x))) (- n))
     (if (<= (/ 1.0 n) 4e-79)
       (/ (/ 1.0 x) n)
       (if (<= (/ 1.0 n) 2e-22)
         (/ (log (/ (+ 1.0 x) x)) n)
         (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-22) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-22) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5e-26:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 5e-134:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 4e-79:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e-22:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-26)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-22)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-26], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-22], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-26

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. log-rec 96.6%

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

      3. mul-1-neg 96.6%

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

      4. distribute-neg-frac 96.6%

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

      5. mul-1-neg 96.6%

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

      6. remove-double-neg 96.6%

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

      7. *-commutative 96.6%

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

   5. Simplified 96.6%

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

   if -5.00000000000000019e-26 < (/.f64 1 n) < 5.0000000000000003e-134

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf 87.9%

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

      1. +-rgt-identity 87.9%

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

      2. +-rgt-identity 87.9%

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

      3. log1p-define 87.9%

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

   5. Simplified 87.9%

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

      1. log1p-undefine 87.9%

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

      2. diff-log 88.0%

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

   7. Applied egg-rr 88.0%

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

      1. clear-num 88.0%

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

      2. log-rec 88.0%

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

   9. Applied egg-rr 88.0%

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

   if 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in n around inf 50.1%

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

      1. +-rgt-identity 50.1%

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

      2. +-rgt-identity 50.1%

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

      3. log1p-define 50.1%

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

   5. Simplified 50.1%

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

   6. Taylor expanded in x around inf 76.1%

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

   if 4e-79 < (/.f64 1 n) < 2.0000000000000001e-22

   1. Initial program 19.1%

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

   3. Taylor expanded in n around inf 79.7%

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

      1. +-rgt-identity 79.7%

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

      2. +-rgt-identity 79.7%

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

      3. log1p-define 79.7%

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

   5. Simplified 79.7%

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

      1. log1p-undefine 79.7%

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

      2. diff-log 79.7%

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

   7. Applied egg-rr 79.7%

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

   if 2.0000000000000001e-22 < (/.f64 1 n)

   1. Initial program 54.1%

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

   3. Taylor expanded in n around 0 54.1%

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

      1. log1p-define 95.1%

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

      2. *-rgt-identity 95.1%

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

      3. associate-/l* 95.1%

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

      4. exp-to-pow 95.1%

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

   5. Simplified 95.1%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot x\right)}^{-3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-26)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-134)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 4e-79)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 4e-8)
           (/ (log (/ (+ 1.0 x) x)) n)
           (if (<= (/ 1.0 n) 5e+157)
             (- (+ 1.0 (/ x n)) t_0)
             (cbrt (pow (* n x) -3.0)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = cbrt(pow((n * x), -3.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.cbrt(Math.pow((n * x), -3.0));
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-26)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 4e-8)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+157)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = cbrt((Float64(n * x) ^ -3.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-26], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-8], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+157], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Power[N[Power[N[(n * x), $MachinePrecision], -3.0], $MachinePrecision], 1/3], $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-26

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. log-rec 96.6%

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

      2. mul-1-neg 96.6%

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

      3. associate-*r/ 96.6%

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

      4. associate-*r* 96.6%

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

      5. metadata-eval 96.6%

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

      6. *-commutative 96.6%

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

      7. associate-/l* 96.6%

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

      8. exp-to-pow 96.6%

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

      9. *-commutative 96.6%

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

   5. Simplified 96.6%

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

   if -5.00000000000000019e-26 < (/.f64 1 n) < 5.0000000000000003e-134

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf 87.9%

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

      1. +-rgt-identity 87.9%

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

      2. +-rgt-identity 87.9%

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

      3. log1p-define 87.9%

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

   5. Simplified 87.9%

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

      1. log1p-undefine 87.9%

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

      2. diff-log 88.0%

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

   7. Applied egg-rr 88.0%

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

      1. clear-num 88.0%

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

      2. log-rec 88.0%

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

   9. Applied egg-rr 88.0%

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

   if 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in n around inf 50.1%

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

      1. +-rgt-identity 50.1%

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

      2. +-rgt-identity 50.1%

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

      3. log1p-define 50.1%

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

   5. Simplified 50.1%

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

   6. Taylor expanded in x around inf 76.1%

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

   if 4e-79 < (/.f64 1 n) < 4.0000000000000001e-8

   1. Initial program 18.4%

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

   3. Taylor expanded in n around inf 76.0%

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

      1. +-rgt-identity 76.0%

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

      2. +-rgt-identity 76.0%

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

      3. log1p-define 76.0%

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

   5. Simplified 76.0%

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

      1. log1p-undefine 76.0%

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

      2. diff-log 76.0%

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

   7. Applied egg-rr 76.0%

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

   if 4.0000000000000001e-8 < (/.f64 1 n) < 4.99999999999999976e157

   1. Initial program 92.4%

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

   3. Taylor expanded in x around 0 93.8%

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

   if 4.99999999999999976e157 < (/.f64 1 n)

   1. Initial program 29.6%

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

   3. Taylor expanded in n around inf 7.3%

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

      1. +-rgt-identity 7.3%

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

      2. +-rgt-identity 7.3%

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

      3. log1p-define 7.3%

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

   5. Simplified 7.3%

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

   6. Taylor expanded in x around inf 58.5%

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

      1. *-commutative 58.5%

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

   8. Simplified 58.5%

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

      1. add-cbrt-cube 74.7%

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

      2. pow1/3 74.7%

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

      3. pow3 74.7%

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

      4. inv-pow 74.7%

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

      5. pow-pow 74.7%

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

      6. metadata-eval 74.7%

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

   10. Applied egg-rr 74.7%

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

       1. unpow1/3 74.7%

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

   12. Simplified 74.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(x \cdot n\right)}^{-3}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot x\right)}^{-3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot x\right)}^{-3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5e-26)
   (/ (exp (/ (log x) n)) (* n x))
   (if (<= (/ 1.0 n) 5e-134)
     (/ (log (/ x (+ 1.0 x))) (- n))
     (if (<= (/ 1.0 n) 4e-79)
       (/ (/ 1.0 x) n)
       (if (<= (/ 1.0 n) 4e-8)
         (/ (log (/ (+ 1.0 x) x)) n)
         (if (<= (/ 1.0 n) 5e+157)
           (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
           (cbrt (pow (* n x) -3.0))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = cbrt(pow((n * x), -3.0));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.cbrt(Math.pow((n * x), -3.0));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-26)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 4e-8)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+157)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = cbrt((Float64(n * x) ^ -3.0));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-26], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-8], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+157], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(n * x), $MachinePrecision], -3.0], $MachinePrecision], 1/3], $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-26

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. log-rec 96.6%

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

      3. mul-1-neg 96.6%

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

      4. distribute-neg-frac 96.6%

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

      5. mul-1-neg 96.6%

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

      6. remove-double-neg 96.6%

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

      7. *-commutative 96.6%

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

   5. Simplified 96.6%

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

   if -5.00000000000000019e-26 < (/.f64 1 n) < 5.0000000000000003e-134

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf 87.9%

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

      1. +-rgt-identity 87.9%

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

      2. +-rgt-identity 87.9%

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

      3. log1p-define 87.9%

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

   5. Simplified 87.9%

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

      1. log1p-undefine 87.9%

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

      2. diff-log 88.0%

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

   7. Applied egg-rr 88.0%

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

      1. clear-num 88.0%

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

      2. log-rec 88.0%

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

   9. Applied egg-rr 88.0%

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

   if 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in n around inf 50.1%

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

      1. +-rgt-identity 50.1%

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

      2. +-rgt-identity 50.1%

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

      3. log1p-define 50.1%

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

   5. Simplified 50.1%

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

   6. Taylor expanded in x around inf 76.1%

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

   if 4e-79 < (/.f64 1 n) < 4.0000000000000001e-8

   1. Initial program 18.4%

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

   3. Taylor expanded in n around inf 76.0%

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

      1. +-rgt-identity 76.0%

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

      2. +-rgt-identity 76.0%

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

      3. log1p-define 76.0%

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

   5. Simplified 76.0%

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

      1. log1p-undefine 76.0%

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

      2. diff-log 76.0%

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

   7. Applied egg-rr 76.0%

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

   if 4.0000000000000001e-8 < (/.f64 1 n) < 4.99999999999999976e157

   1. Initial program 92.4%

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

   3. Taylor expanded in x around 0 93.8%

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

   if 4.99999999999999976e157 < (/.f64 1 n)

   1. Initial program 29.6%

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

   3. Taylor expanded in n around inf 7.3%

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

      1. +-rgt-identity 7.3%

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

      2. +-rgt-identity 7.3%

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

      3. log1p-define 7.3%

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

   5. Simplified 7.3%

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

   6. Taylor expanded in x around inf 58.5%

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

      1. *-commutative 58.5%

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

   8. Simplified 58.5%

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

      1. add-cbrt-cube 74.7%

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

      2. pow1/3 74.7%

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

      3. pow3 74.7%

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

      4. inv-pow 74.7%

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

      5. pow-pow 74.7%

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

      6. metadata-eval 74.7%

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

   10. Applied egg-rr 74.7%

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

       1. unpow1/3 74.7%

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

   12. Simplified 74.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(x \cdot n\right)}^{-3}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot x\right)}^{-3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ t_1 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -500000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+246}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ (+ 1.0 x) x)) n)) (t_1 (/ (/ 1.0 x) n)))
   (if (<= (/ 1.0 n) -500000.0)
     (/ 0.0 n)
     (if (<= (/ 1.0 n) -1e-31)
       t_1
       (if (<= (/ 1.0 n) 5e-134)
         t_0
         (if (<= (/ 1.0 n) 4e-79)
           t_1
           (if (<= (/ 1.0 n) 4e-8)
             t_0
             (if (<= (/ 1.0 n) 5e+246)
               (- 1.0 (pow x (/ 1.0 n)))
               (/ 1.0 (* n x))))))))))

C

double code(double x, double n) {
	double t_0 = log(((1.0 + x) / x)) / n;
	double t_1 = (1.0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -500000.0) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -1e-31) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-134) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+246) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log(((1.0d0 + x) / x)) / n
    t_1 = (1.0d0 / x) / n
    if ((1.0d0 / n) <= (-500000.0d0)) then
        tmp = 0.0d0 / n
    else if ((1.0d0 / n) <= (-1d-31)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-134) then
        tmp = t_0
    else if ((1.0d0 / n) <= 4d-79) then
        tmp = t_1
    else if ((1.0d0 / n) <= 4d-8) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d+246) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    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.log(((1.0 + x) / x)) / n;
	double t_1 = (1.0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -500000.0) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -1e-31) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-134) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = t_1;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e+246) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(((1.0 + x) / x)) / n
	t_1 = (1.0 / x) / n
	tmp = 0
	if (1.0 / n) <= -500000.0:
		tmp = 0.0 / n
	elif (1.0 / n) <= -1e-31:
		tmp = t_1
	elif (1.0 / n) <= 5e-134:
		tmp = t_0
	elif (1.0 / n) <= 4e-79:
		tmp = t_1
	elif (1.0 / n) <= 4e-8:
		tmp = t_0
	elif (1.0 / n) <= 5e+246:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(Float64(Float64(1.0 + x) / x)) / n)
	t_1 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500000.0)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= -1e-31)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 4e-8)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e+246)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(((1.0 + x) / x)) / n;
	t_1 = (1.0 / x) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -500000.0)
		tmp = 0.0 / n;
	elseif ((1.0 / n) <= -1e-31)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-134)
		tmp = t_0;
	elseif ((1.0 / n) <= 4e-79)
		tmp = t_1;
	elseif ((1.0 / n) <= 4e-8)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e+246)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -500000.0], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-31], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-8], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+246], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -5e5

   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 63.3%

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

      1. +-rgt-identity 63.3%

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

      2. +-rgt-identity 63.3%

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

      3. log1p-define 63.3%

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

   5. Simplified 63.3%

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

      1. log1p-undefine 63.3%

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

      2. diff-log 63.3%

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

   7. Applied egg-rr 63.3%

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

   8. Taylor expanded in x around inf 67.2%

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

   if -5e5 < (/.f64 1 n) < -1e-31 or 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 23.7%

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

   3. Taylor expanded in n around inf 45.4%

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

      1. +-rgt-identity 45.4%

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

      2. +-rgt-identity 45.4%

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

      3. log1p-define 45.4%

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

   5. Simplified 45.4%

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

   6. Taylor expanded in x around inf 68.7%

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

   if -1e-31 < (/.f64 1 n) < 5.0000000000000003e-134 or 4e-79 < (/.f64 1 n) < 4.0000000000000001e-8

   1. Initial program 33.6%

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

   3. Taylor expanded in n around inf 86.6%

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

      1. +-rgt-identity 86.6%

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

      2. +-rgt-identity 86.6%

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

      3. log1p-define 86.6%

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

   5. Simplified 86.6%

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

      1. log1p-undefine 86.6%

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

      2. diff-log 86.6%

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

   7. Applied egg-rr 86.6%

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

   if 4.0000000000000001e-8 < (/.f64 1 n) < 4.99999999999999976e246

   1. Initial program 75.0%

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

   3. Taylor expanded in x around 0 71.4%

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

      1. *-rgt-identity 71.4%

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

      2. associate-/l* 71.4%

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

      3. exp-to-pow 71.4%

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

   5. Simplified 71.4%

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

   if 4.99999999999999976e246 < (/.f64 1 n)

   1. Initial program 11.2%

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

   3. Taylor expanded in n around inf 9.6%

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

      1. +-rgt-identity 9.6%

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

      2. +-rgt-identity 9.6%

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

      3. log1p-define 9.6%

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

   5. Simplified 9.6%

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

   6. Taylor expanded in x around inf 91.9%

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

      1. *-commutative 91.9%

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

   8. Simplified 91.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+246}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -500000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+246}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) n)))
   (if (<= (/ 1.0 n) -500000.0)
     (/ 0.0 n)
     (if (<= (/ 1.0 n) -1e-31)
       t_0
       (if (<= (/ 1.0 n) 5e-134)
         (/ (log (/ x (+ 1.0 x))) (- n))
         (if (<= (/ 1.0 n) 4e-79)
           t_0
           (if (<= (/ 1.0 n) 4e-8)
             (/ (log (/ (+ 1.0 x) x)) n)
             (if (<= (/ 1.0 n) 5e+246)
               (- 1.0 (pow x (/ 1.0 n)))
               (/ 1.0 (* n x))))))))))

C

double code(double x, double n) {
	double t_0 = (1.0 / x) / n;
	double tmp;
	if ((1.0 / n) <= -500000.0) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -1e-31) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-134) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+246) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} 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) / n
    if ((1.0d0 / n) <= (-500000.0d0)) then
        tmp = 0.0d0 / n
    else if ((1.0d0 / n) <= (-1d-31)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-134) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 4d-79) then
        tmp = t_0
    else if ((1.0d0 / n) <= 4d-8) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 5d+246) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    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 / x) / n;
	double tmp;
	if ((1.0 / n) <= -500000.0) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -1e-31) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-134) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = t_0;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+246) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (1.0 / x) / n
	tmp = 0
	if (1.0 / n) <= -500000.0:
		tmp = 0.0 / n
	elif (1.0 / n) <= -1e-31:
		tmp = t_0
	elif (1.0 / n) <= 5e-134:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 4e-79:
		tmp = t_0
	elif (1.0 / n) <= 4e-8:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e+246:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / x) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500000.0)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= -1e-31)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 4e-8)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+246)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (1.0 / x) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -500000.0)
		tmp = 0.0 / n;
	elseif ((1.0 / n) <= -1e-31)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-134)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 4e-79)
		tmp = t_0;
	elseif ((1.0 / n) <= 4e-8)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 5e+246)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -500000.0], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-31], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-8], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+246], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5e5

   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 63.3%

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

      1. +-rgt-identity 63.3%

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

      2. +-rgt-identity 63.3%

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

      3. log1p-define 63.3%

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

   5. Simplified 63.3%

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

      1. log1p-undefine 63.3%

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

      2. diff-log 63.3%

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

   7. Applied egg-rr 63.3%

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

   8. Taylor expanded in x around inf 67.2%

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

   if -5e5 < (/.f64 1 n) < -1e-31 or 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 23.7%

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

   3. Taylor expanded in n around inf 45.4%

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

      1. +-rgt-identity 45.4%

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

      2. +-rgt-identity 45.4%

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

      3. log1p-define 45.4%

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

   5. Simplified 45.4%

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

   6. Taylor expanded in x around inf 68.7%

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

   if -1e-31 < (/.f64 1 n) < 5.0000000000000003e-134

   1. Initial program 36.4%

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

   3. Taylor expanded in n around inf 88.5%

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

      1. +-rgt-identity 88.5%

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

      2. +-rgt-identity 88.5%

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

      3. log1p-define 88.5%

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

   5. Simplified 88.5%

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

      1. log1p-undefine 88.5%

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

      2. diff-log 88.6%

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

   7. Applied egg-rr 88.6%

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

      1. clear-num 88.6%

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

      2. log-rec 88.6%

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

   9. Applied egg-rr 88.6%

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

   if 4e-79 < (/.f64 1 n) < 4.0000000000000001e-8

   1. Initial program 18.4%

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

   3. Taylor expanded in n around inf 76.0%

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

      1. +-rgt-identity 76.0%

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

      2. +-rgt-identity 76.0%

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

      3. log1p-define 76.0%

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

   5. Simplified 76.0%

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

      1. log1p-undefine 76.0%

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

      2. diff-log 76.0%

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

   7. Applied egg-rr 76.0%

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

   if 4.0000000000000001e-8 < (/.f64 1 n) < 4.99999999999999976e246

   1. Initial program 75.0%

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

   3. Taylor expanded in x around 0 71.4%

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

      1. *-rgt-identity 71.4%

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

      2. associate-/l* 71.4%

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

      3. exp-to-pow 71.4%

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

   5. Simplified 71.4%

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

   if 4.99999999999999976e246 < (/.f64 1 n)

   1. Initial program 11.2%

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

   3. Taylor expanded in n around inf 9.6%

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

      1. +-rgt-identity 9.6%

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

      2. +-rgt-identity 9.6%

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

      3. log1p-define 9.6%

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

   5. Simplified 9.6%

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

   6. Taylor expanded in x around inf 91.9%

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

      1. *-commutative 91.9%

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

   8. Simplified 91.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+246}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+246}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-26)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-134)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 4e-79)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 4e-8)
           (/ (log (/ (+ 1.0 x) x)) n)
           (if (<= (/ 1.0 n) 5e+246)
             (- (+ 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-26) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+246) {
		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-26)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 5d-134) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 4d-79) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 4d-8) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 5d+246) 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-26) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+246) {
		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-26:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-134:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 4e-79:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 4e-8:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e+246:
		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-26)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 4e-8)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+246)
		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-26)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 5e-134)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 4e-79)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 4e-8)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 5e+246)
		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-26], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-8], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+246], 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-26

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. log-rec 96.6%

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

      2. mul-1-neg 96.6%

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

      3. associate-*r/ 96.6%

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

      4. associate-*r* 96.6%

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

      5. metadata-eval 96.6%

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

      6. *-commutative 96.6%

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

      7. associate-/l* 96.6%

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

      8. exp-to-pow 96.6%

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

      9. *-commutative 96.6%

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

   5. Simplified 96.6%

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

   if -5.00000000000000019e-26 < (/.f64 1 n) < 5.0000000000000003e-134

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf 87.9%

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

      1. +-rgt-identity 87.9%

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

      2. +-rgt-identity 87.9%

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

      3. log1p-define 87.9%

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

   5. Simplified 87.9%

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

      1. log1p-undefine 87.9%

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

      2. diff-log 88.0%

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

   7. Applied egg-rr 88.0%

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

      1. clear-num 88.0%

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

      2. log-rec 88.0%

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

   9. Applied egg-rr 88.0%

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

   if 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in n around inf 50.1%

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

      1. +-rgt-identity 50.1%

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

      2. +-rgt-identity 50.1%

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

      3. log1p-define 50.1%

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

   5. Simplified 50.1%

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

   6. Taylor expanded in x around inf 76.1%

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

   if 4e-79 < (/.f64 1 n) < 4.0000000000000001e-8

   1. Initial program 18.4%

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

   3. Taylor expanded in n around inf 76.0%

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

      1. +-rgt-identity 76.0%

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

      2. +-rgt-identity 76.0%

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

      3. log1p-define 76.0%

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

   5. Simplified 76.0%

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

      1. log1p-undefine 76.0%

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

      2. diff-log 76.0%

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

   7. Applied egg-rr 76.0%

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

   if 4.0000000000000001e-8 < (/.f64 1 n) < 4.99999999999999976e246

   1. Initial program 75.0%

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

   3. Taylor expanded in x around 0 74.3%

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

   if 4.99999999999999976e246 < (/.f64 1 n)

   1. Initial program 11.2%

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

   3. Taylor expanded in n around inf 9.6%

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

      1. +-rgt-identity 9.6%

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

      2. +-rgt-identity 9.6%

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

      3. log1p-define 9.6%

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

   5. Simplified 9.6%

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

   6. Taylor expanded in x around inf 91.9%

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

      1. *-commutative 91.9%

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

   8. Simplified 91.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+246}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 + x}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{t\_1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log t\_0}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(t\_0 + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ x (+ 1.0 x))) (t_1 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-26)
     (/ t_1 (* n x))
     (if (<= (/ 1.0 n) 5e-134)
       (/ (log t_0) (- n))
       (if (<= (/ 1.0 n) 4e-79)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 4e-8)
           (/ (log (/ (+ 1.0 x) x)) n)
           (if (<= (/ 1.0 n) 5e+157)
             (- (+ 1.0 (/ x n)) t_1)
             (/ (log1p (+ t_0 -1.0)) (- n)))))))))

C

double code(double x, double n) {
	double t_0 = x / (1.0 + x);
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = log(t_0) / -n;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = log1p((t_0 + -1.0)) / -n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = x / (1.0 + x);
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -5e-26) {
		tmp = t_1 / (n * x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = Math.log(t_0) / -n;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+157) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = Math.log1p((t_0 + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = x / (1.0 + x)
	t_1 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -5e-26:
		tmp = t_1 / (n * x)
	elif (1.0 / n) <= 5e-134:
		tmp = math.log(t_0) / -n
	elif (1.0 / n) <= 4e-79:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 4e-8:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e+157:
		tmp = (1.0 + (x / n)) - t_1
	else:
		tmp = math.log1p((t_0 + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(x / Float64(1.0 + x))
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5e-26)
		tmp = Float64(t_1 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = Float64(log(t_0) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 4e-8)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+157)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	else
		tmp = Float64(log1p(Float64(t_0 + -1.0)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-26], N[(t$95$1 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], N[(N[Log[t$95$0], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-8], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+157], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[Log[1 + N[(t$95$0 + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -5.00000000000000019e-26

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. log-rec 96.6%

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

      2. mul-1-neg 96.6%

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

      3. associate-*r/ 96.6%

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

      4. associate-*r* 96.6%

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

      5. metadata-eval 96.6%

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

      6. *-commutative 96.6%

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

      7. associate-/l* 96.6%

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

      8. exp-to-pow 96.6%

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

      9. *-commutative 96.6%

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

   5. Simplified 96.6%

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

   if -5.00000000000000019e-26 < (/.f64 1 n) < 5.0000000000000003e-134

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf 87.9%

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

      1. +-rgt-identity 87.9%

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

      2. +-rgt-identity 87.9%

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

      3. log1p-define 87.9%

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

   5. Simplified 87.9%

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

      1. log1p-undefine 87.9%

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

      2. diff-log 88.0%

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

   7. Applied egg-rr 88.0%

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

      1. clear-num 88.0%

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

      2. log-rec 88.0%

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

   9. Applied egg-rr 88.0%

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

   if 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in n around inf 50.1%

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

      1. +-rgt-identity 50.1%

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

      2. +-rgt-identity 50.1%

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

      3. log1p-define 50.1%

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

   5. Simplified 50.1%

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

   6. Taylor expanded in x around inf 76.1%

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

   if 4e-79 < (/.f64 1 n) < 4.0000000000000001e-8

   1. Initial program 18.4%

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

   3. Taylor expanded in n around inf 76.0%

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

      1. +-rgt-identity 76.0%

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

      2. +-rgt-identity 76.0%

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

      3. log1p-define 76.0%

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

   5. Simplified 76.0%

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

      1. log1p-undefine 76.0%

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

      2. diff-log 76.0%

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

   7. Applied egg-rr 76.0%

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

   if 4.0000000000000001e-8 < (/.f64 1 n) < 4.99999999999999976e157

   1. Initial program 92.4%

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

   3. Taylor expanded in x around 0 93.8%

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

   if 4.99999999999999976e157 < (/.f64 1 n)

   1. Initial program 29.6%

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

   3. Taylor expanded in n around inf 7.3%

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

      1. +-rgt-identity 7.3%

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

      2. +-rgt-identity 7.3%

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

      3. log1p-define 7.3%

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

   5. Simplified 7.3%

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

      1. log1p-undefine 7.3%

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

      2. diff-log 7.3%

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

   7. Applied egg-rr 7.3%

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

      1. clear-num 7.3%

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

      2. log-rec 7.3%

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

   9. Applied egg-rr 7.3%

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

       1. log1p-expm1-u 70.7%

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

       2. expm1-undefine 70.7%

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

       3. add-exp-log 70.7%

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

       4. +-commutative 70.7%

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

   11. Applied egg-rr 70.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+157}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\frac{x}{1 + x} + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{t\_0}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+246}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -5e-26)
     (/ t_0 (* n x))
     (if (<= (/ 1.0 n) 5e-134)
       (/ (log (/ x (+ 1.0 x))) (- n))
       (if (<= (/ 1.0 n) 4e-79)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 4e-8)
           (/ (log (/ (+ 1.0 x) x)) n)
           (if (<= (/ 1.0 n) 5e+246) (- 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-26) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+246) {
		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-26)) then
        tmp = t_0 / (n * x)
    else if ((1.0d0 / n) <= 5d-134) then
        tmp = log((x / (1.0d0 + x))) / -n
    else if ((1.0d0 / n) <= 4d-79) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 4d-8) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 5d+246) 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-26) {
		tmp = t_0 / (n * x);
	} else if ((1.0 / n) <= 5e-134) {
		tmp = Math.log((x / (1.0 + x))) / -n;
	} else if ((1.0 / n) <= 4e-79) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 4e-8) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 5e+246) {
		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-26:
		tmp = t_0 / (n * x)
	elif (1.0 / n) <= 5e-134:
		tmp = math.log((x / (1.0 + x))) / -n
	elif (1.0 / n) <= 4e-79:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 4e-8:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 5e+246:
		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-26)
		tmp = Float64(t_0 / Float64(n * x));
	elseif (Float64(1.0 / n) <= 5e-134)
		tmp = Float64(log(Float64(x / Float64(1.0 + x))) / Float64(-n));
	elseif (Float64(1.0 / n) <= 4e-79)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 4e-8)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+246)
		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-26)
		tmp = t_0 / (n * x);
	elseif ((1.0 / n) <= 5e-134)
		tmp = log((x / (1.0 + x))) / -n;
	elseif ((1.0 / n) <= 4e-79)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 4e-8)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 5e+246)
		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-26], N[(t$95$0 / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-134], N[(N[Log[N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-79], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 4e-8], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+246], 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-26

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. log-rec 96.6%

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

      2. mul-1-neg 96.6%

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

      3. associate-*r/ 96.6%

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

      4. associate-*r* 96.6%

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

      5. metadata-eval 96.6%

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

      6. *-commutative 96.6%

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

      7. associate-/l* 96.6%

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

      8. exp-to-pow 96.6%

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

      9. *-commutative 96.6%

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

   5. Simplified 96.6%

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

   if -5.00000000000000019e-26 < (/.f64 1 n) < 5.0000000000000003e-134

   1. Initial program 35.8%

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

   3. Taylor expanded in n around inf 87.9%

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

      1. +-rgt-identity 87.9%

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

      2. +-rgt-identity 87.9%

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

      3. log1p-define 87.9%

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

   5. Simplified 87.9%

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

      1. log1p-undefine 87.9%

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

      2. diff-log 88.0%

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

   7. Applied egg-rr 88.0%

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

      1. clear-num 88.0%

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

      2. log-rec 88.0%

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

   9. Applied egg-rr 88.0%

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

   if 5.0000000000000003e-134 < (/.f64 1 n) < 4e-79

   1. Initial program 26.1%

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

   3. Taylor expanded in n around inf 50.1%

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

      1. +-rgt-identity 50.1%

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

      2. +-rgt-identity 50.1%

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

      3. log1p-define 50.1%

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

   5. Simplified 50.1%

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

   6. Taylor expanded in x around inf 76.1%

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

   if 4e-79 < (/.f64 1 n) < 4.0000000000000001e-8

   1. Initial program 18.4%

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

   3. Taylor expanded in n around inf 76.0%

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

      1. +-rgt-identity 76.0%

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

      2. +-rgt-identity 76.0%

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

      3. log1p-define 76.0%

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

   5. Simplified 76.0%

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

      1. log1p-undefine 76.0%

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

      2. diff-log 76.0%

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

   7. Applied egg-rr 76.0%

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

   if 4.0000000000000001e-8 < (/.f64 1 n) < 4.99999999999999976e246

   1. Initial program 75.0%

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

   3. Taylor expanded in x around 0 71.4%

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

      1. *-rgt-identity 71.4%

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

      2. associate-/l* 71.4%

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

      3. exp-to-pow 71.4%

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

   5. Simplified 71.4%

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

   if 4.99999999999999976e246 < (/.f64 1 n)

   1. Initial program 11.2%

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

   3. Taylor expanded in n around inf 9.6%

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

      1. +-rgt-identity 9.6%

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

      2. +-rgt-identity 9.6%

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

      3. log1p-define 9.6%

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

   5. Simplified 9.6%

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

   6. Taylor expanded in x around inf 91.9%

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

      1. *-commutative 91.9%

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

   8. Simplified 91.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\log \left(\frac{x}{1 + x}\right)}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+246}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 21000:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+72} \lor \neg \left(x \leq 1.42 \cdot 10^{+172}\right):\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 21000.0)
   (/ (- x (log x)) n)
   (if (or (<= x 1.05e+72) (not (<= x 1.42e+172))) (/ 0.0 n) (/ 1.0 (* n x)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 21000.0) {
		tmp = (x - log(x)) / n;
	} else if ((x <= 1.05e+72) || !(x <= 1.42e+172)) {
		tmp = 0.0 / n;
	} 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) :: tmp
    if (x <= 21000.0d0) then
        tmp = (x - log(x)) / n
    else if ((x <= 1.05d+72) .or. (.not. (x <= 1.42d+172))) then
        tmp = 0.0d0 / n
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 21000.0) {
		tmp = (x - Math.log(x)) / n;
	} else if ((x <= 1.05e+72) || !(x <= 1.42e+172)) {
		tmp = 0.0 / n;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 21000.0:
		tmp = (x - math.log(x)) / n
	elif (x <= 1.05e+72) or not (x <= 1.42e+172):
		tmp = 0.0 / n
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 21000.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif ((x <= 1.05e+72) || !(x <= 1.42e+172))
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 21000.0)
		tmp = (x - log(x)) / n;
	elseif ((x <= 1.05e+72) || ~((x <= 1.42e+172)))
		tmp = 0.0 / n;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 21000.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[Or[LessEqual[x, 1.05e+72], N[Not[LessEqual[x, 1.42e+172]], $MachinePrecision]], N[(0.0 / n), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 21000

   1. Initial program 33.0%

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

   3. Taylor expanded in n around inf 60.5%

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

      1. +-rgt-identity 60.5%

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

      2. +-rgt-identity 60.5%

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

      3. log1p-define 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in x around 0 58.9%

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

      1. neg-mul-1 58.9%

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

      2. sub-neg 58.9%

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

   8. Simplified 58.9%

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

   if 21000 < x < 1.0500000000000001e72 or 1.41999999999999994e172 < x

   1. Initial program 83.1%

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

   3. Taylor expanded in n around inf 79.6%

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

      1. +-rgt-identity 79.6%

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

      2. +-rgt-identity 79.6%

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

      3. log1p-define 79.6%

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

   5. Simplified 79.6%

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

      1. log1p-undefine 79.6%

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

      2. diff-log 79.6%

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

   7. Applied egg-rr 79.6%

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

   8. Taylor expanded in x around inf 83.2%

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

   if 1.0500000000000001e72 < x < 1.41999999999999994e172

   1. Initial program 42.6%

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

   3. Taylor expanded in n around inf 42.6%

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

      1. +-rgt-identity 42.6%

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

      2. +-rgt-identity 42.6%

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

      3. log1p-define 42.6%

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

   5. Simplified 42.6%

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

   6. Taylor expanded in x around inf 82.8%

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

      1. *-commutative 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 21000:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+72} \lor \neg \left(x \leq 1.42 \cdot 10^{+172}\right):\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+72} \lor \neg \left(x \leq 5.5 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (log x) (- n))
   (if (or (<= x 1.05e+72) (not (<= x 5.5e+173))) (/ 0.0 n) (/ 1.0 (* n x)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = log(x) / -n;
	} else if ((x <= 1.05e+72) || !(x <= 5.5e+173)) {
		tmp = 0.0 / n;
	} 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) :: tmp
    if (x <= 1.0d0) then
        tmp = log(x) / -n
    else if ((x <= 1.05d+72) .or. (.not. (x <= 5.5d+173))) then
        tmp = 0.0d0 / n
    else
        tmp = 1.0d0 / (n * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = Math.log(x) / -n;
	} else if ((x <= 1.05e+72) || !(x <= 5.5e+173)) {
		tmp = 0.0 / n;
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = math.log(x) / -n
	elif (x <= 1.05e+72) or not (x <= 5.5e+173):
		tmp = 0.0 / n
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(log(x) / Float64(-n));
	elseif ((x <= 1.05e+72) || !(x <= 5.5e+173))
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = log(x) / -n;
	elseif ((x <= 1.05e+72) || ~((x <= 5.5e+173)))
		tmp = 0.0 / n;
	else
		tmp = 1.0 / (n * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[Or[LessEqual[x, 1.05e+72], N[Not[LessEqual[x, 5.5e+173]], $MachinePrecision]], N[(0.0 / n), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 33.4%

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

   3. Taylor expanded in n around inf 60.3%

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

      1. +-rgt-identity 60.3%

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

      2. +-rgt-identity 60.3%

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

      3. log1p-define 60.3%

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

   5. Simplified 60.3%

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

   6. Taylor expanded in x around 0 58.9%

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

      1. neg-mul-1 58.9%

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

   8. Simplified 58.9%

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

   if 1 < x < 1.0500000000000001e72 or 5.50000000000000049e173 < x

   1. Initial program 81.3%

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

   3. Taylor expanded in n around inf 79.6%

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

      1. +-rgt-identity 79.6%

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

      2. +-rgt-identity 79.6%

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

      3. log1p-define 79.6%

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

   5. Simplified 79.6%

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

      1. log1p-undefine 79.6%

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

      2. diff-log 79.7%

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

   7. Applied egg-rr 79.7%

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

   8. Taylor expanded in x around inf 81.3%

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

   if 1.0500000000000001e72 < x < 5.50000000000000049e173

   1. Initial program 42.6%

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

   3. Taylor expanded in n around inf 42.6%

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

      1. +-rgt-identity 42.6%

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

      2. +-rgt-identity 42.6%

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

      3. log1p-define 42.6%

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

   5. Simplified 42.6%

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

   6. Taylor expanded in x around inf 82.8%

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

      1. *-commutative 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+72} \lor \neg \left(x \leq 5.5 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.7% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -500000.0) (/ 0.0 n) (* (/ 1.0 n) (/ 1.0 x))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500000.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / n) * (1.0 / x);
	}
	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) <= (-500000.0d0)) then
        tmp = 0.0d0 / n
    else
        tmp = (1.0d0 / n) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -500000.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / n) * (1.0 / x);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -500000.0:
		tmp = 0.0 / n
	else:
		tmp = (1.0 / n) * (1.0 / x)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500000.0)
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(1.0 / n) * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -500000.0)
		tmp = 0.0 / n;
	else
		tmp = (1.0 / n) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -500000.0], N[(0.0 / n), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -5e5

   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 63.3%

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

      1. +-rgt-identity 63.3%

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

      2. +-rgt-identity 63.3%

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

      3. log1p-define 63.3%

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

   5. Simplified 63.3%

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

      1. log1p-undefine 63.3%

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

      2. diff-log 63.3%

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

   7. Applied egg-rr 63.3%

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

   8. Taylor expanded in x around inf 67.2%

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

   if -5e5 < (/.f64 1 n)

   1. Initial program 36.3%

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

   3. Taylor expanded in n around inf 64.6%

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

      1. +-rgt-identity 64.6%

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

      2. +-rgt-identity 64.6%

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

      3. log1p-define 64.6%

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

   5. Simplified 64.6%

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

   6. Taylor expanded in x around inf 47.6%

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

      1. *-commutative 47.6%

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

   8. Simplified 47.6%

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

      1. inv-pow 47.6%

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

      2. unpow-prod-down 48.0%

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

      3. inv-pow 48.0%

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

      4. inv-pow 48.0%

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

   10. Applied egg-rr 48.0%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n} \cdot \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.1% accurate, 30.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

3. Taylor expanded in n around inf 64.3%

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

   1. +-rgt-identity 64.3%

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

   2. +-rgt-identity 64.3%

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

   3. log1p-define 64.3%

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

5. Simplified 64.3%

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

6. Taylor expanded in x around inf 41.3%

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

   1. *-commutative 41.3%

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

8. Simplified 41.3%

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

   1. inv-pow 41.3%

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

   2. unpow-prod-down 41.6%

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

   3. inv-pow 41.6%

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

   4. inv-pow 41.6%

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

10. Applied egg-rr 41.6%

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

11. Final simplification 41.6%

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

Alternative 13: 39.5% 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 50.5%

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

3. Taylor expanded in n around inf 64.3%

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

   1. +-rgt-identity 64.3%

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

   2. +-rgt-identity 64.3%

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

   3. log1p-define 64.3%

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

5. Simplified 64.3%

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

6. Taylor expanded in x around inf 41.3%

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

   1. *-commutative 41.3%

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

8. Simplified 41.3%

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

9. Final simplification 41.3%

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

Alternative 14: 40.1% 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 50.5%

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

3. Taylor expanded in n around inf 64.3%

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

   1. +-rgt-identity 64.3%

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

   2. +-rgt-identity 64.3%

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

   3. log1p-define 64.3%

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

5. Simplified 64.3%

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

6. Taylor expanded in x around inf 41.6%

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

7. Final simplification 41.6%

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

Alternative 15: 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 50.5%

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

3. Taylor expanded in x around 0 25.7%

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

4. Taylor expanded in x around inf 4.6%

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

5. Final simplification 4.6%

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

Reproduce

herbie shell --seed 2024053 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))