2nthrt (problem 3.4.6)

Percentage Accurate: 52.6% → 80.5%

Time: 28.4s

Alternatives: 19

Speedup: 2.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.6% 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: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ t_1 := \log \left(\frac{x}{x + 1}\right)\\ t_2 := \frac{-t_1}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;t_0 - \frac{0.5}{n \cdot {x}^{2}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t_1 \cdot \frac{1}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x))) (t_1 (log (/ x (+ x 1.0)))) (t_2 (/ (- t_1) n)))
   (if (<= (/ 1.0 n) -1e-30)
     (/ (exp (/ (log x) n)) (* n x))
     (if (<= (/ 1.0 n) -1e-87)
       t_2
       (if (<= (/ 1.0 n) -5e-116)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 2e-112)
           t_2
           (if (<= (/ 1.0 n) 5e-104)
             (- t_0 (/ 0.5 (* n (pow x 2.0))))
             (if (<= (/ 1.0 n) 5e-7)
               (* t_1 (/ 1.0 (- n)))
               (if (<= (/ 1.0 n) 1e+214)
                 (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
                 t_0)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = log((x / (x + 1.0)));
	double t_2 = -t_1 / n;
	double tmp;
	if ((1.0 / n) <= -1e-30) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_2;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = t_0 - (0.5 / (n * pow(x, 2.0)));
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_1 * (1.0 / -n);
	} else if ((1.0 / n) <= 1e+214) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / (n * x)
    t_1 = log((x / (x + 1.0d0)))
    t_2 = -t_1 / n
    if ((1.0d0 / n) <= (-1d-30)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= (-1d-87)) then
        tmp = t_2
    else if ((1.0d0 / n) <= (-5d-116)) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 2d-112) then
        tmp = t_2
    else if ((1.0d0 / n) <= 5d-104) then
        tmp = t_0 - (0.5d0 / (n * (x ** 2.0d0)))
    else if ((1.0d0 / n) <= 5d-7) then
        tmp = t_1 * (1.0d0 / -n)
    else if ((1.0d0 / n) <= 1d+214) then
        tmp = (1.0d0 + (x / n)) - (x ** (1.0d0 / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = Math.log((x / (x + 1.0)));
	double t_2 = -t_1 / n;
	double tmp;
	if ((1.0 / n) <= -1e-30) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_2;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = t_0 - (0.5 / (n * Math.pow(x, 2.0)));
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_1 * (1.0 / -n);
	} else if ((1.0 / n) <= 1e+214) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	t_1 = math.log((x / (x + 1.0)))
	t_2 = -t_1 / n
	tmp = 0
	if (1.0 / n) <= -1e-30:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= -1e-87:
		tmp = t_2
	elif (1.0 / n) <= -5e-116:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e-112:
		tmp = t_2
	elif (1.0 / n) <= 5e-104:
		tmp = t_0 - (0.5 / (n * math.pow(x, 2.0)))
	elif (1.0 / n) <= 5e-7:
		tmp = t_1 * (1.0 / -n)
	elif (1.0 / n) <= 1e+214:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	t_1 = log(Float64(x / Float64(x + 1.0)))
	t_2 = Float64(Float64(-t_1) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e-30)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= -1e-87)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= -5e-116)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-112)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 5e-104)
		tmp = Float64(t_0 - Float64(0.5 / Float64(n * (x ^ 2.0))));
	elseif (Float64(1.0 / n) <= 5e-7)
		tmp = Float64(t_1 * Float64(1.0 / Float64(-n)));
	elseif (Float64(1.0 / n) <= 1e+214)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (n * x);
	t_1 = log((x / (x + 1.0)));
	t_2 = -t_1 / n;
	tmp = 0.0;
	if ((1.0 / n) <= -1e-30)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= -1e-87)
		tmp = t_2;
	elseif ((1.0 / n) <= -5e-116)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 2e-112)
		tmp = t_2;
	elseif ((1.0 / n) <= 5e-104)
		tmp = t_0 - (0.5 / (n * (x ^ 2.0)));
	elseif ((1.0 / n) <= 5e-7)
		tmp = t_1 * (1.0 / -n);
	elseif ((1.0 / n) <= 1e+214)
		tmp = (1.0 + (x / n)) - (x ^ (1.0 / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[((-t$95$1) / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-30], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-87], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-116], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-112], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-104], N[(t$95$0 - N[(0.5 / N[(n * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-7], N[(t$95$1 * N[(1.0 / (-n)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+214], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (/.f64 1 n) < -1e-30

   1. Initial program 92.3%

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

   3. Taylor expanded in x around inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. log-rec 96.4%

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

      3. mul-1-neg 96.4%

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

      4. distribute-neg-frac 96.4%

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

      5. mul-1-neg 96.4%

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

      6. remove-double-neg 96.4%

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

      7. *-commutative 96.4%

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

   5. Simplified 96.4%

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

   if -1e-30 < (/.f64 1 n) < -1.00000000000000002e-87 or -5.0000000000000003e-116 < (/.f64 1 n) < 1.9999999999999999e-112

   1. Initial program 29.0%

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

   3. Taylor expanded in n around inf 89.3%

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

      1. +-rgt-identity 89.3%

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

      2. +-rgt-identity 89.3%

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

      3. log1p-def 89.3%

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

   5. Simplified 89.3%

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

      1. log1p-udef 89.3%

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

      2. diff-log 89.5%

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

   7. Applied egg-rr 89.5%

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

      1. clear-num 89.5%

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

      2. log-rec 89.5%

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

   9. Applied egg-rr 89.5%

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

   if -1.00000000000000002e-87 < (/.f64 1 n) < -5.0000000000000003e-116

   1. Initial program 5.0%

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

   3. Taylor expanded in n around inf 18.7%

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

      1. +-rgt-identity 18.7%

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

      2. +-rgt-identity 18.7%

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

      3. log1p-def 18.7%

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

   5. Simplified 18.7%

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

   6. Taylor expanded in x around inf 86.0%

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

   if 1.9999999999999999e-112 < (/.f64 1 n) < 4.99999999999999979e-104

   1. Initial program 5.7%

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

   3. Taylor expanded in n around inf 11.2%

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

      1. +-rgt-identity 11.2%

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

      2. +-rgt-identity 11.2%

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

      3. log1p-def 11.2%

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

   5. Simplified 11.2%

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

   6. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r/ 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   if 4.99999999999999979e-104 < (/.f64 1 n) < 4.99999999999999977e-7

   1. Initial program 8.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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-def 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. Step-by-step derivation

      1. log1p-udef 60.5%

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

      2. diff-log 60.8%

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

   7. Applied egg-rr 60.8%

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

      1. log-div 60.5%

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

      2. log1p-udef 60.5%

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

      3. frac-2neg 60.5%

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

      4. div-inv 60.6%

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

      5. log1p-udef 60.6%

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

      6. log-div 60.8%

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

      7. neg-log 60.8%

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

      8. clear-num 60.9%

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

   9. Applied egg-rr 60.9%

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

   if 4.99999999999999977e-7 < (/.f64 1 n) < 9.9999999999999995e213

   1. Initial program 77.4%

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

   3. Taylor expanded in x around 0 77.4%

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

   if 9.9999999999999995e213 < (/.f64 1 n)

   1. Initial program 9.7%

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

   3. Taylor expanded in n around inf 8.7%

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

      1. +-rgt-identity 8.7%

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

      2. +-rgt-identity 8.7%

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

      3. log1p-def 8.7%

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

   5. Simplified 8.7%

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

   6. Taylor expanded in x around inf 86.2%

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

      1. *-commutative 86.2%

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

   8. Simplified 86.2%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{-30}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{n \cdot x} - \frac{0.5}{n \cdot {x}^{2}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\log \left(\frac{x}{x + 1}\right) \cdot \frac{1}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ t_1 := \log \left(x + 1\right)\\ \mathbf{if}\;n \leq -1.62 \cdot 10^{+115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.18 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -160000:\\ \;\;\;\;\left(\frac{t_1 - \log x}{n} + \left(\frac{-0.16666666666666666 \cdot {\log x}^{3} - -0.16666666666666666 \cdot {t_1}^{3}}{{n}^{3}} + \left(0.041666666666666664 \cdot \frac{{t_1}^{4}}{{n}^{4}} + 0.5 \cdot \frac{{t_1}^{2}}{{n}^{2}}\right)\right)\right) - \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{elif}\;n \leq 2200000:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log (/ x (+ x 1.0)))) n)) (t_1 (log (+ x 1.0))))
   (if (<= n -1.62e+115)
     t_0
     (if (<= n -1.18e+88)
       (/ (/ 1.0 x) n)
       (if (<= n -160000.0)
         (-
          (+
           (/ (- t_1 (log x)) n)
           (+
            (/
             (-
              (* -0.16666666666666666 (pow (log x) 3.0))
              (* -0.16666666666666666 (pow t_1 3.0)))
             (pow n 3.0))
            (+
             (* 0.041666666666666664 (/ (pow t_1 4.0) (pow n 4.0)))
             (* 0.5 (/ (pow t_1 2.0) (pow n 2.0))))))
          (+
           (* 0.041666666666666664 (/ (pow (log x) 4.0) (pow n 4.0)))
           (* 0.5 (/ (pow (log x) 2.0) (pow n 2.0)))))
         (if (<= n 2200000.0) (- (exp (/ x n)) (pow x (/ 1.0 n))) t_0))))))

C

double code(double x, double n) {
	double t_0 = -log((x / (x + 1.0))) / n;
	double t_1 = log((x + 1.0));
	double tmp;
	if (n <= -1.62e+115) {
		tmp = t_0;
	} else if (n <= -1.18e+88) {
		tmp = (1.0 / x) / n;
	} else if (n <= -160000.0) {
		tmp = (((t_1 - log(x)) / n) + ((((-0.16666666666666666 * pow(log(x), 3.0)) - (-0.16666666666666666 * pow(t_1, 3.0))) / pow(n, 3.0)) + ((0.041666666666666664 * (pow(t_1, 4.0) / pow(n, 4.0))) + (0.5 * (pow(t_1, 2.0) / pow(n, 2.0)))))) - ((0.041666666666666664 * (pow(log(x), 4.0) / pow(n, 4.0))) + (0.5 * (pow(log(x), 2.0) / pow(n, 2.0))));
	} else if (n <= 2200000.0) {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -log((x / (x + 1.0d0))) / n
    t_1 = log((x + 1.0d0))
    if (n <= (-1.62d+115)) then
        tmp = t_0
    else if (n <= (-1.18d+88)) then
        tmp = (1.0d0 / x) / n
    else if (n <= (-160000.0d0)) then
        tmp = (((t_1 - log(x)) / n) + (((((-0.16666666666666666d0) * (log(x) ** 3.0d0)) - ((-0.16666666666666666d0) * (t_1 ** 3.0d0))) / (n ** 3.0d0)) + ((0.041666666666666664d0 * ((t_1 ** 4.0d0) / (n ** 4.0d0))) + (0.5d0 * ((t_1 ** 2.0d0) / (n ** 2.0d0)))))) - ((0.041666666666666664d0 * ((log(x) ** 4.0d0) / (n ** 4.0d0))) + (0.5d0 * ((log(x) ** 2.0d0) / (n ** 2.0d0))))
    else if (n <= 2200000.0d0) then
        tmp = exp((x / n)) - (x ** (1.0d0 / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log((x / (x + 1.0))) / n;
	double t_1 = Math.log((x + 1.0));
	double tmp;
	if (n <= -1.62e+115) {
		tmp = t_0;
	} else if (n <= -1.18e+88) {
		tmp = (1.0 / x) / n;
	} else if (n <= -160000.0) {
		tmp = (((t_1 - Math.log(x)) / n) + ((((-0.16666666666666666 * Math.pow(Math.log(x), 3.0)) - (-0.16666666666666666 * Math.pow(t_1, 3.0))) / Math.pow(n, 3.0)) + ((0.041666666666666664 * (Math.pow(t_1, 4.0) / Math.pow(n, 4.0))) + (0.5 * (Math.pow(t_1, 2.0) / Math.pow(n, 2.0)))))) - ((0.041666666666666664 * (Math.pow(Math.log(x), 4.0) / Math.pow(n, 4.0))) + (0.5 * (Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0))));
	} else if (n <= 2200000.0) {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log((x / (x + 1.0))) / n
	t_1 = math.log((x + 1.0))
	tmp = 0
	if n <= -1.62e+115:
		tmp = t_0
	elif n <= -1.18e+88:
		tmp = (1.0 / x) / n
	elif n <= -160000.0:
		tmp = (((t_1 - math.log(x)) / n) + ((((-0.16666666666666666 * math.pow(math.log(x), 3.0)) - (-0.16666666666666666 * math.pow(t_1, 3.0))) / math.pow(n, 3.0)) + ((0.041666666666666664 * (math.pow(t_1, 4.0) / math.pow(n, 4.0))) + (0.5 * (math.pow(t_1, 2.0) / math.pow(n, 2.0)))))) - ((0.041666666666666664 * (math.pow(math.log(x), 4.0) / math.pow(n, 4.0))) + (0.5 * (math.pow(math.log(x), 2.0) / math.pow(n, 2.0))))
	elif n <= 2200000.0:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(Float64(x / Float64(x + 1.0)))) / n)
	t_1 = log(Float64(x + 1.0))
	tmp = 0.0
	if (n <= -1.62e+115)
		tmp = t_0;
	elseif (n <= -1.18e+88)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (n <= -160000.0)
		tmp = Float64(Float64(Float64(Float64(t_1 - log(x)) / n) + Float64(Float64(Float64(Float64(-0.16666666666666666 * (log(x) ^ 3.0)) - Float64(-0.16666666666666666 * (t_1 ^ 3.0))) / (n ^ 3.0)) + Float64(Float64(0.041666666666666664 * Float64((t_1 ^ 4.0) / (n ^ 4.0))) + Float64(0.5 * Float64((t_1 ^ 2.0) / (n ^ 2.0)))))) - Float64(Float64(0.041666666666666664 * Float64((log(x) ^ 4.0) / (n ^ 4.0))) + Float64(0.5 * Float64((log(x) ^ 2.0) / (n ^ 2.0)))));
	elseif (n <= 2200000.0)
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log((x / (x + 1.0))) / n;
	t_1 = log((x + 1.0));
	tmp = 0.0;
	if (n <= -1.62e+115)
		tmp = t_0;
	elseif (n <= -1.18e+88)
		tmp = (1.0 / x) / n;
	elseif (n <= -160000.0)
		tmp = (((t_1 - log(x)) / n) + ((((-0.16666666666666666 * (log(x) ^ 3.0)) - (-0.16666666666666666 * (t_1 ^ 3.0))) / (n ^ 3.0)) + ((0.041666666666666664 * ((t_1 ^ 4.0) / (n ^ 4.0))) + (0.5 * ((t_1 ^ 2.0) / (n ^ 2.0)))))) - ((0.041666666666666664 * ((log(x) ^ 4.0) / (n ^ 4.0))) + (0.5 * ((log(x) ^ 2.0) / (n ^ 2.0))));
	elseif (n <= 2200000.0)
		tmp = exp((x / n)) - (x ^ (1.0 / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.62e+115], t$95$0, If[LessEqual[n, -1.18e+88], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -160000.0], N[(N[(N[(N[(t$95$1 - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[(N[(N[(-0.16666666666666666 * N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - N[(-0.16666666666666666 * N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.041666666666666664 * N[(N[Power[t$95$1, 4.0], $MachinePrecision] / N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.041666666666666664 * N[(N[Power[N[Log[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[n, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2200000.0], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.62e115 or 2.2e6 < n

   1. Initial program 26.6%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. +-rgt-identity 78.9%

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

      2. +-rgt-identity 78.9%

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

      3. log1p-def 78.9%

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

   5. Simplified 78.9%

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

      1. log1p-udef 78.9%

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

      2. diff-log 79.4%

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

   7. Applied egg-rr 79.4%

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

      1. clear-num 79.4%

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

      2. log-rec 79.6%

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

   9. Applied egg-rr 79.6%

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

   if -1.62e115 < n < -1.1799999999999999e88

   1. Initial program 5.0%

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

   3. Taylor expanded in n around inf 18.7%

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

      1. +-rgt-identity 18.7%

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

      2. +-rgt-identity 18.7%

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

      3. log1p-def 18.7%

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

   5. Simplified 18.7%

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

   6. Taylor expanded in x around inf 86.0%

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

   if -1.1799999999999999e88 < n < -1.6e5

   1. Initial program 10.7%

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

   3. Taylor expanded in n around -inf 77.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \log \left(1 + x\right) - -1 \cdot \log x}{n} + \left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{{n}^{3}} + \left(0.041666666666666664 \cdot \frac{{\log \left(1 + x\right)}^{4}}{{n}^{4}} + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}}\right)\right)\right) - \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right)} \]

   if -1.6e5 < n < 2.2e6

   1. Initial program 87.5%

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

   3. Taylor expanded in n around 0 87.5%

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

      1. log1p-def 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in x around 0 98.1%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.62 \cdot 10^{+115}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;n \leq -1.18 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -160000:\\ \;\;\;\;\left(\frac{\log \left(x + 1\right) - \log x}{n} + \left(\frac{-0.16666666666666666 \cdot {\log x}^{3} - -0.16666666666666666 \cdot {\log \left(x + 1\right)}^{3}}{{n}^{3}} + \left(0.041666666666666664 \cdot \frac{{\log \left(x + 1\right)}^{4}}{{n}^{4}} + 0.5 \cdot \frac{{\log \left(x + 1\right)}^{2}}{{n}^{2}}\right)\right)\right) - \left(0.041666666666666664 \cdot \frac{{\log x}^{4}}{{n}^{4}} + 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{elif}\;n \leq 2200000:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{if}\;n \leq -2.65 \cdot 10^{+120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -720000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n} + \left(\frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\right)\\ \mathbf{elif}\;n \leq 2200000:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log (/ x (+ x 1.0)))) n)))
   (if (<= n -2.65e+120)
     t_0
     (if (<= n -1.3e+88)
       (/ (/ 1.0 x) n)
       (if (<= n -720000.0)
         (+
          (/ (- (log1p x) (log x)) n)
          (+
           (/
            (* 0.16666666666666666 (- (pow (log1p x) 3.0) (pow (log x) 3.0)))
            (pow n 3.0))
           (*
            0.5
            (-
             (/ (pow (log1p x) 2.0) (pow n 2.0))
             (/ (pow (log x) 2.0) (pow n 2.0))))))
         (if (<= n 2200000.0) (- (exp (/ x n)) (pow x (/ 1.0 n))) t_0))))))

C

double code(double x, double n) {
	double t_0 = -log((x / (x + 1.0))) / n;
	double tmp;
	if (n <= -2.65e+120) {
		tmp = t_0;
	} else if (n <= -1.3e+88) {
		tmp = (1.0 / x) / n;
	} else if (n <= -720000.0) {
		tmp = ((log1p(x) - log(x)) / n) + (((0.16666666666666666 * (pow(log1p(x), 3.0) - pow(log(x), 3.0))) / pow(n, 3.0)) + (0.5 * ((pow(log1p(x), 2.0) / pow(n, 2.0)) - (pow(log(x), 2.0) / pow(n, 2.0)))));
	} else if (n <= 2200000.0) {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = -Math.log((x / (x + 1.0))) / n;
	double tmp;
	if (n <= -2.65e+120) {
		tmp = t_0;
	} else if (n <= -1.3e+88) {
		tmp = (1.0 / x) / n;
	} else if (n <= -720000.0) {
		tmp = ((Math.log1p(x) - Math.log(x)) / n) + (((0.16666666666666666 * (Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0))) / Math.pow(n, 3.0)) + (0.5 * ((Math.pow(Math.log1p(x), 2.0) / Math.pow(n, 2.0)) - (Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0)))));
	} else if (n <= 2200000.0) {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log((x / (x + 1.0))) / n
	tmp = 0
	if n <= -2.65e+120:
		tmp = t_0
	elif n <= -1.3e+88:
		tmp = (1.0 / x) / n
	elif n <= -720000.0:
		tmp = ((math.log1p(x) - math.log(x)) / n) + (((0.16666666666666666 * (math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0))) / math.pow(n, 3.0)) + (0.5 * ((math.pow(math.log1p(x), 2.0) / math.pow(n, 2.0)) - (math.pow(math.log(x), 2.0) / math.pow(n, 2.0)))))
	elif n <= 2200000.0:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(Float64(x / Float64(x + 1.0)))) / n)
	tmp = 0.0
	if (n <= -2.65e+120)
		tmp = t_0;
	elseif (n <= -1.3e+88)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (n <= -720000.0)
		tmp = Float64(Float64(Float64(log1p(x) - log(x)) / n) + Float64(Float64(Float64(0.16666666666666666 * Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0))) / (n ^ 3.0)) + Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) / (n ^ 2.0)) - Float64((log(x) ^ 2.0) / (n ^ 2.0))))));
	elseif (n <= 2200000.0)
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[n, -2.65e+120], t$95$0, If[LessEqual[n, -1.3e+88], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -720000.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[(N[(0.16666666666666666 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2200000.0], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.64999999999999993e120 or 2.2e6 < n

   1. Initial program 26.6%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. +-rgt-identity 78.9%

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

      2. +-rgt-identity 78.9%

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

      3. log1p-def 78.9%

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

   5. Simplified 78.9%

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

      1. log1p-udef 78.9%

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

      2. diff-log 79.4%

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

   7. Applied egg-rr 79.4%

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

      1. clear-num 79.4%

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

      2. log-rec 79.6%

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

   9. Applied egg-rr 79.6%

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

   if -2.64999999999999993e120 < n < -1.3e88

   1. Initial program 5.0%

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

   3. Taylor expanded in n around inf 18.7%

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

      1. +-rgt-identity 18.7%

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

      2. +-rgt-identity 18.7%

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

      3. log1p-def 18.7%

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

   5. Simplified 18.7%

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

   6. Taylor expanded in x around inf 86.0%

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

   if -1.3e88 < n < -7.2e5

   1. Initial program 7.2%

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

   3. Taylor expanded in n around -inf 76.4%

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

      1. associate--l+ 76.5%

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

      2. associate-*r/ 76.5%

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

      3. mul-1-neg 76.5%

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

      4. distribute-lft-out-- 76.5%

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

      5. mul-1-neg 76.5%

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

      6. remove-double-neg 76.5%

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

      7. log1p-def 76.5%

         \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} + \left(\left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{{n}^{3}} + 0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}}\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right) \]

      8. associate--l+ 76.5%

         \[\leadsto \frac{\mathsf{log1p}\left(x\right) - \log x}{n} + \color{blue}{\left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{{n}^{3}} + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right)\right)} \]

   5. Simplified 76.5%

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

   if -7.2e5 < n < 2.2e6

   1. Initial program 87.5%

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

   3. Taylor expanded in n around 0 87.5%

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

      1. log1p-def 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around 0 98.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.65 \cdot 10^{+120}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -720000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n} + \left(\frac{0.16666666666666666 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}\right)}{{n}^{3}} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\right)\\ \mathbf{elif}\;n \leq 2200000:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -15500000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{elif}\;n \leq 2200000:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log (/ x (+ x 1.0)))) n)))
   (if (<= n -1.15e+121)
     t_0
     (if (<= n -1.3e+88)
       (/ (/ 1.0 x) n)
       (if (<= n -15500000.0)
         (+
          (/ (- (log1p x) (log x)) n)
          (*
           0.5
           (-
            (/ (pow (log1p x) 2.0) (pow n 2.0))
            (/ (pow (log x) 2.0) (pow n 2.0)))))
         (if (<= n 2200000.0) (- (exp (/ x n)) (pow x (/ 1.0 n))) t_0))))))

C

double code(double x, double n) {
	double t_0 = -log((x / (x + 1.0))) / n;
	double tmp;
	if (n <= -1.15e+121) {
		tmp = t_0;
	} else if (n <= -1.3e+88) {
		tmp = (1.0 / x) / n;
	} else if (n <= -15500000.0) {
		tmp = ((log1p(x) - log(x)) / n) + (0.5 * ((pow(log1p(x), 2.0) / pow(n, 2.0)) - (pow(log(x), 2.0) / pow(n, 2.0))));
	} else if (n <= 2200000.0) {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = -Math.log((x / (x + 1.0))) / n;
	double tmp;
	if (n <= -1.15e+121) {
		tmp = t_0;
	} else if (n <= -1.3e+88) {
		tmp = (1.0 / x) / n;
	} else if (n <= -15500000.0) {
		tmp = ((Math.log1p(x) - Math.log(x)) / n) + (0.5 * ((Math.pow(Math.log1p(x), 2.0) / Math.pow(n, 2.0)) - (Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0))));
	} else if (n <= 2200000.0) {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log((x / (x + 1.0))) / n
	tmp = 0
	if n <= -1.15e+121:
		tmp = t_0
	elif n <= -1.3e+88:
		tmp = (1.0 / x) / n
	elif n <= -15500000.0:
		tmp = ((math.log1p(x) - math.log(x)) / n) + (0.5 * ((math.pow(math.log1p(x), 2.0) / math.pow(n, 2.0)) - (math.pow(math.log(x), 2.0) / math.pow(n, 2.0))))
	elif n <= 2200000.0:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(Float64(x / Float64(x + 1.0)))) / n)
	tmp = 0.0
	if (n <= -1.15e+121)
		tmp = t_0;
	elseif (n <= -1.3e+88)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (n <= -15500000.0)
		tmp = Float64(Float64(Float64(log1p(x) - log(x)) / n) + Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) / (n ^ 2.0)) - Float64((log(x) ^ 2.0) / (n ^ 2.0)))));
	elseif (n <= 2200000.0)
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[n, -1.15e+121], t$95$0, If[LessEqual[n, -1.3e+88], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, -15500000.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2200000.0], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.1499999999999999e121 or 2.2e6 < n

   1. Initial program 26.6%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. +-rgt-identity 78.9%

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

      2. +-rgt-identity 78.9%

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

      3. log1p-def 78.9%

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

   5. Simplified 78.9%

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

      1. log1p-udef 78.9%

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

      2. diff-log 79.4%

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

   7. Applied egg-rr 79.4%

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

      1. clear-num 79.4%

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

      2. log-rec 79.6%

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

   9. Applied egg-rr 79.6%

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

   if -1.1499999999999999e121 < n < -1.3e88

   1. Initial program 5.0%

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

   3. Taylor expanded in n around inf 18.7%

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

      1. +-rgt-identity 18.7%

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

      2. +-rgt-identity 18.7%

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

      3. log1p-def 18.7%

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

   5. Simplified 18.7%

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

   6. Taylor expanded in x around inf 86.0%

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

   if -1.3e88 < n < -1.55e7

   1. Initial program 7.2%

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

   3. Taylor expanded in n around inf 75.8%

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

      1. associate--l+ 75.0%

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

      2. +-commutative 75.0%

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

      3. associate--r+ 75.8%

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

      4. div-sub 76.1%

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

      5. remove-double-neg 76.1%

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

      6. mul-1-neg 76.1%

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

      7. distribute-lft-out-- 76.1%

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

      8. distribute-neg-frac 76.1%

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

      9. mul-1-neg 76.1%

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

   5. Simplified 76.1%

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

   if -1.55e7 < n < 2.2e6

   1. Initial program 87.5%

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

   3. Taylor expanded in n around 0 87.5%

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

      1. log1p-def 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around 0 98.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{+121}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -15500000:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n} + 0.5 \cdot \left(\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{elif}\;n \leq 2200000:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{if}\;n \leq -1.62 \cdot 10^{+115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -35000000000 \lor \neg \left(n \leq 6.2 \cdot 10^{+24}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log (/ x (+ x 1.0)))) n)))
   (if (<= n -1.62e+115)
     t_0
     (if (<= n -1.3e+88)
       (/ (/ 1.0 x) n)
       (if (or (<= n -35000000000.0) (not (<= n 6.2e+24)))
         t_0
         (- (exp (/ (log1p x) n)) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = -log((x / (x + 1.0))) / n;
	double tmp;
	if (n <= -1.62e+115) {
		tmp = t_0;
	} else if (n <= -1.3e+88) {
		tmp = (1.0 / x) / n;
	} else if ((n <= -35000000000.0) || !(n <= 6.2e+24)) {
		tmp = t_0;
	} else {
		tmp = exp((log1p(x) / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = -Math.log((x / (x + 1.0))) / n;
	double tmp;
	if (n <= -1.62e+115) {
		tmp = t_0;
	} else if (n <= -1.3e+88) {
		tmp = (1.0 / x) / n;
	} else if ((n <= -35000000000.0) || !(n <= 6.2e+24)) {
		tmp = t_0;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log((x / (x + 1.0))) / n
	tmp = 0
	if n <= -1.62e+115:
		tmp = t_0
	elif n <= -1.3e+88:
		tmp = (1.0 / x) / n
	elif (n <= -35000000000.0) or not (n <= 6.2e+24):
		tmp = t_0
	else:
		tmp = math.exp((math.log1p(x) / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(Float64(x / Float64(x + 1.0)))) / n)
	tmp = 0.0
	if (n <= -1.62e+115)
		tmp = t_0;
	elseif (n <= -1.3e+88)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif ((n <= -35000000000.0) || !(n <= 6.2e+24))
		tmp = t_0;
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[n, -1.62e+115], t$95$0, If[LessEqual[n, -1.3e+88], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[Or[LessEqual[n, -35000000000.0], N[Not[LessEqual[n, 6.2e+24]], $MachinePrecision]], t$95$0, 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 3 regimes

2. if n < -1.62e115 or -1.3e88 < n < -3.5e10 or 6.20000000000000022e24 < n

   1. Initial program 23.9%

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

   3. Taylor expanded in n around inf 81.0%

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

      1. +-rgt-identity 81.0%

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

      2. +-rgt-identity 81.0%

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

      3. log1p-def 81.0%

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

   5. Simplified 81.0%

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

      1. log1p-udef 81.0%

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

      2. diff-log 81.5%

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

   7. Applied egg-rr 81.5%

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

      1. clear-num 81.5%

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

      2. log-rec 81.6%

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

   9. Applied egg-rr 81.6%

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

   if -1.62e115 < n < -1.3e88

   1. Initial program 5.0%

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

   3. Taylor expanded in n around inf 18.7%

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

      1. +-rgt-identity 18.7%

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

      2. +-rgt-identity 18.7%

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

      3. log1p-def 18.7%

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

   5. Simplified 18.7%

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

   6. Taylor expanded in x around inf 86.0%

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

   if -3.5e10 < n < 6.20000000000000022e24

   1. Initial program 83.7%

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

   3. Taylor expanded in n around 0 83.7%

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

      1. log1p-def 93.7%

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

   5. Simplified 93.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.62 \cdot 10^{+115}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -35000000000 \lor \neg \left(n \leq 6.2 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{if}\;n \leq -3.1 \cdot 10^{+116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -2400000000 \lor \neg \left(n \leq 2200000\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- (log (/ x (+ x 1.0)))) n)))
   (if (<= n -3.1e+116)
     t_0
     (if (<= n -1.3e+88)
       (/ (/ 1.0 x) n)
       (if (or (<= n -2400000000.0) (not (<= n 2200000.0)))
         t_0
         (- (exp (/ x n)) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double t_0 = -log((x / (x + 1.0))) / n;
	double tmp;
	if (n <= -3.1e+116) {
		tmp = t_0;
	} else if (n <= -1.3e+88) {
		tmp = (1.0 / x) / n;
	} else if ((n <= -2400000000.0) || !(n <= 2200000.0)) {
		tmp = t_0;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -log((x / (x + 1.0d0))) / n
    if (n <= (-3.1d+116)) then
        tmp = t_0
    else if (n <= (-1.3d+88)) then
        tmp = (1.0d0 / x) / n
    else if ((n <= (-2400000000.0d0)) .or. (.not. (n <= 2200000.0d0))) then
        tmp = t_0
    else
        tmp = exp((x / n)) - (x ** (1.0d0 / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = -Math.log((x / (x + 1.0))) / n;
	double tmp;
	if (n <= -3.1e+116) {
		tmp = t_0;
	} else if (n <= -1.3e+88) {
		tmp = (1.0 / x) / n;
	} else if ((n <= -2400000000.0) || !(n <= 2200000.0)) {
		tmp = t_0;
	} else {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = -math.log((x / (x + 1.0))) / n
	tmp = 0
	if n <= -3.1e+116:
		tmp = t_0
	elif n <= -1.3e+88:
		tmp = (1.0 / x) / n
	elif (n <= -2400000000.0) or not (n <= 2200000.0):
		tmp = t_0
	else:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-log(Float64(x / Float64(x + 1.0)))) / n)
	tmp = 0.0
	if (n <= -3.1e+116)
		tmp = t_0;
	elseif (n <= -1.3e+88)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif ((n <= -2400000000.0) || !(n <= 2200000.0))
		tmp = t_0;
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = -log((x / (x + 1.0))) / n;
	tmp = 0.0;
	if (n <= -3.1e+116)
		tmp = t_0;
	elseif (n <= -1.3e+88)
		tmp = (1.0 / x) / n;
	elseif ((n <= -2400000000.0) || ~((n <= 2200000.0)))
		tmp = t_0;
	else
		tmp = exp((x / n)) - (x ^ (1.0 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[((-N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision]}, If[LessEqual[n, -3.1e+116], t$95$0, If[LessEqual[n, -1.3e+88], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[Or[LessEqual[n, -2400000000.0], N[Not[LessEqual[n, 2200000.0]], $MachinePrecision]], t$95$0, N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.09999999999999996e116 or -1.3e88 < n < -2.4e9 or 2.2e6 < n

   1. Initial program 23.4%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. +-rgt-identity 78.9%

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

      2. +-rgt-identity 78.9%

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

      3. log1p-def 78.9%

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

   5. Simplified 78.9%

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

      1. log1p-udef 78.9%

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

      2. diff-log 79.3%

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

   7. Applied egg-rr 79.3%

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

      1. clear-num 79.3%

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

      2. log-rec 79.5%

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

   9. Applied egg-rr 79.5%

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

   if -3.09999999999999996e116 < n < -1.3e88

   1. Initial program 5.0%

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

   3. Taylor expanded in n around inf 18.7%

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

      1. +-rgt-identity 18.7%

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

      2. +-rgt-identity 18.7%

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

      3. log1p-def 18.7%

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

   5. Simplified 18.7%

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

   6. Taylor expanded in x around inf 86.0%

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

   if -2.4e9 < n < 2.2e6

   1. Initial program 86.6%

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

   3. Taylor expanded in n around 0 86.6%

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

      1. log1p-def 96.9%

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

   5. Simplified 96.9%

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

   6. Taylor expanded in x around 0 96.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{+116}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -2400000000 \lor \neg \left(n \leq 2200000\right):\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{x + 1}\right)\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ t_2 := 1 - t_1\\ t_3 := \frac{-t_0}{n}\\ t_4 := \frac{1}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+243}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t_0 \cdot \frac{1}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (/ x (+ x 1.0))))
        (t_1 (pow x (/ 1.0 n)))
        (t_2 (- 1.0 t_1))
        (t_3 (/ (- t_0) n))
        (t_4 (/ 1.0 (* n x))))
   (if (<= (/ 1.0 n) -1e+243)
     (/ 0.0 n)
     (if (<= (/ 1.0 n) -2e+164)
       t_2
       (if (<= (/ 1.0 n) -2e+25)
         (/ 0.0 n)
         (if (<= (/ 1.0 n) -1e-9)
           t_2
           (if (<= (/ 1.0 n) -1e-87)
             t_3
             (if (<= (/ 1.0 n) -5e-116)
               (/ (/ 1.0 x) n)
               (if (<= (/ 1.0 n) 2e-112)
                 t_3
                 (if (<= (/ 1.0 n) 5e-104)
                   t_4
                   (if (<= (/ 1.0 n) 5e-7)
                     (* t_0 (/ 1.0 (- n)))
                     (if (<= (/ 1.0 n) 1e+214)
                       (- (+ 1.0 (/ x n)) t_1)
                       t_4))))))))))))

C

double code(double x, double n) {
	double t_0 = log((x / (x + 1.0)));
	double t_1 = pow(x, (1.0 / n));
	double t_2 = 1.0 - t_1;
	double t_3 = -t_0 / n;
	double t_4 = 1.0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e+243) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -2e+164) {
		tmp = t_2;
	} else if ((1.0 / n) <= -2e+25) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -1e-9) {
		tmp = t_2;
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_3;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_3;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = t_4;
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_0 * (1.0 / -n);
	} else if ((1.0 / n) <= 1e+214) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = log((x / (x + 1.0d0)))
    t_1 = x ** (1.0d0 / n)
    t_2 = 1.0d0 - t_1
    t_3 = -t_0 / n
    t_4 = 1.0d0 / (n * x)
    if ((1.0d0 / n) <= (-1d+243)) then
        tmp = 0.0d0 / n
    else if ((1.0d0 / n) <= (-2d+164)) then
        tmp = t_2
    else if ((1.0d0 / n) <= (-2d+25)) then
        tmp = 0.0d0 / n
    else if ((1.0d0 / n) <= (-1d-9)) then
        tmp = t_2
    else if ((1.0d0 / n) <= (-1d-87)) then
        tmp = t_3
    else if ((1.0d0 / n) <= (-5d-116)) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 2d-112) then
        tmp = t_3
    else if ((1.0d0 / n) <= 5d-104) then
        tmp = t_4
    else if ((1.0d0 / n) <= 5d-7) then
        tmp = t_0 * (1.0d0 / -n)
    else if ((1.0d0 / n) <= 1d+214) then
        tmp = (1.0d0 + (x / n)) - t_1
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log((x / (x + 1.0)));
	double t_1 = Math.pow(x, (1.0 / n));
	double t_2 = 1.0 - t_1;
	double t_3 = -t_0 / n;
	double t_4 = 1.0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e+243) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -2e+164) {
		tmp = t_2;
	} else if ((1.0 / n) <= -2e+25) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -1e-9) {
		tmp = t_2;
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_3;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_3;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = t_4;
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_0 * (1.0 / -n);
	} else if ((1.0 / n) <= 1e+214) {
		tmp = (1.0 + (x / n)) - t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log((x / (x + 1.0)))
	t_1 = math.pow(x, (1.0 / n))
	t_2 = 1.0 - t_1
	t_3 = -t_0 / n
	t_4 = 1.0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -1e+243:
		tmp = 0.0 / n
	elif (1.0 / n) <= -2e+164:
		tmp = t_2
	elif (1.0 / n) <= -2e+25:
		tmp = 0.0 / n
	elif (1.0 / n) <= -1e-9:
		tmp = t_2
	elif (1.0 / n) <= -1e-87:
		tmp = t_3
	elif (1.0 / n) <= -5e-116:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e-112:
		tmp = t_3
	elif (1.0 / n) <= 5e-104:
		tmp = t_4
	elif (1.0 / n) <= 5e-7:
		tmp = t_0 * (1.0 / -n)
	elif (1.0 / n) <= 1e+214:
		tmp = (1.0 + (x / n)) - t_1
	else:
		tmp = t_4
	return tmp

Julia

function code(x, n)
	t_0 = log(Float64(x / Float64(x + 1.0)))
	t_1 = x ^ Float64(1.0 / n)
	t_2 = Float64(1.0 - t_1)
	t_3 = Float64(Float64(-t_0) / n)
	t_4 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+243)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= -2e+164)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= -2e+25)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= -1e-9)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= -1e-87)
		tmp = t_3;
	elseif (Float64(1.0 / n) <= -5e-116)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-112)
		tmp = t_3;
	elseif (Float64(1.0 / n) <= 5e-104)
		tmp = t_4;
	elseif (Float64(1.0 / n) <= 5e-7)
		tmp = Float64(t_0 * Float64(1.0 / Float64(-n)));
	elseif (Float64(1.0 / n) <= 1e+214)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log((x / (x + 1.0)));
	t_1 = x ^ (1.0 / n);
	t_2 = 1.0 - t_1;
	t_3 = -t_0 / n;
	t_4 = 1.0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -1e+243)
		tmp = 0.0 / n;
	elseif ((1.0 / n) <= -2e+164)
		tmp = t_2;
	elseif ((1.0 / n) <= -2e+25)
		tmp = 0.0 / n;
	elseif ((1.0 / n) <= -1e-9)
		tmp = t_2;
	elseif ((1.0 / n) <= -1e-87)
		tmp = t_3;
	elseif ((1.0 / n) <= -5e-116)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 2e-112)
		tmp = t_3;
	elseif ((1.0 / n) <= 5e-104)
		tmp = t_4;
	elseif ((1.0 / n) <= 5e-7)
		tmp = t_0 * (1.0 / -n);
	elseif ((1.0 / n) <= 1e+214)
		tmp = (1.0 + (x / n)) - t_1;
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[((-t$95$0) / n), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+243], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+164], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+25], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-9], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-87], t$95$3, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-116], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-112], t$95$3, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-104], t$95$4, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-7], N[(t$95$0 * N[(1.0 / (-n)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+214], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], t$95$4]]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (/.f64 1 n) < -1.0000000000000001e243 or -2e164 < (/.f64 1 n) < -2.00000000000000018e25

   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 62.1%

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

      1. +-rgt-identity 62.1%

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

      2. +-rgt-identity 62.1%

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

      3. log1p-def 62.1%

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

   5. Simplified 62.1%

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

      1. log1p-udef 62.1%

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

      2. diff-log 62.1%

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

   7. Applied egg-rr 62.1%

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

   8. Taylor expanded in x around inf 71.5%

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

   if -1.0000000000000001e243 < (/.f64 1 n) < -2e164 or -2.00000000000000018e25 < (/.f64 1 n) < -1.00000000000000006e-9

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0 67.4%

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

   if -1.00000000000000006e-9 < (/.f64 1 n) < -1.00000000000000002e-87 or -5.0000000000000003e-116 < (/.f64 1 n) < 1.9999999999999999e-112

   1. Initial program 28.1%

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

   3. Taylor expanded in n around inf 87.0%

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

      1. +-rgt-identity 87.0%

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

      2. +-rgt-identity 87.0%

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

      3. log1p-def 87.0%

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

   5. Simplified 87.0%

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

      1. log1p-udef 87.0%

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

      2. diff-log 87.2%

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

   7. Applied egg-rr 87.2%

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

      1. clear-num 87.2%

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

      2. log-rec 87.3%

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

   9. Applied egg-rr 87.3%

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

   if -1.00000000000000002e-87 < (/.f64 1 n) < -5.0000000000000003e-116

   1. Initial program 5.0%

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

   3. Taylor expanded in n around inf 18.7%

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

      1. +-rgt-identity 18.7%

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

      2. +-rgt-identity 18.7%

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

      3. log1p-def 18.7%

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

   5. Simplified 18.7%

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

   6. Taylor expanded in x around inf 86.0%

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

   if 1.9999999999999999e-112 < (/.f64 1 n) < 4.99999999999999979e-104 or 9.9999999999999995e213 < (/.f64 1 n)

   1. Initial program 7.9%

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

   3. Taylor expanded in n around inf 9.9%

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

      1. +-rgt-identity 9.9%

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

      2. +-rgt-identity 9.9%

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

      3. log1p-def 9.9%

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

   5. Simplified 9.9%

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

   6. Taylor expanded in x around inf 90.4%

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

      1. *-commutative 90.4%

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

   8. Simplified 90.4%

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

   if 4.99999999999999979e-104 < (/.f64 1 n) < 4.99999999999999977e-7

   1. Initial program 8.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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-def 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. Step-by-step derivation

      1. log1p-udef 60.5%

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

      2. diff-log 60.8%

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

   7. Applied egg-rr 60.8%

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

      1. log-div 60.5%

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

      2. log1p-udef 60.5%

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

      3. frac-2neg 60.5%

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

      4. div-inv 60.6%

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

      5. log1p-udef 60.6%

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

      6. log-div 60.8%

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

      7. neg-log 60.8%

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

      8. clear-num 60.9%

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

   9. Applied egg-rr 60.9%

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

   if 4.99999999999999977e-7 < (/.f64 1 n) < 9.9999999999999995e213

   1. Initial program 77.4%

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

   3. Taylor expanded in x around 0 77.4%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 7 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+243}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+164}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\log \left(\frac{x}{x + 1}\right) \cdot \frac{1}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ t_1 := \frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ t_2 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+243}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x)))
        (t_1 (/ (log (+ 1.0 (/ 1.0 x))) n))
        (t_2 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -1e+243)
     (/ 0.0 n)
     (if (<= (/ 1.0 n) -2e+164)
       t_2
       (if (<= (/ 1.0 n) -2e+25)
         (/ 0.0 n)
         (if (<= (/ 1.0 n) -1e-9)
           t_2
           (if (<= (/ 1.0 n) -1e-87)
             t_1
             (if (<= (/ 1.0 n) -5e-116)
               (/ (/ 1.0 x) n)
               (if (<= (/ 1.0 n) 2e-112)
                 t_1
                 (if (<= (/ 1.0 n) 5e-104)
                   t_0
                   (if (<= (/ 1.0 n) 5e-7)
                     t_1
                     (if (<= (/ 1.0 n) 1e+214) t_2 t_0))))))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = log((1.0 + (1.0 / x))) / n;
	double t_2 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+243) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -2e+164) {
		tmp = t_2;
	} else if ((1.0 / n) <= -2e+25) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -1e-9) {
		tmp = t_2;
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+214) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / (n * x)
    t_1 = log((1.0d0 + (1.0d0 / x))) / n
    t_2 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-1d+243)) then
        tmp = 0.0d0 / n
    else if ((1.0d0 / n) <= (-2d+164)) then
        tmp = t_2
    else if ((1.0d0 / n) <= (-2d+25)) then
        tmp = 0.0d0 / n
    else if ((1.0d0 / n) <= (-1d-9)) then
        tmp = t_2
    else if ((1.0d0 / n) <= (-1d-87)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-116)) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 2d-112) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-104) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-7) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+214) then
        tmp = t_2
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = Math.log((1.0 + (1.0 / x))) / n;
	double t_2 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -1e+243) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -2e+164) {
		tmp = t_2;
	} else if ((1.0 / n) <= -2e+25) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -1e-9) {
		tmp = t_2;
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+214) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	t_1 = math.log((1.0 + (1.0 / x))) / n
	t_2 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -1e+243:
		tmp = 0.0 / n
	elif (1.0 / n) <= -2e+164:
		tmp = t_2
	elif (1.0 / n) <= -2e+25:
		tmp = 0.0 / n
	elif (1.0 / n) <= -1e-9:
		tmp = t_2
	elif (1.0 / n) <= -1e-87:
		tmp = t_1
	elif (1.0 / n) <= -5e-116:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e-112:
		tmp = t_1
	elif (1.0 / n) <= 5e-104:
		tmp = t_0
	elif (1.0 / n) <= 5e-7:
		tmp = t_1
	elif (1.0 / n) <= 1e+214:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	t_1 = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n)
	t_2 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+243)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= -2e+164)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= -2e+25)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= -1e-9)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= -1e-87)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-116)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-112)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-104)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-7)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+214)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (n * x);
	t_1 = log((1.0 + (1.0 / x))) / n;
	t_2 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -1e+243)
		tmp = 0.0 / n;
	elseif ((1.0 / n) <= -2e+164)
		tmp = t_2;
	elseif ((1.0 / n) <= -2e+25)
		tmp = 0.0 / n;
	elseif ((1.0 / n) <= -1e-9)
		tmp = t_2;
	elseif ((1.0 / n) <= -1e-87)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-116)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 2e-112)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-104)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-7)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+214)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+243], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+164], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+25], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-9], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-87], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-116], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-112], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-104], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-7], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+214], t$95$2, t$95$0]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -1.0000000000000001e243 or -2e164 < (/.f64 1 n) < -2.00000000000000018e25

   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 62.1%

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

      1. +-rgt-identity 62.1%

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

      2. +-rgt-identity 62.1%

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

      3. log1p-def 62.1%

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

   5. Simplified 62.1%

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

      1. log1p-udef 62.1%

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

      2. diff-log 62.1%

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

   7. Applied egg-rr 62.1%

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

   8. Taylor expanded in x around inf 71.5%

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

   if -1.0000000000000001e243 < (/.f64 1 n) < -2e164 or -2.00000000000000018e25 < (/.f64 1 n) < -1.00000000000000006e-9 or 4.99999999999999977e-7 < (/.f64 1 n) < 9.9999999999999995e213

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0 71.5%

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

   if -1.00000000000000006e-9 < (/.f64 1 n) < -1.00000000000000002e-87 or -5.0000000000000003e-116 < (/.f64 1 n) < 1.9999999999999999e-112 or 4.99999999999999979e-104 < (/.f64 1 n) < 4.99999999999999977e-7

   1. Initial program 24.2%

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

   3. Taylor expanded in n around inf 81.9%

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

      1. +-rgt-identity 81.9%

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

      2. +-rgt-identity 81.9%

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

      3. log1p-def 81.9%

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

   5. Simplified 81.9%

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

      1. log1p-udef 81.9%

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

      2. diff-log 82.1%

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

   7. Applied egg-rr 82.1%

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

   8. Taylor expanded in x around 0 82.2%

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

   if -1.00000000000000002e-87 < (/.f64 1 n) < -5.0000000000000003e-116

   1. Initial program 5.0%

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

   3. Taylor expanded in n around inf 18.7%

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

      1. +-rgt-identity 18.7%

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

      2. +-rgt-identity 18.7%

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

      3. log1p-def 18.7%

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

   5. Simplified 18.7%

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

   6. Taylor expanded in x around inf 86.0%

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

   if 1.9999999999999999e-112 < (/.f64 1 n) < 4.99999999999999979e-104 or 9.9999999999999995e213 < (/.f64 1 n)

   1. Initial program 7.9%

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

   3. Taylor expanded in n around inf 9.9%

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

      1. +-rgt-identity 9.9%

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

      2. +-rgt-identity 9.9%

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

      3. log1p-def 9.9%

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

   5. Simplified 9.9%

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

   6. Taylor expanded in x around inf 90.4%

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

      1. *-commutative 90.4%

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

   8. Simplified 90.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+243}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+164}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ t_2 := \frac{1}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+243}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n))))
        (t_1 (/ (- (log (/ x (+ x 1.0)))) n))
        (t_2 (/ 1.0 (* n x))))
   (if (<= (/ 1.0 n) -1e+243)
     (/ 0.0 n)
     (if (<= (/ 1.0 n) -2e+164)
       t_0
       (if (<= (/ 1.0 n) -2e+25)
         (/ 0.0 n)
         (if (<= (/ 1.0 n) -1e-9)
           t_0
           (if (<= (/ 1.0 n) -1e-87)
             t_1
             (if (<= (/ 1.0 n) -5e-116)
               (/ (/ 1.0 x) n)
               (if (<= (/ 1.0 n) 2e-112)
                 t_1
                 (if (<= (/ 1.0 n) 5e-104)
                   t_2
                   (if (<= (/ 1.0 n) 5e-7)
                     t_1
                     (if (<= (/ 1.0 n) 1e+214) t_0 t_2))))))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = -log((x / (x + 1.0))) / n;
	double t_2 = 1.0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e+243) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -2e+164) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e+25) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -1e-9) {
		tmp = t_0;
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+214) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = -log((x / (x + 1.0d0))) / n
    t_2 = 1.0d0 / (n * x)
    if ((1.0d0 / n) <= (-1d+243)) then
        tmp = 0.0d0 / n
    else if ((1.0d0 / n) <= (-2d+164)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-2d+25)) then
        tmp = 0.0d0 / n
    else if ((1.0d0 / n) <= (-1d-9)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-1d-87)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-116)) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 2d-112) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d-104) then
        tmp = t_2
    else if ((1.0d0 / n) <= 5d-7) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+214) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = -Math.log((x / (x + 1.0))) / n;
	double t_2 = 1.0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e+243) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -2e+164) {
		tmp = t_0;
	} else if ((1.0 / n) <= -2e+25) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -1e-9) {
		tmp = t_0;
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+214) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = -math.log((x / (x + 1.0))) / n
	t_2 = 1.0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -1e+243:
		tmp = 0.0 / n
	elif (1.0 / n) <= -2e+164:
		tmp = t_0
	elif (1.0 / n) <= -2e+25:
		tmp = 0.0 / n
	elif (1.0 / n) <= -1e-9:
		tmp = t_0
	elif (1.0 / n) <= -1e-87:
		tmp = t_1
	elif (1.0 / n) <= -5e-116:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e-112:
		tmp = t_1
	elif (1.0 / n) <= 5e-104:
		tmp = t_2
	elif (1.0 / n) <= 5e-7:
		tmp = t_1
	elif (1.0 / n) <= 1e+214:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(Float64(-log(Float64(x / Float64(x + 1.0)))) / n)
	t_2 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+243)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= -2e+164)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -2e+25)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= -1e-9)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -1e-87)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-116)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-112)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e-104)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 5e-7)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+214)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = -log((x / (x + 1.0))) / n;
	t_2 = 1.0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -1e+243)
		tmp = 0.0 / n;
	elseif ((1.0 / n) <= -2e+164)
		tmp = t_0;
	elseif ((1.0 / n) <= -2e+25)
		tmp = 0.0 / n;
	elseif ((1.0 / n) <= -1e-9)
		tmp = t_0;
	elseif ((1.0 / n) <= -1e-87)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-116)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 2e-112)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e-104)
		tmp = t_2;
	elseif ((1.0 / n) <= 5e-7)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+214)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+243], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+164], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+25], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-9], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-87], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-116], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-112], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-104], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-7], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+214], t$95$0, t$95$2]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 1 n) < -1.0000000000000001e243 or -2e164 < (/.f64 1 n) < -2.00000000000000018e25

   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 62.1%

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

      1. +-rgt-identity 62.1%

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

      2. +-rgt-identity 62.1%

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

      3. log1p-def 62.1%

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

   5. Simplified 62.1%

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

      1. log1p-udef 62.1%

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

      2. diff-log 62.1%

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

   7. Applied egg-rr 62.1%

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

   8. Taylor expanded in x around inf 71.5%

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

   if -1.0000000000000001e243 < (/.f64 1 n) < -2e164 or -2.00000000000000018e25 < (/.f64 1 n) < -1.00000000000000006e-9 or 4.99999999999999977e-7 < (/.f64 1 n) < 9.9999999999999995e213

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0 71.5%

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

   if -1.00000000000000006e-9 < (/.f64 1 n) < -1.00000000000000002e-87 or -5.0000000000000003e-116 < (/.f64 1 n) < 1.9999999999999999e-112 or 4.99999999999999979e-104 < (/.f64 1 n) < 4.99999999999999977e-7

   1. Initial program 24.2%

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

   3. Taylor expanded in n around inf 81.9%

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

      1. +-rgt-identity 81.9%

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

      2. +-rgt-identity 81.9%

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

      3. log1p-def 81.9%

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

   5. Simplified 81.9%

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

      1. log1p-udef 81.9%

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

      2. diff-log 82.1%

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

   7. Applied egg-rr 82.1%

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

      1. clear-num 82.1%

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

      2. log-rec 82.2%

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

   9. Applied egg-rr 82.2%

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

   if -1.00000000000000002e-87 < (/.f64 1 n) < -5.0000000000000003e-116

   1. Initial program 5.0%

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

   3. Taylor expanded in n around inf 18.7%

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

      1. +-rgt-identity 18.7%

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

      2. +-rgt-identity 18.7%

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

      3. log1p-def 18.7%

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

   5. Simplified 18.7%

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

   6. Taylor expanded in x around inf 86.0%

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

   if 1.9999999999999999e-112 < (/.f64 1 n) < 4.99999999999999979e-104 or 9.9999999999999995e213 < (/.f64 1 n)

   1. Initial program 7.9%

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

   3. Taylor expanded in n around inf 9.9%

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

      1. +-rgt-identity 9.9%

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

      2. +-rgt-identity 9.9%

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

      3. log1p-def 9.9%

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

   5. Simplified 9.9%

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

   6. Taylor expanded in x around inf 90.4%

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

      1. *-commutative 90.4%

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

   8. Simplified 90.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+243}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+164}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{x + 1}\right)\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_2 := \frac{-t_0}{n}\\ t_3 := \frac{1}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+243}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t_0 \cdot \frac{1}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (/ x (+ x 1.0))))
        (t_1 (- 1.0 (pow x (/ 1.0 n))))
        (t_2 (/ (- t_0) n))
        (t_3 (/ 1.0 (* n x))))
   (if (<= (/ 1.0 n) -1e+243)
     (/ 0.0 n)
     (if (<= (/ 1.0 n) -2e+164)
       t_1
       (if (<= (/ 1.0 n) -2e+25)
         (/ 0.0 n)
         (if (<= (/ 1.0 n) -1e-9)
           t_1
           (if (<= (/ 1.0 n) -1e-87)
             t_2
             (if (<= (/ 1.0 n) -5e-116)
               (/ (/ 1.0 x) n)
               (if (<= (/ 1.0 n) 2e-112)
                 t_2
                 (if (<= (/ 1.0 n) 5e-104)
                   t_3
                   (if (<= (/ 1.0 n) 5e-7)
                     (* t_0 (/ 1.0 (- n)))
                     (if (<= (/ 1.0 n) 1e+214) t_1 t_3))))))))))))

C

double code(double x, double n) {
	double t_0 = log((x / (x + 1.0)));
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double t_2 = -t_0 / n;
	double t_3 = 1.0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e+243) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -2e+164) {
		tmp = t_1;
	} else if ((1.0 / n) <= -2e+25) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -1e-9) {
		tmp = t_1;
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_2;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = t_3;
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_0 * (1.0 / -n);
	} else if ((1.0 / n) <= 1e+214) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = log((x / (x + 1.0d0)))
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    t_2 = -t_0 / n
    t_3 = 1.0d0 / (n * x)
    if ((1.0d0 / n) <= (-1d+243)) then
        tmp = 0.0d0 / n
    else if ((1.0d0 / n) <= (-2d+164)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-2d+25)) then
        tmp = 0.0d0 / n
    else if ((1.0d0 / n) <= (-1d-9)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-1d-87)) then
        tmp = t_2
    else if ((1.0d0 / n) <= (-5d-116)) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 2d-112) then
        tmp = t_2
    else if ((1.0d0 / n) <= 5d-104) then
        tmp = t_3
    else if ((1.0d0 / n) <= 5d-7) then
        tmp = t_0 * (1.0d0 / -n)
    else if ((1.0d0 / n) <= 1d+214) then
        tmp = t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log((x / (x + 1.0)));
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double t_2 = -t_0 / n;
	double t_3 = 1.0 / (n * x);
	double tmp;
	if ((1.0 / n) <= -1e+243) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -2e+164) {
		tmp = t_1;
	} else if ((1.0 / n) <= -2e+25) {
		tmp = 0.0 / n;
	} else if ((1.0 / n) <= -1e-9) {
		tmp = t_1;
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_2;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = t_3;
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_0 * (1.0 / -n);
	} else if ((1.0 / n) <= 1e+214) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log((x / (x + 1.0)))
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	t_2 = -t_0 / n
	t_3 = 1.0 / (n * x)
	tmp = 0
	if (1.0 / n) <= -1e+243:
		tmp = 0.0 / n
	elif (1.0 / n) <= -2e+164:
		tmp = t_1
	elif (1.0 / n) <= -2e+25:
		tmp = 0.0 / n
	elif (1.0 / n) <= -1e-9:
		tmp = t_1
	elif (1.0 / n) <= -1e-87:
		tmp = t_2
	elif (1.0 / n) <= -5e-116:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e-112:
		tmp = t_2
	elif (1.0 / n) <= 5e-104:
		tmp = t_3
	elif (1.0 / n) <= 5e-7:
		tmp = t_0 * (1.0 / -n)
	elif (1.0 / n) <= 1e+214:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, n)
	t_0 = log(Float64(x / Float64(x + 1.0)))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_2 = Float64(Float64(-t_0) / n)
	t_3 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+243)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= -2e+164)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -2e+25)
		tmp = Float64(0.0 / n);
	elseif (Float64(1.0 / n) <= -1e-9)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -1e-87)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= -5e-116)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-112)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 5e-104)
		tmp = t_3;
	elseif (Float64(1.0 / n) <= 5e-7)
		tmp = Float64(t_0 * Float64(1.0 / Float64(-n)));
	elseif (Float64(1.0 / n) <= 1e+214)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log((x / (x + 1.0)));
	t_1 = 1.0 - (x ^ (1.0 / n));
	t_2 = -t_0 / n;
	t_3 = 1.0 / (n * x);
	tmp = 0.0;
	if ((1.0 / n) <= -1e+243)
		tmp = 0.0 / n;
	elseif ((1.0 / n) <= -2e+164)
		tmp = t_1;
	elseif ((1.0 / n) <= -2e+25)
		tmp = 0.0 / n;
	elseif ((1.0 / n) <= -1e-9)
		tmp = t_1;
	elseif ((1.0 / n) <= -1e-87)
		tmp = t_2;
	elseif ((1.0 / n) <= -5e-116)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 2e-112)
		tmp = t_2;
	elseif ((1.0 / n) <= 5e-104)
		tmp = t_3;
	elseif ((1.0 / n) <= 5e-7)
		tmp = t_0 * (1.0 / -n);
	elseif ((1.0 / n) <= 1e+214)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Log[N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-t$95$0) / n), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+243], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+164], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+25], N[(0.0 / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-9], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-87], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-116], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-112], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-104], t$95$3, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-7], N[(t$95$0 * N[(1.0 / (-n)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+214], t$95$1, t$95$3]]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -1.0000000000000001e243 or -2e164 < (/.f64 1 n) < -2.00000000000000018e25

   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 62.1%

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

      1. +-rgt-identity 62.1%

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

      2. +-rgt-identity 62.1%

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

      3. log1p-def 62.1%

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

   5. Simplified 62.1%

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

      1. log1p-udef 62.1%

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

      2. diff-log 62.1%

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

   7. Applied egg-rr 62.1%

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

   8. Taylor expanded in x around inf 71.5%

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

   if -1.0000000000000001e243 < (/.f64 1 n) < -2e164 or -2.00000000000000018e25 < (/.f64 1 n) < -1.00000000000000006e-9 or 4.99999999999999977e-7 < (/.f64 1 n) < 9.9999999999999995e213

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0 71.5%

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

   if -1.00000000000000006e-9 < (/.f64 1 n) < -1.00000000000000002e-87 or -5.0000000000000003e-116 < (/.f64 1 n) < 1.9999999999999999e-112

   1. Initial program 28.1%

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

   3. Taylor expanded in n around inf 87.0%

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

      1. +-rgt-identity 87.0%

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

      2. +-rgt-identity 87.0%

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

      3. log1p-def 87.0%

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

   5. Simplified 87.0%

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

      1. log1p-udef 87.0%

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

      2. diff-log 87.2%

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

   7. Applied egg-rr 87.2%

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

      1. clear-num 87.2%

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

      2. log-rec 87.3%

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

   9. Applied egg-rr 87.3%

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

   if -1.00000000000000002e-87 < (/.f64 1 n) < -5.0000000000000003e-116

   1. Initial program 5.0%

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

   3. Taylor expanded in n around inf 18.7%

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

      1. +-rgt-identity 18.7%

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

      2. +-rgt-identity 18.7%

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

      3. log1p-def 18.7%

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

   5. Simplified 18.7%

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

   6. Taylor expanded in x around inf 86.0%

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

   if 1.9999999999999999e-112 < (/.f64 1 n) < 4.99999999999999979e-104 or 9.9999999999999995e213 < (/.f64 1 n)

   1. Initial program 7.9%

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

   3. Taylor expanded in n around inf 9.9%

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

      1. +-rgt-identity 9.9%

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

      2. +-rgt-identity 9.9%

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

      3. log1p-def 9.9%

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

   5. Simplified 9.9%

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

   6. Taylor expanded in x around inf 90.4%

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

      1. *-commutative 90.4%

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

   8. Simplified 90.4%

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

   if 4.99999999999999979e-104 < (/.f64 1 n) < 4.99999999999999977e-7

   1. Initial program 8.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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-def 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. Step-by-step derivation

      1. log1p-udef 60.5%

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

      2. diff-log 60.8%

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

   7. Applied egg-rr 60.8%

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

      1. log-div 60.5%

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

      2. log1p-udef 60.5%

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

      3. frac-2neg 60.5%

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

      4. div-inv 60.6%

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

      5. log1p-udef 60.6%

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

      6. log-div 60.8%

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

      7. neg-log 60.8%

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

      8. clear-num 60.9%

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

   9. Applied egg-rr 60.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+243}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+164}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\log \left(\frac{x}{x + 1}\right) \cdot \frac{1}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ t_1 := \frac{x}{x + 1}\\ t_2 := \log t_1\\ t_3 := \frac{-t_2}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-1 + t_1\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t_2 \cdot \frac{1}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x)))
        (t_1 (/ x (+ x 1.0)))
        (t_2 (log t_1))
        (t_3 (/ (- t_2) n)))
   (if (<= (/ 1.0 n) -500.0)
     (/ (- (log1p (+ -1.0 t_1))) n)
     (if (<= (/ 1.0 n) -1e-87)
       t_3
       (if (<= (/ 1.0 n) -5e-116)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 2e-112)
           t_3
           (if (<= (/ 1.0 n) 5e-104)
             t_0
             (if (<= (/ 1.0 n) 5e-7)
               (* t_2 (/ 1.0 (- n)))
               (if (<= (/ 1.0 n) 1e+214)
                 (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
                 t_0)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = x / (x + 1.0);
	double t_2 = log(t_1);
	double t_3 = -t_2 / n;
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = -log1p((-1.0 + t_1)) / n;
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_3;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_3;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_2 * (1.0 / -n);
	} else if ((1.0 / n) <= 1e+214) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = x / (x + 1.0);
	double t_2 = Math.log(t_1);
	double t_3 = -t_2 / n;
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = -Math.log1p((-1.0 + t_1)) / n;
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_3;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_3;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_2 * (1.0 / -n);
	} else if ((1.0 / n) <= 1e+214) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	t_1 = x / (x + 1.0)
	t_2 = math.log(t_1)
	t_3 = -t_2 / n
	tmp = 0
	if (1.0 / n) <= -500.0:
		tmp = -math.log1p((-1.0 + t_1)) / n
	elif (1.0 / n) <= -1e-87:
		tmp = t_3
	elif (1.0 / n) <= -5e-116:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e-112:
		tmp = t_3
	elif (1.0 / n) <= 5e-104:
		tmp = t_0
	elif (1.0 / n) <= 5e-7:
		tmp = t_2 * (1.0 / -n)
	elif (1.0 / n) <= 1e+214:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	t_1 = Float64(x / Float64(x + 1.0))
	t_2 = log(t_1)
	t_3 = Float64(Float64(-t_2) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500.0)
		tmp = Float64(Float64(-log1p(Float64(-1.0 + t_1))) / n);
	elseif (Float64(1.0 / n) <= -1e-87)
		tmp = t_3;
	elseif (Float64(1.0 / n) <= -5e-116)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-112)
		tmp = t_3;
	elseif (Float64(1.0 / n) <= 5e-104)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-7)
		tmp = Float64(t_2 * Float64(1.0 / Float64(-n)));
	elseif (Float64(1.0 / n) <= 1e+214)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[((-t$95$2) / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -500.0], N[((-N[Log[1 + N[(-1.0 + t$95$1), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-87], t$95$3, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-116], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-112], t$95$3, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-104], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-7], N[(t$95$2 * N[(1.0 / (-n)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+214], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 1 n) < -500

   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 48.8%

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

      1. +-rgt-identity 48.8%

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

      2. +-rgt-identity 48.8%

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

      3. log1p-def 48.8%

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

   5. Simplified 48.8%

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

      1. log1p-udef 48.8%

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

      2. diff-log 48.8%

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

   7. Applied egg-rr 48.8%

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

      1. clear-num 48.8%

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

      2. log-rec 48.8%

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

   9. Applied egg-rr 48.8%

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

       1. add-sqr-sqrt 46.5%

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

       2. sqrt-unprod 46.8%

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

       3. sqr-neg 46.8%

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

       4. sqrt-unprod 46.8%

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

       5. log1p-expm1-u 46.8%

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

       6. add-sqr-sqrt 46.8%

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

       7. neg-log 46.8%

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

       8. clear-num 46.8%

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

       9. diff-log 46.8%

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

       10. log1p-udef 46.8%

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

       11. log1p-udef 46.8%

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

       12. diff-log 46.8%

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

       13. clear-num 46.8%

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

       14. neg-log 46.8%

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

       15. add-sqr-sqrt 46.8%

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

       16. sqrt-unprod 46.8%

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

   11. Applied egg-rr 93.2%

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

   if -500 < (/.f64 1 n) < -1.00000000000000002e-87 or -5.0000000000000003e-116 < (/.f64 1 n) < 1.9999999999999999e-112

   1. Initial program 28.6%

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

   3. Taylor expanded in n around inf 84.9%

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

      1. +-rgt-identity 84.9%

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

      2. +-rgt-identity 84.9%

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

      3. log1p-def 84.9%

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

   5. Simplified 84.9%

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

      1. log1p-udef 84.9%

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

      2. diff-log 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. clear-num 85.1%

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

      2. log-rec 85.1%

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

   9. Applied egg-rr 85.1%

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

   if -1.00000000000000002e-87 < (/.f64 1 n) < -5.0000000000000003e-116

   1. Initial program 5.0%

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

   3. Taylor expanded in n around inf 18.7%

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

      1. +-rgt-identity 18.7%

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

      2. +-rgt-identity 18.7%

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

      3. log1p-def 18.7%

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

   5. Simplified 18.7%

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

   6. Taylor expanded in x around inf 86.0%

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

   if 1.9999999999999999e-112 < (/.f64 1 n) < 4.99999999999999979e-104 or 9.9999999999999995e213 < (/.f64 1 n)

   1. Initial program 7.9%

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

   3. Taylor expanded in n around inf 9.9%

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

      1. +-rgt-identity 9.9%

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

      2. +-rgt-identity 9.9%

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

      3. log1p-def 9.9%

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

   5. Simplified 9.9%

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

   6. Taylor expanded in x around inf 90.4%

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

      1. *-commutative 90.4%

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

   8. Simplified 90.4%

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

   if 4.99999999999999979e-104 < (/.f64 1 n) < 4.99999999999999977e-7

   1. Initial program 8.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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-def 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. Step-by-step derivation

      1. log1p-udef 60.5%

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

      2. diff-log 60.8%

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

   7. Applied egg-rr 60.8%

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

      1. log-div 60.5%

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

      2. log1p-udef 60.5%

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

      3. frac-2neg 60.5%

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

      4. div-inv 60.6%

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

      5. log1p-udef 60.6%

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

      6. log-div 60.8%

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

      7. neg-log 60.8%

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

      8. clear-num 60.9%

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

   9. Applied egg-rr 60.9%

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

   if 4.99999999999999977e-7 < (/.f64 1 n) < 9.9999999999999995e213

   1. Initial program 77.4%

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

   3. Taylor expanded in x around 0 77.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-1 + \frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\log \left(\frac{x}{x + 1}\right) \cdot \frac{1}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := \log t_0\\ t_2 := \frac{-t_1}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-1 + t_0\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t_1 \cdot \frac{1}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))) (t_1 (log t_0)) (t_2 (/ (- t_1) n)))
   (if (<= (/ 1.0 n) -500.0)
     (/ (- (log1p (+ -1.0 t_0))) n)
     (if (<= (/ 1.0 n) -1e-87)
       t_2
       (if (<= (/ 1.0 n) -5e-116)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 2e-112)
           t_2
           (if (<= (/ 1.0 n) 5e-104)
             (/ (- (/ 1.0 x) (/ 0.5 (pow x 2.0))) n)
             (if (<= (/ 1.0 n) 5e-7)
               (* t_1 (/ 1.0 (- n)))
               (if (<= (/ 1.0 n) 1e+214)
                 (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
                 (/ 1.0 (* n x)))))))))))

C

double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double t_1 = log(t_0);
	double t_2 = -t_1 / n;
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = -log1p((-1.0 + t_0)) / n;
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_2;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = ((1.0 / x) - (0.5 / pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_1 * (1.0 / -n);
	} else if ((1.0 / n) <= 1e+214) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = x / (x + 1.0);
	double t_1 = Math.log(t_0);
	double t_2 = -t_1 / n;
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = -Math.log1p((-1.0 + t_0)) / n;
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_2;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_2;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = ((1.0 / x) - (0.5 / Math.pow(x, 2.0))) / n;
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_1 * (1.0 / -n);
	} else if ((1.0 / n) <= 1e+214) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = 1.0 / (n * x);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = x / (x + 1.0)
	t_1 = math.log(t_0)
	t_2 = -t_1 / n
	tmp = 0
	if (1.0 / n) <= -500.0:
		tmp = -math.log1p((-1.0 + t_0)) / n
	elif (1.0 / n) <= -1e-87:
		tmp = t_2
	elif (1.0 / n) <= -5e-116:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e-112:
		tmp = t_2
	elif (1.0 / n) <= 5e-104:
		tmp = ((1.0 / x) - (0.5 / math.pow(x, 2.0))) / n
	elif (1.0 / n) <= 5e-7:
		tmp = t_1 * (1.0 / -n)
	elif (1.0 / n) <= 1e+214:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = 1.0 / (n * x)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(x / Float64(x + 1.0))
	t_1 = log(t_0)
	t_2 = Float64(Float64(-t_1) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500.0)
		tmp = Float64(Float64(-log1p(Float64(-1.0 + t_0))) / n);
	elseif (Float64(1.0 / n) <= -1e-87)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= -5e-116)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-112)
		tmp = t_2;
	elseif (Float64(1.0 / n) <= 5e-104)
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(0.5 / (x ^ 2.0))) / n);
	elseif (Float64(1.0 / n) <= 5e-7)
		tmp = Float64(t_1 * Float64(1.0 / Float64(-n)));
	elseif (Float64(1.0 / n) <= 1e+214)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(1.0 / Float64(n * x));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[((-t$95$1) / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -500.0], N[((-N[Log[1 + N[(-1.0 + t$95$0), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-87], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-116], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-112], t$95$2, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-104], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-7], N[(t$95$1 * N[(1.0 / (-n)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+214], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (/.f64 1 n) < -500

   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 48.8%

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

      1. +-rgt-identity 48.8%

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

      2. +-rgt-identity 48.8%

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

      3. log1p-def 48.8%

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

   5. Simplified 48.8%

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

      1. log1p-udef 48.8%

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

      2. diff-log 48.8%

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

   7. Applied egg-rr 48.8%

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

      1. clear-num 48.8%

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

      2. log-rec 48.8%

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

   9. Applied egg-rr 48.8%

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

       1. add-sqr-sqrt 46.5%

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

       2. sqrt-unprod 46.8%

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

       3. sqr-neg 46.8%

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

       4. sqrt-unprod 46.8%

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

       5. log1p-expm1-u 46.8%

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

       6. add-sqr-sqrt 46.8%

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

       7. neg-log 46.8%

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

       8. clear-num 46.8%

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

       9. diff-log 46.8%

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

       10. log1p-udef 46.8%

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

       11. log1p-udef 46.8%

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

       12. diff-log 46.8%

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

       13. clear-num 46.8%

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

       14. neg-log 46.8%

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

       15. add-sqr-sqrt 46.8%

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

       16. sqrt-unprod 46.8%

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

   11. Applied egg-rr 93.2%

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

   if -500 < (/.f64 1 n) < -1.00000000000000002e-87 or -5.0000000000000003e-116 < (/.f64 1 n) < 1.9999999999999999e-112

   1. Initial program 28.6%

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

   3. Taylor expanded in n around inf 84.9%

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

      1. +-rgt-identity 84.9%

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

      2. +-rgt-identity 84.9%

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

      3. log1p-def 84.9%

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

   5. Simplified 84.9%

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

      1. log1p-udef 84.9%

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

      2. diff-log 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. clear-num 85.1%

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

      2. log-rec 85.1%

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

   9. Applied egg-rr 85.1%

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

   if -1.00000000000000002e-87 < (/.f64 1 n) < -5.0000000000000003e-116

   1. Initial program 5.0%

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

   3. Taylor expanded in n around inf 18.7%

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

      1. +-rgt-identity 18.7%

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

      2. +-rgt-identity 18.7%

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

      3. log1p-def 18.7%

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

   5. Simplified 18.7%

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

   6. Taylor expanded in x around inf 86.0%

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

   if 1.9999999999999999e-112 < (/.f64 1 n) < 4.99999999999999979e-104

   1. Initial program 5.7%

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

   3. Taylor expanded in n around inf 11.2%

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

      1. +-rgt-identity 11.2%

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

      2. +-rgt-identity 11.2%

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

      3. log1p-def 11.2%

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

   5. Simplified 11.2%

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

   6. Taylor expanded in x around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

   8. Simplified 99.5%

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

   if 4.99999999999999979e-104 < (/.f64 1 n) < 4.99999999999999977e-7

   1. Initial program 8.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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-def 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. Step-by-step derivation

      1. log1p-udef 60.5%

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

      2. diff-log 60.8%

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

   7. Applied egg-rr 60.8%

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

      1. log-div 60.5%

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

      2. log1p-udef 60.5%

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

      3. frac-2neg 60.5%

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

      4. div-inv 60.6%

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

      5. log1p-udef 60.6%

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

      6. log-div 60.8%

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

      7. neg-log 60.8%

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

      8. clear-num 60.9%

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

   9. Applied egg-rr 60.9%

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

   if 4.99999999999999977e-7 < (/.f64 1 n) < 9.9999999999999995e213

   1. Initial program 77.4%

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

   3. Taylor expanded in x around 0 77.4%

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

   if 9.9999999999999995e213 < (/.f64 1 n)

   1. Initial program 9.7%

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

   3. Taylor expanded in n around inf 8.7%

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

      1. +-rgt-identity 8.7%

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

      2. +-rgt-identity 8.7%

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

      3. log1p-def 8.7%

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

   5. Simplified 8.7%

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

   6. Taylor expanded in x around inf 86.2%

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

      1. *-commutative 86.2%

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

   8. Simplified 86.2%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-1 + \frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\log \left(\frac{x}{x + 1}\right) \cdot \frac{1}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ t_1 := \frac{x}{x + 1}\\ t_2 := \log t_1\\ t_3 := \frac{-t_2}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-1 + t_1\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;t_0 - \frac{0.5}{n \cdot {x}^{2}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t_2 \cdot \frac{1}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x)))
        (t_1 (/ x (+ x 1.0)))
        (t_2 (log t_1))
        (t_3 (/ (- t_2) n)))
   (if (<= (/ 1.0 n) -500.0)
     (/ (- (log1p (+ -1.0 t_1))) n)
     (if (<= (/ 1.0 n) -1e-87)
       t_3
       (if (<= (/ 1.0 n) -5e-116)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 2e-112)
           t_3
           (if (<= (/ 1.0 n) 5e-104)
             (- t_0 (/ 0.5 (* n (pow x 2.0))))
             (if (<= (/ 1.0 n) 5e-7)
               (* t_2 (/ 1.0 (- n)))
               (if (<= (/ 1.0 n) 1e+214)
                 (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
                 t_0)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = x / (x + 1.0);
	double t_2 = log(t_1);
	double t_3 = -t_2 / n;
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = -log1p((-1.0 + t_1)) / n;
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_3;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_3;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = t_0 - (0.5 / (n * pow(x, 2.0)));
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_2 * (1.0 / -n);
	} else if ((1.0 / n) <= 1e+214) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double t_1 = x / (x + 1.0);
	double t_2 = Math.log(t_1);
	double t_3 = -t_2 / n;
	double tmp;
	if ((1.0 / n) <= -500.0) {
		tmp = -Math.log1p((-1.0 + t_1)) / n;
	} else if ((1.0 / n) <= -1e-87) {
		tmp = t_3;
	} else if ((1.0 / n) <= -5e-116) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-112) {
		tmp = t_3;
	} else if ((1.0 / n) <= 5e-104) {
		tmp = t_0 - (0.5 / (n * Math.pow(x, 2.0)));
	} else if ((1.0 / n) <= 5e-7) {
		tmp = t_2 * (1.0 / -n);
	} else if ((1.0 / n) <= 1e+214) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	t_1 = x / (x + 1.0)
	t_2 = math.log(t_1)
	t_3 = -t_2 / n
	tmp = 0
	if (1.0 / n) <= -500.0:
		tmp = -math.log1p((-1.0 + t_1)) / n
	elif (1.0 / n) <= -1e-87:
		tmp = t_3
	elif (1.0 / n) <= -5e-116:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e-112:
		tmp = t_3
	elif (1.0 / n) <= 5e-104:
		tmp = t_0 - (0.5 / (n * math.pow(x, 2.0)))
	elif (1.0 / n) <= 5e-7:
		tmp = t_2 * (1.0 / -n)
	elif (1.0 / n) <= 1e+214:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	t_1 = Float64(x / Float64(x + 1.0))
	t_2 = log(t_1)
	t_3 = Float64(Float64(-t_2) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -500.0)
		tmp = Float64(Float64(-log1p(Float64(-1.0 + t_1))) / n);
	elseif (Float64(1.0 / n) <= -1e-87)
		tmp = t_3;
	elseif (Float64(1.0 / n) <= -5e-116)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-112)
		tmp = t_3;
	elseif (Float64(1.0 / n) <= 5e-104)
		tmp = Float64(t_0 - Float64(0.5 / Float64(n * (x ^ 2.0))));
	elseif (Float64(1.0 / n) <= 5e-7)
		tmp = Float64(t_2 * Float64(1.0 / Float64(-n)));
	elseif (Float64(1.0 / n) <= 1e+214)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[((-t$95$2) / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -500.0], N[((-N[Log[1 + N[(-1.0 + t$95$1), $MachinePrecision]], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e-87], t$95$3, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-116], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-112], t$95$3, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-104], N[(t$95$0 - N[(0.5 / N[(n * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-7], N[(t$95$2 * N[(1.0 / (-n)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+214], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (/.f64 1 n) < -500

   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 48.8%

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

      1. +-rgt-identity 48.8%

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

      2. +-rgt-identity 48.8%

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

      3. log1p-def 48.8%

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

   5. Simplified 48.8%

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

      1. log1p-udef 48.8%

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

      2. diff-log 48.8%

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

   7. Applied egg-rr 48.8%

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

      1. clear-num 48.8%

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

      2. log-rec 48.8%

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

   9. Applied egg-rr 48.8%

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

       1. add-sqr-sqrt 46.5%

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

       2. sqrt-unprod 46.8%

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

       3. sqr-neg 46.8%

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

       4. sqrt-unprod 46.8%

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

       5. log1p-expm1-u 46.8%

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

       6. add-sqr-sqrt 46.8%

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

       7. neg-log 46.8%

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

       8. clear-num 46.8%

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

       9. diff-log 46.8%

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

       10. log1p-udef 46.8%

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

       11. log1p-udef 46.8%

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

       12. diff-log 46.8%

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

       13. clear-num 46.8%

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

       14. neg-log 46.8%

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

       15. add-sqr-sqrt 46.8%

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

       16. sqrt-unprod 46.8%

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

   11. Applied egg-rr 93.2%

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

   if -500 < (/.f64 1 n) < -1.00000000000000002e-87 or -5.0000000000000003e-116 < (/.f64 1 n) < 1.9999999999999999e-112

   1. Initial program 28.6%

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

   3. Taylor expanded in n around inf 84.9%

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

      1. +-rgt-identity 84.9%

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

      2. +-rgt-identity 84.9%

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

      3. log1p-def 84.9%

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

   5. Simplified 84.9%

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

      1. log1p-udef 84.9%

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

      2. diff-log 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. clear-num 85.1%

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

      2. log-rec 85.1%

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

   9. Applied egg-rr 85.1%

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

   if -1.00000000000000002e-87 < (/.f64 1 n) < -5.0000000000000003e-116

   1. Initial program 5.0%

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

   3. Taylor expanded in n around inf 18.7%

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

      1. +-rgt-identity 18.7%

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

      2. +-rgt-identity 18.7%

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

      3. log1p-def 18.7%

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

   5. Simplified 18.7%

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

   6. Taylor expanded in x around inf 86.0%

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

   if 1.9999999999999999e-112 < (/.f64 1 n) < 4.99999999999999979e-104

   1. Initial program 5.7%

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

   3. Taylor expanded in n around inf 11.2%

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

      1. +-rgt-identity 11.2%

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

      2. +-rgt-identity 11.2%

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

      3. log1p-def 11.2%

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

   5. Simplified 11.2%

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

   6. Taylor expanded in x around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r/ 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   if 4.99999999999999979e-104 < (/.f64 1 n) < 4.99999999999999977e-7

   1. Initial program 8.0%

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   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-def 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. Step-by-step derivation

      1. log1p-udef 60.5%

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

      2. diff-log 60.8%

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

   7. Applied egg-rr 60.8%

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

      1. log-div 60.5%

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

      2. log1p-udef 60.5%

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

      3. frac-2neg 60.5%

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

      4. div-inv 60.6%

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

      5. log1p-udef 60.6%

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

      6. log-div 60.8%

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

      7. neg-log 60.8%

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

      8. clear-num 60.9%

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

   9. Applied egg-rr 60.9%

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

   if 4.99999999999999977e-7 < (/.f64 1 n) < 9.9999999999999995e213

   1. Initial program 77.4%

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

   3. Taylor expanded in x around 0 77.4%

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

   if 9.9999999999999995e213 < (/.f64 1 n)

   1. Initial program 9.7%

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

   3. Taylor expanded in n around inf 8.7%

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

      1. +-rgt-identity 8.7%

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

      2. +-rgt-identity 8.7%

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

      3. log1p-def 8.7%

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

   5. Simplified 8.7%

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

   6. Taylor expanded in x around inf 86.2%

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

      1. *-commutative 86.2%

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

   8. Simplified 86.2%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -500:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(-1 + \frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-87}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\log \left(\frac{x}{x + 1}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{n \cdot x} - \frac{0.5}{n \cdot {x}^{2}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\log \left(\frac{x}{x + 1}\right) \cdot \frac{1}{-n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.62 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-43}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+175}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.62e-74)
   (/ (- (log x)) n)
   (if (<= x 7.2e-43)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 1.0)
       (/ (- x (log x)) n)
       (if (<= x 1.1e+175) (/ (/ 1.0 x) n) (/ 0.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.62e-74) {
		tmp = -log(x) / n;
	} else if (x <= 7.2e-43) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else if (x <= 1.1e+175) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.62d-74) then
        tmp = -log(x) / n
    else if (x <= 7.2d-43) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else if (x <= 1.1d+175) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.62e-74) {
		tmp = -Math.log(x) / n;
	} else if (x <= 7.2e-43) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 1.1e+175) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.62e-74:
		tmp = -math.log(x) / n
	elif x <= 7.2e-43:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 1.0:
		tmp = (x - math.log(x)) / n
	elif x <= 1.1e+175:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.62e-74)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 7.2e-43)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 1.1e+175)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.62e-74)
		tmp = -log(x) / n;
	elseif (x <= 7.2e-43)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 1.0)
		tmp = (x - log(x)) / n;
	elseif (x <= 1.1e+175)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.62e-74], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 7.2e-43], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 1.1e+175], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.62000000000000007e-74

   1. Initial program 37.7%

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

   3. Taylor expanded in n around inf 59.0%

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

      1. +-rgt-identity 59.0%

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

      2. +-rgt-identity 59.0%

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

      3. log1p-def 59.0%

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

   5. Simplified 59.0%

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

   6. Taylor expanded in x around 0 59.0%

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

      1. neg-mul-1 59.0%

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

   8. Simplified 59.0%

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

   if 1.62000000000000007e-74 < x < 7.1999999999999998e-43

   1. Initial program 79.4%

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

   3. Taylor expanded in x around 0 79.4%

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

   if 7.1999999999999998e-43 < x < 1

   1. Initial program 21.1%

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

   3. Taylor expanded in n around inf 75.9%

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

      1. +-rgt-identity 75.9%

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

      2. +-rgt-identity 75.9%

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

      3. log1p-def 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in x around 0 67.7%

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

      1. neg-mul-1 67.7%

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

      2. unsub-neg 67.7%

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

   8. Simplified 67.7%

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

   if 1 < x < 1.1e175

   1. Initial program 51.9%

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

   3. Taylor expanded in n around inf 51.7%

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

      1. +-rgt-identity 51.7%

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

      2. +-rgt-identity 51.7%

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

      3. log1p-def 51.7%

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

   5. Simplified 51.7%

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

   6. Taylor expanded in x around inf 64.2%

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

   if 1.1e175 < x

   1. Initial program 81.4%

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

   3. Taylor expanded in n around inf 81.4%

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

      1. +-rgt-identity 81.4%

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

      2. +-rgt-identity 81.4%

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

      3. log1p-def 81.4%

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

   5. Simplified 81.4%

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

      1. log1p-udef 81.4%

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

      2. diff-log 81.4%

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

   7. Applied egg-rr 81.4%

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

   8. Taylor expanded in x around inf 81.4%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.62 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-43}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+175}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (/ (- x (log x)) n)
   (if (<= x 2.05e+173) (/ (/ 1.0 x) n) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else if (x <= 2.05e+173) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else if (x <= 2.05d+173) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 2.05e+173) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = (x - math.log(x)) / n
	elif x <= 2.05e+173:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 2.05e+173)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = (x - log(x)) / n;
	elseif (x <= 2.05e+173)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 2.05e+173], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 40.9%

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

   3. Taylor expanded in n around inf 57.2%

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

      1. +-rgt-identity 57.2%

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

      2. +-rgt-identity 57.2%

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

      3. log1p-def 57.2%

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

   5. Simplified 57.2%

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

   6. Taylor expanded in x around 0 56.5%

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

      1. neg-mul-1 56.5%

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

      2. unsub-neg 56.5%

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

   8. Simplified 56.5%

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

   if 1 < x < 2.04999999999999988e173

   1. Initial program 51.9%

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

   3. Taylor expanded in n around inf 51.7%

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

      1. +-rgt-identity 51.7%

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

      2. +-rgt-identity 51.7%

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

      3. log1p-def 51.7%

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

   5. Simplified 51.7%

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

   6. Taylor expanded in x around inf 64.2%

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

   if 2.04999999999999988e173 < x

   1. Initial program 81.4%

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

   3. Taylor expanded in n around inf 81.4%

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

      1. +-rgt-identity 81.4%

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

      2. +-rgt-identity 81.4%

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

      3. log1p-def 81.4%

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

   5. Simplified 81.4%

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

      1. log1p-udef 81.4%

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

      2. diff-log 81.4%

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

   7. Applied egg-rr 81.4%

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

   8. Taylor expanded in x around inf 81.4%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.55)
   (/ (- (log x)) n)
   (if (<= x 1.4e+173) (/ (/ 1.0 x) n) (/ 0.0 n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -log(x) / n;
	} else if (x <= 1.4e+173) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.55d0) then
        tmp = -log(x) / n
    else if (x <= 1.4d+173) then
        tmp = (1.0d0 / x) / n
    else
        tmp = 0.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.55) {
		tmp = -Math.log(x) / n;
	} else if (x <= 1.4e+173) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = 0.0 / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.55:
		tmp = -math.log(x) / n
	elif x <= 1.4e+173:
		tmp = (1.0 / x) / n
	else:
		tmp = 0.0 / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.55)
		tmp = Float64(Float64(-log(x)) / n);
	elseif (x <= 1.4e+173)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(0.0 / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.55)
		tmp = -log(x) / n;
	elseif (x <= 1.4e+173)
		tmp = (1.0 / x) / n;
	else
		tmp = 0.0 / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.55], N[((-N[Log[x], $MachinePrecision]) / n), $MachinePrecision], If[LessEqual[x, 1.4e+173], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(0.0 / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.55000000000000004

   1. Initial program 40.9%

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

   3. Taylor expanded in n around inf 57.2%

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

      1. +-rgt-identity 57.2%

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

      2. +-rgt-identity 57.2%

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

      3. log1p-def 57.2%

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

   5. Simplified 57.2%

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

   6. Taylor expanded in x around 0 55.6%

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

      1. neg-mul-1 55.6%

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

   8. Simplified 55.6%

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

   if 0.55000000000000004 < x < 1.39999999999999991e173

   1. Initial program 51.9%

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

   3. Taylor expanded in n around inf 51.7%

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

      1. +-rgt-identity 51.7%

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

      2. +-rgt-identity 51.7%

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

      3. log1p-def 51.7%

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

   5. Simplified 51.7%

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

   6. Taylor expanded in x around inf 64.2%

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

   if 1.39999999999999991e173 < x

   1. Initial program 81.4%

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

   3. Taylor expanded in n around inf 81.4%

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

      1. +-rgt-identity 81.4%

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

      2. +-rgt-identity 81.4%

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

      3. log1p-def 81.4%

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

   5. Simplified 81.4%

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

      1. log1p-udef 81.4%

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

      2. diff-log 81.4%

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

   7. Applied egg-rr 81.4%

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

   8. Taylor expanded in x around inf 81.4%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.55:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.4% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -10000000.0) (/ 0.0 n) (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -10000000.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-10000000.0d0)) then
        tmp = 0.0d0 / n
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -10000000.0) {
		tmp = 0.0 / n;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -10000000.0:
		tmp = 0.0 / n
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -10000000.0)
		tmp = Float64(0.0 / n);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -10000000.0)
		tmp = 0.0 / n;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -10000000.0], N[(0.0 / n), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 n) < -1e7

   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 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-def 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. Step-by-step derivation

      1. log1p-udef 50.1%

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

      2. diff-log 50.1%

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

   7. Applied egg-rr 50.1%

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

   8. Taylor expanded in x around inf 54.3%

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

   if -1e7 < (/.f64 1 n)

   1. Initial program 30.9%

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

   3. Taylor expanded in n around inf 62.2%

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

      1. +-rgt-identity 62.2%

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

      2. +-rgt-identity 62.2%

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

      3. log1p-def 62.2%

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

   5. Simplified 62.2%

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

   6. Taylor expanded in x around inf 40.6%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -10000000:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 40.2% 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 49.6%

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

3. Taylor expanded in n around inf 58.9%

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

   1. +-rgt-identity 58.9%

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

   2. +-rgt-identity 58.9%

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

   3. log1p-def 58.9%

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

5. Simplified 58.9%

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

6. Taylor expanded in x around inf 34.9%

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

   1. *-commutative 34.9%

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

8. Simplified 34.9%

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

9. Final simplification 34.9%

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

Alternative 19: 40.7% 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 49.6%

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

3. Taylor expanded in n around inf 58.9%

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

   1. +-rgt-identity 58.9%

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

   2. +-rgt-identity 58.9%

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

   3. log1p-def 58.9%

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

5. Simplified 58.9%

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

6. Taylor expanded in x around inf 35.0%

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

7. Final simplification 35.0%

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

Reproduce

herbie shell --seed 2024011 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))