2nthrt (problem 3.4.6)

Percentage Accurate: 53.7% → 85.9%

Time: 55.9s

Alternatives: 15

Speedup: 1.7×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-48}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5000:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-48)
     (/ (exp (/ (log x) n)) (* n x))
     (if (<= (/ 1.0 n) 1e-10)
       (/
        (+
         (log1p x)
         (- (* 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n)) (log x)))
        n)
       (if (<= (/ 1.0 n) 5000.0) (/ (/ t_0 n) x) (- (exp (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-48) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1e-10) {
		tmp = (log1p(x) + ((0.5 * ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n)) - log(x))) / n;
	} else if ((1.0 / n) <= 5000.0) {
		tmp = (t_0 / n) / x;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-48) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 1e-10) {
		tmp = (Math.log1p(x) + ((0.5 * ((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) / n)) - Math.log(x))) / n;
	} else if ((1.0 / n) <= 5000.0) {
		tmp = (t_0 / n) / x;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-48:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 1e-10:
		tmp = (math.log1p(x) + ((0.5 * ((math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)) / n)) - math.log(x))) / n
	elif (1.0 / n) <= 5000.0:
		tmp = (t_0 / n) / x
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-48)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 1e-10)
		tmp = Float64(Float64(log1p(x) + Float64(Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n)) - log(x))) / n);
	elseif (Float64(1.0 / n) <= 5000.0)
		tmp = Float64(Float64(t_0 / n) / x);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-48], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-10], N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5000.0], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999999e-48

   1. Initial program 86.8%

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

   3. Taylor expanded in x around inf 93.7%

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

      1. mul-1-neg 93.7%

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

      2. log-rec 93.7%

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

      3. mul-1-neg 93.7%

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

      4. distribute-neg-frac 93.7%

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

      5. mul-1-neg 93.7%

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

      6. remove-double-neg 93.7%

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

      7. *-commutative 93.7%

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

   5. Simplified 93.7%

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

   if -3.9999999999999999e-48 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000004e-10

   1. Initial program 35.6%

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

   3. Taylor expanded in n around inf 80.5%

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

      1. associate--l+ 80.5%

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

      2. log1p-define 80.5%

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

      3. +-commutative 80.5%

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

      4. associate--r+ 80.5%

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

      5. distribute-lft-out-- 80.5%

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

      6. div-sub 80.5%

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

      7. log1p-define 80.5%

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

   5. Simplified 80.5%

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

   if 1.00000000000000004e-10 < (/.f64 #s(literal 1 binary64) n) < 5e3

   1. Initial program 6.0%

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

   3. Taylor expanded in x around inf 99.4%

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

      1. associate-/r* 99.4%

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

      2. mul-1-neg 99.4%

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

      3. log-rec 99.4%

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

      4. mul-1-neg 99.4%

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

      5. distribute-neg-frac 99.4%

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

      6. mul-1-neg 99.4%

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

      7. remove-double-neg 99.4%

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

      8. *-rgt-identity 99.4%

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

      9. associate-/l* 99.4%

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

      10. exp-to-pow 99.4%

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

   5. Simplified 99.4%

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

   if 5e3 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 43.5%

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

   3. Taylor expanded in n around 0 43.5%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-48}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-10}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-48}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5000:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-48)
     (/ (exp (/ (log x) n)) (* n x))
     (if (<= (/ 1.0 n) 2e-57)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5000.0) (/ (/ t_0 n) x) (- (exp (/ x n)) t_0))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-48) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-57) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5000.0) {
		tmp = (t_0 / n) / x;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-48)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 2d-57) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5000.0d0) then
        tmp = (t_0 / n) / x
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-48) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-57) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5000.0) {
		tmp = (t_0 / n) / x;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-48:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-57:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5000.0:
		tmp = (t_0 / n) / x
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-48)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-57)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5000.0)
		tmp = Float64(Float64(t_0 / n) / x);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-48)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 2e-57)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5000.0)
		tmp = (t_0 / n) / x;
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-48], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-57], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5000.0], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999999e-48

   1. Initial program 86.8%

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

   3. Taylor expanded in x around inf 93.7%

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

      1. mul-1-neg 93.7%

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

      2. log-rec 93.7%

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

      3. mul-1-neg 93.7%

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

      4. distribute-neg-frac 93.7%

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

      5. mul-1-neg 93.7%

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

      6. remove-double-neg 93.7%

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

      7. *-commutative 93.7%

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

   5. Simplified 93.7%

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

   if -3.9999999999999999e-48 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-57

   1. Initial program 37.8%

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

   3. Taylor expanded in n around inf 83.7%

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

      1. log1p-define 83.7%

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

   5. Simplified 83.7%

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

      1. log1p-undefine 83.7%

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

      2. diff-log 83.8%

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

      3. +-commutative 83.8%

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

   7. Applied egg-rr 83.8%

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

   if 1.99999999999999991e-57 < (/.f64 #s(literal 1 binary64) n) < 5e3

   1. Initial program 14.1%

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

   3. Taylor expanded in x around inf 64.4%

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

      1. associate-/r* 64.5%

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

      2. mul-1-neg 64.5%

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

      3. log-rec 64.5%

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

      4. mul-1-neg 64.5%

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

      5. distribute-neg-frac 64.5%

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

      6. mul-1-neg 64.5%

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

      7. remove-double-neg 64.5%

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

      8. *-rgt-identity 64.5%

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

      9. associate-/l* 64.5%

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

      10. exp-to-pow 64.5%

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

   5. Simplified 64.5%

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

   if 5e3 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 43.5%

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

   3. Taylor expanded in n around 0 43.5%

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

      1. log1p-define 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. exp-to-pow 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-48}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-48}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5000:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+141}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-48)
     (/ (exp (/ (log x) n)) (* n x))
     (if (<= (/ 1.0 n) 2e-57)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5000.0)
         (/ (/ t_0 n) x)
         (if (<= (/ 1.0 n) 1e+141)
           (- (+ (/ x n) 1.0) t_0)
           (/
            (+
             (/ 1.0 n)
             (/ (/ (+ (* n -0.5) (* n (/ 0.3333333333333333 x))) (* n n)) x))
            x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-48) {
		tmp = exp((log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-57) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5000.0) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e+141) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-4d-48)) then
        tmp = exp((log(x) / n)) / (n * x)
    else if ((1.0d0 / n) <= 2d-57) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5000.0d0) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 1d+141) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = ((1.0d0 / n) + ((((n * (-0.5d0)) + (n * (0.3333333333333333d0 / x))) / (n * n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-48) {
		tmp = Math.exp((Math.log(x) / n)) / (n * x);
	} else if ((1.0 / n) <= 2e-57) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5000.0) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e+141) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-48:
		tmp = math.exp((math.log(x) / n)) / (n * x)
	elif (1.0 / n) <= 2e-57:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5000.0:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e+141:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-48)
		tmp = Float64(exp(Float64(log(x) / n)) / Float64(n * x));
	elseif (Float64(1.0 / n) <= 2e-57)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5000.0)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e+141)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(n * -0.5) + Float64(n * Float64(0.3333333333333333 / x))) / Float64(n * n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -4e-48)
		tmp = exp((log(x) / n)) / (n * x);
	elseif ((1.0 / n) <= 2e-57)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5000.0)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 1e+141)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-48], N[(N[Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-57], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5000.0], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+141], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(n * -0.5), $MachinePrecision] + N[(n * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999999e-48

   1. Initial program 86.8%

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

   3. Taylor expanded in x around inf 93.7%

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

      1. mul-1-neg 93.7%

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

      2. log-rec 93.7%

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

      3. mul-1-neg 93.7%

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

      4. distribute-neg-frac 93.7%

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

      5. mul-1-neg 93.7%

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

      6. remove-double-neg 93.7%

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

      7. *-commutative 93.7%

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

   5. Simplified 93.7%

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

   if -3.9999999999999999e-48 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-57

   1. Initial program 37.8%

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

   3. Taylor expanded in n around inf 83.7%

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

      1. log1p-define 83.7%

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

   5. Simplified 83.7%

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

      1. log1p-undefine 83.7%

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

      2. diff-log 83.8%

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

      3. +-commutative 83.8%

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

   7. Applied egg-rr 83.8%

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

   if 1.99999999999999991e-57 < (/.f64 #s(literal 1 binary64) n) < 5e3

   1. Initial program 14.1%

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

   3. Taylor expanded in x around inf 64.4%

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

      1. associate-/r* 64.5%

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

      2. mul-1-neg 64.5%

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

      3. log-rec 64.5%

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

      4. mul-1-neg 64.5%

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

      5. distribute-neg-frac 64.5%

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

      6. mul-1-neg 64.5%

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

      7. remove-double-neg 64.5%

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

      8. *-rgt-identity 64.5%

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

      9. associate-/l* 64.5%

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

      10. exp-to-pow 64.5%

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

   5. Simplified 64.5%

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

   if 5e3 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000002e141

   1. Initial program 76.6%

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

   3. Taylor expanded in x around 0 73.2%

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

   if 1.00000000000000002e141 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf 6.5%

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

      1. log1p-define 6.5%

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

   5. Simplified 6.5%

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

   6. Taylor expanded in x around -inf 82.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 82.5%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

      2. mul-1-neg 82.5%

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

      3. associate-*r/ 82.5%

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

      4. metadata-eval 82.5%

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

      5. *-commutative 82.5%

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

      6. associate-*r/ 82.5%

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

      7. metadata-eval 82.5%

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

   8. Simplified 82.5%

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

      1. associate-/r* 82.5%

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

      2. frac-2neg 82.5%

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

      3. frac-sub 86.8%

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

      4. metadata-eval 86.8%

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

   10. Applied egg-rr 86.8%

       \[\leadsto -\frac{\left(-\frac{\color{blue}{\frac{\frac{0.3333333333333333}{x} \cdot \left(-n\right) - n \cdot -0.5}{n \cdot \left(-n\right)}}}{x}\right) - \frac{1}{n}}{x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-48}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{n \cdot x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+141}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\frac{--0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-237}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-76}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+141}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+14)
   (/ (- -0.3333333333333333) (* n (pow x 3.0)))
   (if (<= (/ 1.0 n) 2e-237)
     (/ (/ 1.0 n) x)
     (if (<= (/ 1.0 n) 2e-76)
       (/ (- x (log x)) n)
       (if (<= (/ 1.0 n) 1e-18)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 1e+141)
           (- 1.0 (pow x (/ 1.0 n)))
           (/
            (+
             (/ 1.0 n)
             (/ (/ (+ (* n -0.5) (* n (/ 0.3333333333333333 x))) (* n n)) x))
            x)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+14) {
		tmp = -(-0.3333333333333333) / (n * pow(x, 3.0));
	} else if ((1.0 / n) <= 2e-237) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e-76) {
		tmp = (x - log(x)) / n;
	} else if ((1.0 / n) <= 1e-18) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 1e+141) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d+14)) then
        tmp = -(-0.3333333333333333d0) / (n * (x ** 3.0d0))
    else if ((1.0d0 / n) <= 2d-237) then
        tmp = (1.0d0 / n) / x
    else if ((1.0d0 / n) <= 2d-76) then
        tmp = (x - log(x)) / n
    else if ((1.0d0 / n) <= 1d-18) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 1d+141) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = ((1.0d0 / n) + ((((n * (-0.5d0)) + (n * (0.3333333333333333d0 / x))) / (n * n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+14) {
		tmp = -(-0.3333333333333333) / (n * Math.pow(x, 3.0));
	} else if ((1.0 / n) <= 2e-237) {
		tmp = (1.0 / n) / x;
	} else if ((1.0 / n) <= 2e-76) {
		tmp = (x - Math.log(x)) / n;
	} else if ((1.0 / n) <= 1e-18) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 1e+141) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+14:
		tmp = -(-0.3333333333333333) / (n * math.pow(x, 3.0))
	elif (1.0 / n) <= 2e-237:
		tmp = (1.0 / n) / x
	elif (1.0 / n) <= 2e-76:
		tmp = (x - math.log(x)) / n
	elif (1.0 / n) <= 1e-18:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 1e+141:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+14)
		tmp = Float64(Float64(-(-0.3333333333333333)) / Float64(n * (x ^ 3.0)));
	elseif (Float64(1.0 / n) <= 2e-237)
		tmp = Float64(Float64(1.0 / n) / x);
	elseif (Float64(1.0 / n) <= 2e-76)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (Float64(1.0 / n) <= 1e-18)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 1e+141)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(n * -0.5) + Float64(n * Float64(0.3333333333333333 / x))) / Float64(n * n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e+14)
		tmp = -(-0.3333333333333333) / (n * (x ^ 3.0));
	elseif ((1.0 / n) <= 2e-237)
		tmp = (1.0 / n) / x;
	elseif ((1.0 / n) <= 2e-76)
		tmp = (x - log(x)) / n;
	elseif ((1.0 / n) <= 1e-18)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 1e+141)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+14], N[((--0.3333333333333333) / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-237], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-76], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-18], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+141], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(n * -0.5), $MachinePrecision] + N[(n * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1e14

   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 48.6%

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

      1. log1p-define 48.6%

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

   5. Simplified 48.6%

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

   6. Taylor expanded in x around -inf 51.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 51.2%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

      2. mul-1-neg 51.2%

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

      3. associate-*r/ 51.2%

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

      4. metadata-eval 51.2%

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

      5. *-commutative 51.2%

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

      6. associate-*r/ 51.2%

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

      7. metadata-eval 51.2%

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

   8. Simplified 51.2%

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

   9. Taylor expanded in x around 0 83.4%

      \[\leadsto -\color{blue}{\frac{-0.3333333333333333}{n \cdot {x}^{3}}} \]

   if -1e14 < (/.f64 #s(literal 1 binary64) n) < 2e-237

   1. Initial program 39.9%

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

   3. Taylor expanded in n around inf 77.7%

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

      1. log1p-define 77.7%

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

   5. Simplified 77.7%

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

   6. Taylor expanded in x around inf 57.7%

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

      1. *-commutative 57.7%

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

   8. Simplified 57.7%

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

   9. Taylor expanded in x around 0 57.7%

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

       1. associate-/r* 58.0%

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

   11. Simplified 58.0%

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

   if 2e-237 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999985e-76

   1. Initial program 23.0%

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

   3. Taylor expanded in n around inf 83.2%

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

      1. log1p-define 83.2%

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

   5. Simplified 83.2%

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

   6. Taylor expanded in x around 0 64.0%

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

   if 1.99999999999999985e-76 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-18

   1. Initial program 22.3%

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

   3. Taylor expanded in n around inf 53.2%

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

      1. log1p-define 53.2%

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

   5. Simplified 53.2%

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

   6. Taylor expanded in x around inf 70.0%

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

   if 1.0000000000000001e-18 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000002e141

   1. Initial program 59.4%

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

   3. Taylor expanded in x around 0 55.0%

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

      1. *-rgt-identity 55.0%

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

      2. associate-*l/ 55.0%

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

      3. associate-/l* 55.0%

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

      4. exp-to-pow 55.0%

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

   5. Simplified 55.0%

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

   if 1.00000000000000002e141 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf 6.5%

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

      1. log1p-define 6.5%

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

   5. Simplified 6.5%

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

   6. Taylor expanded in x around -inf 82.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 82.5%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

      2. mul-1-neg 82.5%

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

      3. associate-*r/ 82.5%

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

      4. metadata-eval 82.5%

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

      5. *-commutative 82.5%

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

      6. associate-*r/ 82.5%

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

      7. metadata-eval 82.5%

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

   8. Simplified 82.5%

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

      1. associate-/r* 82.5%

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

      2. frac-2neg 82.5%

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

      3. frac-sub 86.8%

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

      4. metadata-eval 86.8%

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

   10. Applied egg-rr 86.8%

       \[\leadsto -\frac{\left(-\frac{\color{blue}{\frac{\frac{0.3333333333333333}{x} \cdot \left(-n\right) - n \cdot -0.5}{n \cdot \left(-n\right)}}}{x}\right) - \frac{1}{n}}{x} \]
3. Recombined 6 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\frac{--0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-237}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-76}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+141}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+141}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -4e-48)
     t_1
     (if (<= (/ 1.0 n) 2e-57)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5000.0)
         t_1
         (if (<= (/ 1.0 n) 1e+141)
           (- (+ (/ x n) 1.0) t_0)
           (/
            (+
             (/ 1.0 n)
             (/ (/ (+ (* n -0.5) (* n (/ 0.3333333333333333 x))) (* n n)) x))
            x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -4e-48) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-57) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+141) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / n) / x
    if ((1.0d0 / n) <= (-4d-48)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-57) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5000.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+141) then
        tmp = ((x / n) + 1.0d0) - t_0
    else
        tmp = ((1.0d0 / n) + ((((n * (-0.5d0)) + (n * (0.3333333333333333d0 / x))) / (n * n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -4e-48) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-57) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+141) {
		tmp = ((x / n) + 1.0) - t_0;
	} else {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -4e-48:
		tmp = t_1
	elif (1.0 / n) <= 2e-57:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5000.0:
		tmp = t_1
	elif (1.0 / n) <= 1e+141:
		tmp = ((x / n) + 1.0) - t_0
	else:
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-48)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-57)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5000.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+141)
		tmp = Float64(Float64(Float64(x / n) + 1.0) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(n * -0.5) + Float64(n * Float64(0.3333333333333333 / x))) / Float64(n * n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / n) / x;
	tmp = 0.0;
	if ((1.0 / n) <= -4e-48)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-57)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5000.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+141)
		tmp = ((x / n) + 1.0) - t_0;
	else
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-48], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-57], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5000.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+141], N[(N[(N[(x / n), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(n * -0.5), $MachinePrecision] + N[(n * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999999e-48 or 1.99999999999999991e-57 < (/.f64 #s(literal 1 binary64) n) < 5e3

   1. Initial program 72.2%

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

   3. Taylor expanded in x around inf 87.8%

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

      1. associate-/r* 87.8%

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

      2. mul-1-neg 87.8%

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

      3. log-rec 87.8%

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

      4. mul-1-neg 87.8%

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

      5. distribute-neg-frac 87.8%

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

      6. mul-1-neg 87.8%

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

      7. remove-double-neg 87.8%

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

      8. *-rgt-identity 87.8%

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

      9. associate-/l* 87.8%

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

      10. exp-to-pow 87.8%

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

   5. Simplified 87.8%

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

   if -3.9999999999999999e-48 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-57

   1. Initial program 37.8%

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

   3. Taylor expanded in n around inf 83.7%

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

      1. log1p-define 83.7%

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

   5. Simplified 83.7%

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

      1. log1p-undefine 83.7%

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

      2. diff-log 83.8%

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

      3. +-commutative 83.8%

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

   7. Applied egg-rr 83.8%

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

   if 5e3 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000002e141

   1. Initial program 76.6%

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

   3. Taylor expanded in x around 0 73.2%

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

   if 1.00000000000000002e141 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf 6.5%

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

      1. log1p-define 6.5%

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

   5. Simplified 6.5%

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

   6. Taylor expanded in x around -inf 82.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 82.5%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

      2. mul-1-neg 82.5%

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

      3. associate-*r/ 82.5%

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

      4. metadata-eval 82.5%

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

      5. *-commutative 82.5%

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

      6. associate-*r/ 82.5%

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

      7. metadata-eval 82.5%

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

   8. Simplified 82.5%

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

      1. associate-/r* 82.5%

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

      2. frac-2neg 82.5%

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

      3. frac-sub 86.8%

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

      4. metadata-eval 86.8%

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

   10. Applied egg-rr 86.8%

       \[\leadsto -\frac{\left(-\frac{\color{blue}{\frac{\frac{0.3333333333333333}{x} \cdot \left(-n\right) - n \cdot -0.5}{n \cdot \left(-n\right)}}}{x}\right) - \frac{1}{n}}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+141}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\frac{t\_0}{n}}{x}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+141}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (/ t_0 n) x)))
   (if (<= (/ 1.0 n) -4e-48)
     t_1
     (if (<= (/ 1.0 n) 2e-57)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5000.0)
         t_1
         (if (<= (/ 1.0 n) 1e+141)
           (- 1.0 t_0)
           (/
            (+
             (/ 1.0 n)
             (/ (/ (+ (* n -0.5) (* n (/ 0.3333333333333333 x))) (* n n)) x))
            x)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -4e-48) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-57) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+141) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = (t_0 / n) / x
    if ((1.0d0 / n) <= (-4d-48)) then
        tmp = t_1
    else if ((1.0d0 / n) <= 2d-57) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5000.0d0) then
        tmp = t_1
    else if ((1.0d0 / n) <= 1d+141) then
        tmp = 1.0d0 - t_0
    else
        tmp = ((1.0d0 / n) + ((((n * (-0.5d0)) + (n * (0.3333333333333333d0 / x))) / (n * n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = (t_0 / n) / x;
	double tmp;
	if ((1.0 / n) <= -4e-48) {
		tmp = t_1;
	} else if ((1.0 / n) <= 2e-57) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5000.0) {
		tmp = t_1;
	} else if ((1.0 / n) <= 1e+141) {
		tmp = 1.0 - t_0;
	} else {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = (t_0 / n) / x
	tmp = 0
	if (1.0 / n) <= -4e-48:
		tmp = t_1
	elif (1.0 / n) <= 2e-57:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5000.0:
		tmp = t_1
	elif (1.0 / n) <= 1e+141:
		tmp = 1.0 - t_0
	else:
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(Float64(t_0 / n) / x)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-48)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 2e-57)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5000.0)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 1e+141)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(n * -0.5) + Float64(n * Float64(0.3333333333333333 / x))) / Float64(n * n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = (t_0 / n) / x;
	tmp = 0.0;
	if ((1.0 / n) <= -4e-48)
		tmp = t_1;
	elseif ((1.0 / n) <= 2e-57)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5000.0)
		tmp = t_1;
	elseif ((1.0 / n) <= 1e+141)
		tmp = 1.0 - t_0;
	else
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-48], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-57], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5000.0], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+141], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(n * -0.5), $MachinePrecision] + N[(n * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999999e-48 or 1.99999999999999991e-57 < (/.f64 #s(literal 1 binary64) n) < 5e3

   1. Initial program 72.2%

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

   3. Taylor expanded in x around inf 87.8%

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

      1. associate-/r* 87.8%

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

      2. mul-1-neg 87.8%

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

      3. log-rec 87.8%

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

      4. mul-1-neg 87.8%

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

      5. distribute-neg-frac 87.8%

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

      6. mul-1-neg 87.8%

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

      7. remove-double-neg 87.8%

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

      8. *-rgt-identity 87.8%

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

      9. associate-/l* 87.8%

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

      10. exp-to-pow 87.8%

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

   5. Simplified 87.8%

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

   if -3.9999999999999999e-48 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-57

   1. Initial program 37.8%

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

   3. Taylor expanded in n around inf 83.7%

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

      1. log1p-define 83.7%

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

   5. Simplified 83.7%

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

      1. log1p-undefine 83.7%

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

      2. diff-log 83.8%

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

      3. +-commutative 83.8%

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

   7. Applied egg-rr 83.8%

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

   if 5e3 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000002e141

   1. Initial program 76.6%

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

   3. Taylor expanded in x around 0 71.2%

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

      1. *-rgt-identity 71.2%

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

      2. associate-*l/ 71.2%

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

      3. associate-/l* 71.2%

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

      4. exp-to-pow 71.2%

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

   5. Simplified 71.2%

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

   if 1.00000000000000002e141 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf 6.5%

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

      1. log1p-define 6.5%

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

   5. Simplified 6.5%

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

   6. Taylor expanded in x around -inf 82.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 82.5%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

      2. mul-1-neg 82.5%

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

      3. associate-*r/ 82.5%

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

      4. metadata-eval 82.5%

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

      5. *-commutative 82.5%

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

      6. associate-*r/ 82.5%

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

      7. metadata-eval 82.5%

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

   8. Simplified 82.5%

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

      1. associate-/r* 82.5%

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

      2. frac-2neg 82.5%

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

      3. frac-sub 86.8%

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

      4. metadata-eval 86.8%

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

   10. Applied egg-rr 86.8%

       \[\leadsto -\frac{\left(-\frac{\color{blue}{\frac{\frac{0.3333333333333333}{x} \cdot \left(-n\right) - n \cdot -0.5}{n \cdot \left(-n\right)}}}{x}\right) - \frac{1}{n}}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5000:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+141}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 3.5 \cdot 10^{-273}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-217}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+195}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))) (t_1 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 3.5e-273)
     t_0
     (if (<= x 2.8e-217)
       t_1
       (if (<= x 3.1e-189)
         t_0
         (if (<= x 2.5e-169)
           t_1
           (if (<= x 9.2e-32)
             t_0
             (if (<= x 1.02e+195)
               (/
                (+
                 (/ 1.0 n)
                 (/
                  (/ (+ (* n -0.5) (* n (/ 0.3333333333333333 x))) (* n n))
                  x))
                x)
               0.0))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 3.5e-273) {
		tmp = t_0;
	} else if (x <= 2.8e-217) {
		tmp = t_1;
	} else if (x <= 3.1e-189) {
		tmp = t_0;
	} else if (x <= 2.5e-169) {
		tmp = t_1;
	} else if (x <= 9.2e-32) {
		tmp = t_0;
	} else if (x <= 1.02e+195) {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	} else {
		tmp = 0.0;
	}
	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) / -n
    t_1 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 3.5d-273) then
        tmp = t_0
    else if (x <= 2.8d-217) then
        tmp = t_1
    else if (x <= 3.1d-189) then
        tmp = t_0
    else if (x <= 2.5d-169) then
        tmp = t_1
    else if (x <= 9.2d-32) then
        tmp = t_0
    else if (x <= 1.02d+195) then
        tmp = ((1.0d0 / n) + ((((n * (-0.5d0)) + (n * (0.3333333333333333d0 / x))) / (n * n)) / x)) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 3.5e-273) {
		tmp = t_0;
	} else if (x <= 2.8e-217) {
		tmp = t_1;
	} else if (x <= 3.1e-189) {
		tmp = t_0;
	} else if (x <= 2.5e-169) {
		tmp = t_1;
	} else if (x <= 9.2e-32) {
		tmp = t_0;
	} else if (x <= 1.02e+195) {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 3.5e-273:
		tmp = t_0
	elif x <= 2.8e-217:
		tmp = t_1
	elif x <= 3.1e-189:
		tmp = t_0
	elif x <= 2.5e-169:
		tmp = t_1
	elif x <= 9.2e-32:
		tmp = t_0
	elif x <= 1.02e+195:
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 3.5e-273)
		tmp = t_0;
	elseif (x <= 2.8e-217)
		tmp = t_1;
	elseif (x <= 3.1e-189)
		tmp = t_0;
	elseif (x <= 2.5e-169)
		tmp = t_1;
	elseif (x <= 9.2e-32)
		tmp = t_0;
	elseif (x <= 1.02e+195)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(n * -0.5) + Float64(n * Float64(0.3333333333333333 / x))) / Float64(n * n)) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 3.5e-273)
		tmp = t_0;
	elseif (x <= 2.8e-217)
		tmp = t_1;
	elseif (x <= 3.1e-189)
		tmp = t_0;
	elseif (x <= 2.5e-169)
		tmp = t_1;
	elseif (x <= 9.2e-32)
		tmp = t_0;
	elseif (x <= 1.02e+195)
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.5e-273], t$95$0, If[LessEqual[x, 2.8e-217], t$95$1, If[LessEqual[x, 3.1e-189], t$95$0, If[LessEqual[x, 2.5e-169], t$95$1, If[LessEqual[x, 9.2e-32], t$95$0, If[LessEqual[x, 1.02e+195], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(n * -0.5), $MachinePrecision] + N[(n * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 3.49999999999999992e-273 or 2.8e-217 < x < 3.1e-189 or 2.5000000000000001e-169 < x < 9.2000000000000002e-32

   1. Initial program 31.4%

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

   3. Taylor expanded in n around inf 62.3%

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

      1. log1p-define 62.3%

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

   5. Simplified 62.3%

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

   6. Taylor expanded in x around 0 62.3%

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

      1. neg-mul-1 62.3%

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

   8. Simplified 62.3%

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

   if 3.49999999999999992e-273 < x < 2.8e-217 or 3.1e-189 < x < 2.5000000000000001e-169

   1. Initial program 65.5%

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

   3. Taylor expanded in x around 0 65.5%

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

      1. *-rgt-identity 65.5%

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

      2. associate-*l/ 65.5%

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

      3. associate-/l* 65.5%

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

      4. exp-to-pow 65.5%

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

   5. Simplified 65.5%

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

   if 9.2000000000000002e-32 < x < 1.02e195

   1. Initial program 50.5%

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

   3. Taylor expanded in n around inf 50.7%

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

      1. log1p-define 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in x around -inf 68.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 68.2%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

      2. mul-1-neg 68.2%

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

      3. associate-*r/ 68.2%

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

      4. metadata-eval 68.2%

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

      5. *-commutative 68.2%

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

      6. associate-*r/ 68.2%

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

      7. metadata-eval 68.2%

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

   8. Simplified 68.2%

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

      1. associate-/r* 68.2%

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

      2. frac-2neg 68.2%

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

      3. frac-sub 70.5%

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

      4. metadata-eval 70.5%

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

   10. Applied egg-rr 70.5%

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

   if 1.02e195 < x

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0 59.0%

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

      1. *-rgt-identity 59.0%

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

      2. associate-*l/ 59.0%

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

      3. associate-/l* 59.0%

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

      4. exp-to-pow 59.0%

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

   5. Simplified 59.0%

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

      1. add-sqr-sqrt 59.0%

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

      2. unpow-prod-down 59.0%

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

   7. Applied egg-rr 59.0%

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

      1. pow-sqr 59.0%

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

      2. associate-*r/ 59.0%

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

      3. metadata-eval 59.0%

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

   9. Simplified 59.0%

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

   10. Taylor expanded in n around inf 88.5%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-273}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-217}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-189}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-169}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+195}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ t_1 := \frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \mathbf{if}\;x \leq 2.4 \cdot 10^{-228}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-217}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+195}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n)))
        (t_1 (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* n x))))
   (if (<= x 2.4e-228)
     t_0
     (if (<= x 2.8e-217)
       t_1
       (if (<= x 4.2e-189)
         t_0
         (if (<= x 1.3e-169)
           t_1
           (if (<= x 9.2e-32)
             t_0
             (if (<= x 1.5e+195)
               (/
                (+
                 (/ 1.0 n)
                 (/
                  (/ (+ (* n -0.5) (* n (/ 0.3333333333333333 x))) (* n n))
                  x))
                x)
               0.0))))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	double tmp;
	if (x <= 2.4e-228) {
		tmp = t_0;
	} else if (x <= 2.8e-217) {
		tmp = t_1;
	} else if (x <= 4.2e-189) {
		tmp = t_0;
	} else if (x <= 1.3e-169) {
		tmp = t_1;
	} else if (x <= 9.2e-32) {
		tmp = t_0;
	} else if (x <= 1.5e+195) {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	} else {
		tmp = 0.0;
	}
	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) / -n
    t_1 = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (n * x)
    if (x <= 2.4d-228) then
        tmp = t_0
    else if (x <= 2.8d-217) then
        tmp = t_1
    else if (x <= 4.2d-189) then
        tmp = t_0
    else if (x <= 1.3d-169) then
        tmp = t_1
    else if (x <= 9.2d-32) then
        tmp = t_0
    else if (x <= 1.5d+195) then
        tmp = ((1.0d0 / n) + ((((n * (-0.5d0)) + (n * (0.3333333333333333d0 / x))) / (n * n)) / x)) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	double tmp;
	if (x <= 2.4e-228) {
		tmp = t_0;
	} else if (x <= 2.8e-217) {
		tmp = t_1;
	} else if (x <= 4.2e-189) {
		tmp = t_0;
	} else if (x <= 1.3e-169) {
		tmp = t_1;
	} else if (x <= 9.2e-32) {
		tmp = t_0;
	} else if (x <= 1.5e+195) {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x)
	tmp = 0
	if x <= 2.4e-228:
		tmp = t_0
	elif x <= 2.8e-217:
		tmp = t_1
	elif x <= 4.2e-189:
		tmp = t_0
	elif x <= 1.3e-169:
		tmp = t_1
	elif x <= 9.2e-32:
		tmp = t_0
	elif x <= 1.5e+195:
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	t_1 = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(n * x))
	tmp = 0.0
	if (x <= 2.4e-228)
		tmp = t_0;
	elseif (x <= 2.8e-217)
		tmp = t_1;
	elseif (x <= 4.2e-189)
		tmp = t_0;
	elseif (x <= 1.3e-169)
		tmp = t_1;
	elseif (x <= 9.2e-32)
		tmp = t_0;
	elseif (x <= 1.5e+195)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(n * -0.5) + Float64(n * Float64(0.3333333333333333 / x))) / Float64(n * n)) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	t_1 = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	tmp = 0.0;
	if (x <= 2.4e-228)
		tmp = t_0;
	elseif (x <= 2.8e-217)
		tmp = t_1;
	elseif (x <= 4.2e-189)
		tmp = t_0;
	elseif (x <= 1.3e-169)
		tmp = t_1;
	elseif (x <= 9.2e-32)
		tmp = t_0;
	elseif (x <= 1.5e+195)
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.4e-228], t$95$0, If[LessEqual[x, 2.8e-217], t$95$1, If[LessEqual[x, 4.2e-189], t$95$0, If[LessEqual[x, 1.3e-169], t$95$1, If[LessEqual[x, 9.2e-32], t$95$0, If[LessEqual[x, 1.5e+195], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(n * -0.5), $MachinePrecision] + N[(n * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.40000000000000002e-228 or 2.8e-217 < x < 4.20000000000000033e-189 or 1.30000000000000007e-169 < x < 9.2000000000000002e-32

   1. Initial program 34.1%

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

   3. Taylor expanded in n around inf 58.4%

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

      1. log1p-define 58.4%

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

   5. Simplified 58.4%

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

   6. Taylor expanded in x around 0 58.4%

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

      1. neg-mul-1 58.4%

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

   8. Simplified 58.4%

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

   if 2.40000000000000002e-228 < x < 2.8e-217 or 4.20000000000000033e-189 < x < 1.30000000000000007e-169

   1. Initial program 84.7%

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

   3. Taylor expanded in n around inf 10.3%

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

      1. log1p-define 10.3%

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

   5. Simplified 10.3%

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

   6. Taylor expanded in x around -inf 74.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   7. Taylor expanded in n around 0 74.5%

      \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{n \cdot x}} \]
   8. Step-by-step derivation

      1. sub-neg 74.5%

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

      2. associate-*r/ 74.5%

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

      3. sub-neg 74.5%

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

      4. metadata-eval 74.5%

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

      5. distribute-lft-in 74.5%

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

      6. neg-mul-1 74.5%

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

      7. associate-*r/ 74.5%

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

      8. metadata-eval 74.5%

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

      9. distribute-neg-frac 74.5%

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

      10. metadata-eval 74.5%

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

      11. metadata-eval 74.5%

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

      12. metadata-eval 74.5%

          \[\leadsto -1 \cdot \frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + \color{blue}{-1}}{n \cdot x} \]

      13. *-commutative 74.5%

          \[\leadsto -1 \cdot \frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + -1}{\color{blue}{x \cdot n}} \]

   9. Simplified 74.5%

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + -1}{x \cdot n}} \]

   if 9.2000000000000002e-32 < x < 1.5e195

   1. Initial program 50.5%

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

   3. Taylor expanded in n around inf 50.7%

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

      1. log1p-define 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in x around -inf 68.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 68.2%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

      2. mul-1-neg 68.2%

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

      3. associate-*r/ 68.2%

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

      4. metadata-eval 68.2%

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

      5. *-commutative 68.2%

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

      6. associate-*r/ 68.2%

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

      7. metadata-eval 68.2%

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

   8. Simplified 68.2%

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

      1. associate-/r* 68.2%

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

      2. frac-2neg 68.2%

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

      3. frac-sub 70.5%

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

      4. metadata-eval 70.5%

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

   10. Applied egg-rr 70.5%

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

   if 1.5e195 < x

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0 59.0%

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

      1. *-rgt-identity 59.0%

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

      2. associate-*l/ 59.0%

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

      3. associate-/l* 59.0%

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

      4. exp-to-pow 59.0%

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

   5. Simplified 59.0%

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

      1. add-sqr-sqrt 59.0%

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

      2. unpow-prod-down 59.0%

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

   7. Applied egg-rr 59.0%

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

      1. pow-sqr 59.0%

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

      2. associate-*r/ 59.0%

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

      3. metadata-eval 59.0%

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

   9. Simplified 59.0%

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

   10. Taylor expanded in n around inf 88.5%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-228}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-217}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-189}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-169}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+195}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1:\\ \;\;\;\;\frac{--0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5000:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+141}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1.0)
   (/ (- -0.3333333333333333) (* n (pow x 3.0)))
   (if (<= (/ 1.0 n) 5000.0)
     (/ (log (/ (+ x 1.0) x)) n)
     (if (<= (/ 1.0 n) 1e+141)
       (- 1.0 (pow x (/ 1.0 n)))
       (/
        (+
         (/ 1.0 n)
         (/ (/ (+ (* n -0.5) (* n (/ 0.3333333333333333 x))) (* n n)) x))
        x)))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.0) {
		tmp = -(-0.3333333333333333) / (n * pow(x, 3.0));
	} else if ((1.0 / n) <= 5000.0) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+141) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1.0d0)) then
        tmp = -(-0.3333333333333333d0) / (n * (x ** 3.0d0))
    else if ((1.0d0 / n) <= 5000.0d0) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 1d+141) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = ((1.0d0 / n) + ((((n * (-0.5d0)) + (n * (0.3333333333333333d0 / x))) / (n * n)) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1.0) {
		tmp = -(-0.3333333333333333) / (n * Math.pow(x, 3.0));
	} else if ((1.0 / n) <= 5000.0) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 1e+141) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1.0:
		tmp = -(-0.3333333333333333) / (n * math.pow(x, 3.0))
	elif (1.0 / n) <= 5000.0:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 1e+141:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1.0)
		tmp = Float64(Float64(-(-0.3333333333333333)) / Float64(n * (x ^ 3.0)));
	elseif (Float64(1.0 / n) <= 5000.0)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+141)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(n * -0.5) + Float64(n * Float64(0.3333333333333333 / x))) / Float64(n * n)) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1.0)
		tmp = -(-0.3333333333333333) / (n * (x ^ 3.0));
	elseif ((1.0 / n) <= 5000.0)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 1e+141)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1.0], N[((--0.3333333333333333) / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5000.0], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+141], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(n * -0.5), $MachinePrecision] + N[(n * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1

   1. Initial program 99.8%

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

   3. Taylor expanded in n around inf 47.9%

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

      1. log1p-define 47.9%

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

   5. Simplified 47.9%

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

   6. Taylor expanded in x around -inf 50.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 50.5%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

      2. mul-1-neg 50.5%

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

      3. associate-*r/ 50.5%

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

      4. metadata-eval 50.5%

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

      5. *-commutative 50.5%

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

      6. associate-*r/ 50.5%

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

      7. metadata-eval 50.5%

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

   8. Simplified 50.5%

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

   9. Taylor expanded in x around 0 82.2%

      \[\leadsto -\color{blue}{\frac{-0.3333333333333333}{n \cdot {x}^{3}}} \]

   if -1 < (/.f64 #s(literal 1 binary64) n) < 5e3

   1. Initial program 33.2%

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

   3. Taylor expanded in n around inf 74.4%

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

      1. log1p-define 74.4%

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

   5. Simplified 74.4%

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

      1. log1p-undefine 74.4%

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

      2. diff-log 74.5%

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

      3. +-commutative 74.5%

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

   7. Applied egg-rr 74.5%

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

   if 5e3 < (/.f64 #s(literal 1 binary64) n) < 1.00000000000000002e141

   1. Initial program 76.6%

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

   3. Taylor expanded in x around 0 71.2%

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

      1. *-rgt-identity 71.2%

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

      2. associate-*l/ 71.2%

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

      3. associate-/l* 71.2%

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

      4. exp-to-pow 71.2%

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

   5. Simplified 71.2%

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

   if 1.00000000000000002e141 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 16.3%

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

   3. Taylor expanded in n around inf 6.5%

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

      1. log1p-define 6.5%

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

   5. Simplified 6.5%

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

   6. Taylor expanded in x around -inf 82.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 82.5%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

      2. mul-1-neg 82.5%

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

      3. associate-*r/ 82.5%

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

      4. metadata-eval 82.5%

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

      5. *-commutative 82.5%

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

      6. associate-*r/ 82.5%

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

      7. metadata-eval 82.5%

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

   8. Simplified 82.5%

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

      1. associate-/r* 82.5%

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

      2. frac-2neg 82.5%

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

      3. frac-sub 86.8%

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

      4. metadata-eval 86.8%

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

   10. Applied egg-rr 86.8%

       \[\leadsto -\frac{\left(-\frac{\color{blue}{\frac{\frac{0.3333333333333333}{x} \cdot \left(-n\right) - n \cdot -0.5}{n \cdot \left(-n\right)}}}{x}\right) - \frac{1}{n}}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1:\\ \;\;\;\;\frac{--0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5000:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+141}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+150}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{0.5}{n \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 5.5e+123)
   (/ (- 1.0 (/ (+ 0.5 (/ -0.3333333333333333 x)) x)) (* n x))
   (if (<= x 4.1e+150)
     0.0
     (if (<= x 5.6e+194) (/ (- (/ 1.0 n) (/ 0.5 (* n x))) x) 0.0))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5.5e+123) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	} else if (x <= 4.1e+150) {
		tmp = 0.0;
	} else if (x <= 5.6e+194) {
		tmp = ((1.0 / n) - (0.5 / (n * x))) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 5.5d+123) then
        tmp = (1.0d0 - ((0.5d0 + ((-0.3333333333333333d0) / x)) / x)) / (n * x)
    else if (x <= 4.1d+150) then
        tmp = 0.0d0
    else if (x <= 5.6d+194) then
        tmp = ((1.0d0 / n) - (0.5d0 / (n * x))) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 5.5e+123) {
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	} else if (x <= 4.1e+150) {
		tmp = 0.0;
	} else if (x <= 5.6e+194) {
		tmp = ((1.0 / n) - (0.5 / (n * x))) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 5.5e+123:
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x)
	elif x <= 4.1e+150:
		tmp = 0.0
	elif x <= 5.6e+194:
		tmp = ((1.0 / n) - (0.5 / (n * x))) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 5.5e+123)
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x)) / Float64(n * x));
	elseif (x <= 4.1e+150)
		tmp = 0.0;
	elseif (x <= 5.6e+194)
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(0.5 / Float64(n * x))) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 5.5e+123)
		tmp = (1.0 - ((0.5 + (-0.3333333333333333 / x)) / x)) / (n * x);
	elseif (x <= 4.1e+150)
		tmp = 0.0;
	elseif (x <= 5.6e+194)
		tmp = ((1.0 / n) - (0.5 / (n * x))) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 5.5e+123], N[(N[(1.0 - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.1e+150], 0.0, If[LessEqual[x, 5.6e+194], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(0.5 / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.5000000000000002e123

   1. Initial program 41.0%

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

   3. Taylor expanded in n around inf 48.9%

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

      1. log1p-define 48.9%

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

   5. Simplified 48.9%

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

   6. Taylor expanded in x around -inf 44.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   7. Taylor expanded in n around 0 44.8%

      \[\leadsto -1 \cdot \color{blue}{\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{n \cdot x}} \]
   8. Step-by-step derivation

      1. sub-neg 44.8%

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

      2. associate-*r/ 44.8%

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

      3. sub-neg 44.8%

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

      4. metadata-eval 44.8%

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

      5. distribute-lft-in 44.8%

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

      6. neg-mul-1 44.8%

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

      7. associate-*r/ 44.8%

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

      8. metadata-eval 44.8%

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

      9. distribute-neg-frac 44.8%

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

      10. metadata-eval 44.8%

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

      11. metadata-eval 44.8%

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

      12. metadata-eval 44.8%

          \[\leadsto -1 \cdot \frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + \color{blue}{-1}}{n \cdot x} \]

      13. *-commutative 44.8%

          \[\leadsto -1 \cdot \frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + -1}{\color{blue}{x \cdot n}} \]

   9. Simplified 44.8%

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{\frac{-0.3333333333333333}{x} + 0.5}{x} + -1}{x \cdot n}} \]

   if 5.5000000000000002e123 < x < 4.09999999999999994e150 or 5.60000000000000021e194 < x

   1. Initial program 83.9%

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

   3. Taylor expanded in x around 0 49.4%

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

      1. *-rgt-identity 49.4%

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

      2. associate-*l/ 49.4%

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

      3. associate-/l* 49.4%

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

      4. exp-to-pow 49.4%

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

   5. Simplified 49.4%

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

      1. add-sqr-sqrt 49.4%

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

      2. unpow-prod-down 49.4%

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

   7. Applied egg-rr 49.4%

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

      1. pow-sqr 49.4%

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

      2. associate-*r/ 49.4%

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

      3. metadata-eval 49.4%

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

   9. Simplified 49.4%

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

   10. Taylor expanded in n around inf 83.9%

       \[\leadsto 1 - \color{blue}{1} \]

   if 4.09999999999999994e150 < x < 5.60000000000000021e194

   1. Initial program 59.4%

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

   3. Taylor expanded in n around inf 59.4%

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

      1. log1p-define 59.4%

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

   5. Simplified 59.4%

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

   6. Taylor expanded in x around inf 92.9%

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

      1. associate-*r/ 92.9%

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

      2. metadata-eval 92.9%

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

      3. *-commutative 92.9%

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

   8. Simplified 92.9%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{1 - \frac{0.5 + \frac{-0.3333333333333333}{x}}{x}}{n \cdot x}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+150}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{0.5}{n \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+195}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.5e+195)
   (/
    (+
     (/ 1.0 n)
     (/ (/ (+ (* n -0.5) (* n (/ 0.3333333333333333 x))) (* n n)) x))
    x)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.5e+195) {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.5d+195) then
        tmp = ((1.0d0 / n) + ((((n * (-0.5d0)) + (n * (0.3333333333333333d0 / x))) / (n * n)) / x)) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.5e+195) {
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.5e+195:
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.5e+195)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(Float64(n * -0.5) + Float64(n * Float64(0.3333333333333333 / x))) / Float64(n * n)) / x)) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.5e+195)
		tmp = ((1.0 / n) + ((((n * -0.5) + (n * (0.3333333333333333 / x))) / (n * n)) / x)) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.5e+195], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(N[(n * -0.5), $MachinePrecision] + N[(n * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e195

   1. Initial program 44.6%

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

   3. Taylor expanded in n around inf 51.3%

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

      1. log1p-define 51.3%

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

   5. Simplified 51.3%

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

   6. Taylor expanded in x around -inf 49.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 49.2%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

      2. mul-1-neg 49.2%

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

      3. associate-*r/ 49.2%

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

      4. metadata-eval 49.2%

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

      5. *-commutative 49.2%

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

      6. associate-*r/ 49.2%

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

      7. metadata-eval 49.2%

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

   8. Simplified 49.2%

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

      1. associate-/r* 49.2%

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

      2. frac-2neg 49.2%

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

      3. frac-sub 49.8%

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

      4. metadata-eval 49.8%

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

   10. Applied egg-rr 49.8%

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

   if 1.5e195 < x

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0 59.0%

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

      1. *-rgt-identity 59.0%

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

      2. associate-*l/ 59.0%

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

      3. associate-/l* 59.0%

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

      4. exp-to-pow 59.0%

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

   5. Simplified 59.0%

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

      1. add-sqr-sqrt 59.0%

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

      2. unpow-prod-down 59.0%

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

   7. Applied egg-rr 59.0%

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

      1. pow-sqr 59.0%

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

      2. associate-*r/ 59.0%

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

      3. metadata-eval 59.0%

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

   9. Simplified 59.0%

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

   10. Taylor expanded in n around inf 88.5%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+195}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{n \cdot -0.5 + n \cdot \frac{0.3333333333333333}{x}}{n \cdot n}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.2% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.06 \cdot 10^{-32} \lor \neg \left(n \leq -2.2 \cdot 10^{-153} \lor \neg \left(n \leq -1.08 \cdot 10^{-219}\right) \land n \leq -1.1 \cdot 10^{-263}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -1.06e-32)
         (not
          (or (<= n -2.2e-153)
              (and (not (<= n -1.08e-219)) (<= n -1.1e-263)))))
   (/ 1.0 (* n x))
   0.0))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -1.06e-32) || !((n <= -2.2e-153) || (!(n <= -1.08e-219) && (n <= -1.1e-263)))) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-1.06d-32)) .or. (.not. (n <= (-2.2d-153)) .or. (.not. (n <= (-1.08d-219))) .and. (n <= (-1.1d-263)))) then
        tmp = 1.0d0 / (n * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -1.06e-32) || !((n <= -2.2e-153) || (!(n <= -1.08e-219) && (n <= -1.1e-263)))) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -1.06e-32) or not ((n <= -2.2e-153) or (not (n <= -1.08e-219) and (n <= -1.1e-263))):
		tmp = 1.0 / (n * x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -1.06e-32) || !((n <= -2.2e-153) || (!(n <= -1.08e-219) && (n <= -1.1e-263))))
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -1.06e-32) || ~(((n <= -2.2e-153) || (~((n <= -1.08e-219)) && (n <= -1.1e-263)))))
		tmp = 1.0 / (n * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -1.06e-32], N[Not[Or[LessEqual[n, -2.2e-153], And[N[Not[LessEqual[n, -1.08e-219]], $MachinePrecision], LessEqual[n, -1.1e-263]]]], $MachinePrecision]], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.05999999999999994e-32 or -2.20000000000000001e-153 < n < -1.08e-219 or -1.1e-263 < n

   1. Initial program 42.5%

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

   3. Taylor expanded in n around inf 56.3%

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

      1. log1p-define 56.3%

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

   5. Simplified 56.3%

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

   6. Taylor expanded in x around inf 48.5%

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

      1. *-commutative 48.5%

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

   8. Simplified 48.5%

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

   if -1.05999999999999994e-32 < n < -2.20000000000000001e-153 or -1.08e-219 < n < -1.1e-263

   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 x around 0 39.5%

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

      1. *-rgt-identity 39.5%

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

      2. associate-*l/ 39.5%

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

      3. associate-/l* 39.5%

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

      4. exp-to-pow 39.5%

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

   5. Simplified 39.5%

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

      1. add-sqr-sqrt 39.5%

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

      2. unpow-prod-down 39.5%

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

   7. Applied egg-rr 39.5%

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

      1. pow-sqr 39.5%

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

      2. associate-*r/ 39.5%

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

      3. metadata-eval 39.5%

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

   9. Simplified 39.5%

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

   10. Taylor expanded in n around inf 63.1%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.06 \cdot 10^{-32} \lor \neg \left(n \leq -2.2 \cdot 10^{-153} \lor \neg \left(n \leq -1.08 \cdot 10^{-219}\right) \land n \leq -1.1 \cdot 10^{-263}\right):\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.8% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.3 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{0.5 + \frac{-0.3333333333333333}{x}}{n \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 5.3e+194)
   (/ (- (/ 1.0 n) (/ (+ 0.5 (/ -0.3333333333333333 x)) (* n x))) x)
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 5.3e+194) {
		tmp = ((1.0 / n) - ((0.5 + (-0.3333333333333333 / x)) / (n * x))) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 5.3d+194) then
        tmp = ((1.0d0 / n) - ((0.5d0 + ((-0.3333333333333333d0) / x)) / (n * x))) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 5.3e+194) {
		tmp = ((1.0 / n) - ((0.5 + (-0.3333333333333333 / x)) / (n * x))) / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 5.3e+194:
		tmp = ((1.0 / n) - ((0.5 + (-0.3333333333333333 / x)) / (n * x))) / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 5.3e+194)
		tmp = Float64(Float64(Float64(1.0 / n) - Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / Float64(n * x))) / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 5.3e+194)
		tmp = ((1.0 / n) - ((0.5 + (-0.3333333333333333 / x)) / (n * x))) / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 5.3e+194], N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / N[(n * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 5.30000000000000005e194

   1. Initial program 44.6%

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

   3. Taylor expanded in n around inf 51.3%

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

      1. log1p-define 51.3%

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

   5. Simplified 51.3%

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

   6. Taylor expanded in x around -inf 49.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 49.2%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

      2. mul-1-neg 49.2%

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

      3. associate-*r/ 49.2%

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

      4. metadata-eval 49.2%

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

      5. *-commutative 49.2%

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

      6. associate-*r/ 49.2%

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

      7. metadata-eval 49.2%

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

   8. Simplified 49.2%

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

   9. Taylor expanded in x around inf 37.1%

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

   11. Simplified 49.2%

       \[\leadsto -\color{blue}{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x \cdot n} + \frac{-1}{n}}{x}} \]

   if 5.30000000000000005e194 < x

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0 59.0%

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

      1. *-rgt-identity 59.0%

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

      2. associate-*l/ 59.0%

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

      3. associate-/l* 59.0%

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

      4. exp-to-pow 59.0%

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

   5. Simplified 59.0%

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

      1. add-sqr-sqrt 59.0%

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

      2. unpow-prod-down 59.0%

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

   7. Applied egg-rr 59.0%

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

      1. pow-sqr 59.0%

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

      2. associate-*r/ 59.0%

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

      3. metadata-eval 59.0%

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

   9. Simplified 59.0%

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

   10. Taylor expanded in n around inf 88.5%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.3 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{1}{n} - \frac{0.5 + \frac{-0.3333333333333333}{x}}{n \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.6% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+44}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -1e+44) 0.0 (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+44) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-1d+44)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -1e+44) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -1e+44:
		tmp = 0.0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -1e+44)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -1e+44)
		tmp = 0.0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+44], 0.0, N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.0000000000000001e44

   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 x around 0 51.6%

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

      1. *-rgt-identity 51.6%

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

      2. associate-*l/ 51.6%

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

      3. associate-/l* 51.6%

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

      4. exp-to-pow 51.6%

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

   5. Simplified 51.6%

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

      1. add-sqr-sqrt 51.6%

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

      2. unpow-prod-down 51.6%

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

   7. Applied egg-rr 51.6%

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

      1. pow-sqr 51.6%

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

      2. associate-*r/ 51.6%

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

      3. metadata-eval 51.6%

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

   9. Simplified 51.6%

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

   10. Taylor expanded in n around inf 50.8%

       \[\leadsto 1 - \color{blue}{1} \]

   if -1.0000000000000001e44 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 37.2%

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

   3. Taylor expanded in n around inf 58.5%

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

      1. log1p-define 58.5%

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

   5. Simplified 58.5%

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

   6. Taylor expanded in x around inf 46.7%

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

      1. *-commutative 46.7%

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

   8. Simplified 46.7%

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

   9. Taylor expanded in x around 0 46.7%

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

       1. associate-/r* 47.3%

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

   11. Simplified 47.3%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1 \cdot 10^{+44}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 31.0% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 51.5%

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

3. Taylor expanded in x around 0 40.0%

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

   1. *-rgt-identity 40.0%

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

   2. associate-*l/ 40.0%

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

   3. associate-/l* 40.0%

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

   4. exp-to-pow 40.0%

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

5. Simplified 40.0%

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

   1. add-sqr-sqrt 40.0%

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

   2. unpow-prod-down 40.0%

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

7. Applied egg-rr 40.0%

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

   1. pow-sqr 40.0%

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

   2. associate-*r/ 40.0%

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

   3. metadata-eval 40.0%

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

9. Simplified 40.0%

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

10. Taylor expanded in n around inf 30.7%

    \[\leadsto 1 - \color{blue}{1} \]

11. Final simplification 30.7%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024091 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))