2nthrt (problem 3.4.6)

Percentage Accurate: 55.1% → 88.2%

Time: 50.3s

Alternatives: 21

Speedup: 1.8×

• 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: 55.1% 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: 88.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 6.2 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \frac{-0.5 + \frac{0.5}{n}}{n}, \frac{1}{n}\right) - \mathsf{expm1}\left(t\_0\right)\\ \mathbf{elif}\;x \leq 32000000:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) - \frac{0.5 \cdot \left({\log x}^{2} - {\left(\mathsf{log1p}\left(x\right)\right)}^{2}\right) - 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{t\_0}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= x 6.2e-53)
     (- (* x (fma x (/ (+ -0.5 (/ 0.5 n)) n) (/ 1.0 n))) (expm1 t_0))
     (if (<= x 32000000.0)
       (/
        (-
         (-
          (log1p x)
          (/
           (-
            (* 0.5 (- (pow (log x) 2.0) (pow (log1p x) 2.0)))
            (*
             0.16666666666666666
             (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n)))
           n))
         (log x))
        n)
       (/ (exp t_0) (* x n))))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if (x <= 6.2e-53) {
		tmp = (x * fma(x, ((-0.5 + (0.5 / n)) / n), (1.0 / n))) - expm1(t_0);
	} else if (x <= 32000000.0) {
		tmp = ((log1p(x) - (((0.5 * (pow(log(x), 2.0) - pow(log1p(x), 2.0))) - (0.16666666666666666 * ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n))) / n)) - log(x)) / n;
	} else {
		tmp = exp(t_0) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (x <= 6.2e-53)
		tmp = Float64(Float64(x * fma(x, Float64(Float64(-0.5 + Float64(0.5 / n)) / n), Float64(1.0 / n))) - expm1(t_0));
	elseif (x <= 32000000.0)
		tmp = Float64(Float64(Float64(log1p(x) - Float64(Float64(Float64(0.5 * Float64((log(x) ^ 2.0) - (log1p(x) ^ 2.0))) - Float64(0.16666666666666666 * Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n))) / n)) - log(x)) / n);
	else
		tmp = Float64(exp(t_0) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 6.2e-53], N[(N[(x * N[(x * N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Exp[t$95$0] - 1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 32000000.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] - N[(N[(N[(0.5 * N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[t$95$0], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.20000000000000031e-53

   1. Initial program 44.5%

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

   3. Taylor expanded in x around 0 44.5%

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

   4. Taylor expanded in n around inf 44.7%

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

   5. Taylor expanded in x around 0 44.5%

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

   7. Simplified 87.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \frac{-0.5 + \frac{0.5}{n}}{n}, \frac{1}{n}\right) - \mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 6.20000000000000031e-53 < x < 3.2e7

   1. Initial program 39.7%

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

   3. Taylor expanded in n around -inf 92.0%

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

   5. Simplified 92.0%

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

   if 3.2e7 < x

   1. Initial program 69.6%

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

   3. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. *-commutative 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \frac{-0.5 + \frac{0.5}{n}}{n}, \frac{1}{n}\right) - \mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{elif}\;x \leq 32000000:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) - \frac{0.5 \cdot \left({\log x}^{2} - {\left(\mathsf{log1p}\left(x\right)\right)}^{2}\right) - 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{n}\\ \mathbf{if}\;x \leq 0.002:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \frac{-0.5 + \frac{0.5}{n}}{n}, \frac{1}{n}\right) - \mathsf{expm1}\left(t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{t\_0}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) n)))
   (if (<= x 0.002)
     (- (* x (fma x (/ (+ -0.5 (/ 0.5 n)) n) (/ 1.0 n))) (expm1 t_0))
     (/ (exp t_0) (* x n)))))

C

double code(double x, double n) {
	double t_0 = log(x) / n;
	double tmp;
	if (x <= 0.002) {
		tmp = (x * fma(x, ((-0.5 + (0.5 / n)) / n), (1.0 / n))) - expm1(t_0);
	} else {
		tmp = exp(t_0) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(log(x) / n)
	tmp = 0.0
	if (x <= 0.002)
		tmp = Float64(Float64(x * fma(x, Float64(Float64(-0.5 + Float64(0.5 / n)) / n), Float64(1.0 / n))) - expm1(t_0));
	else
		tmp = Float64(exp(t_0) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 0.002], N[(N[(x * N[(x * N[(N[(-0.5 + N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(1.0 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(Exp[t$95$0] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Exp[t$95$0], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-3

   1. Initial program 43.1%

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

   3. Taylor expanded in x around 0 41.5%

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

   4. Taylor expanded in n around inf 41.7%

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

   5. Taylor expanded in x around 0 41.5%

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

   7. Simplified 87.3%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \frac{-0.5 + \frac{0.5}{n}}{n}, \frac{1}{n}\right) - \mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 2e-3 < x

   1. Initial program 68.9%

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

   3. Taylor expanded in x around inf 97.1%

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

      1. mul-1-neg 97.1%

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

      2. log-rec 97.1%

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

      3. mul-1-neg 97.1%

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

      4. distribute-neg-frac 97.1%

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

      5. mul-1-neg 97.1%

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

      6. remove-double-neg 97.1%

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

      7. *-commutative 97.1%

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

   5. Simplified 97.1%

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 82.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x - \mathsf{log1p}\left(x\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\log x}{x \cdot n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} - x \cdot \left(0.5 \cdot \frac{1}{n} + 0.5 \cdot \frac{-1}{{n}^{2}}\right)\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-42)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -5e-104)
       (/ -1.0 (/ n (- (log x) (log1p x))))
       (if (<= (/ 1.0 n) -5e-145)
         (/ (+ (/ 1.0 x) (/ (log x) (* x n))) n)
         (if (<= (/ 1.0 n) 2e-13)
           (/ (log (/ (+ x 1.0) x)) n)
           (-
            (+
             1.0
             (*
              x
              (-
               (/ 1.0 n)
               (* x (+ (* 0.5 (/ 1.0 n)) (* 0.5 (/ -1.0 (pow n 2.0))))))))
            t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = -1.0 / (n / (log(x) - log1p(x)));
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 / x) + (log(x) / (x * n))) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) - (x * ((0.5 * (1.0 / n)) + (0.5 * (-1.0 / pow(n, 2.0)))))))) - 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-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = -1.0 / (n / (Math.log(x) - Math.log1p(x)));
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 / x) + (Math.log(x) / (x * n))) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) - (x * ((0.5 * (1.0 / n)) + (0.5 * (-1.0 / Math.pow(n, 2.0)))))))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-42:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -5e-104:
		tmp = -1.0 / (n / (math.log(x) - math.log1p(x)))
	elif (1.0 / n) <= -5e-145:
		tmp = ((1.0 / x) + (math.log(x) / (x * n))) / n
	elif (1.0 / n) <= 2e-13:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = (1.0 + (x * ((1.0 / n) - (x * ((0.5 * (1.0 / n)) + (0.5 * (-1.0 / math.pow(n, 2.0)))))))) - 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-42)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e-104)
		tmp = Float64(-1.0 / Float64(n / Float64(log(x) - log1p(x))));
	elseif (Float64(1.0 / n) <= -5e-145)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(log(x) / Float64(x * n))) / n);
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) - Float64(x * Float64(Float64(0.5 * Float64(1.0 / n)) + Float64(0.5 * Float64(-1.0 / (n ^ 2.0)))))))) - 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-42], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-104], N[(-1.0 / N[(n / N[(N[Log[x], $MachinePrecision] - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] - N[(x * N[(N[(0.5 * N[(1.0 / n), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000015e-42

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. associate-*r/ 98.9%

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

      3. exp-to-pow 98.9%

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

      4. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   if -4.00000000000000015e-42 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999979e-104

   1. Initial program 17.4%

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

   3. Taylor expanded in n around inf 95.7%

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

      1. log1p-define 95.7%

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

   5. Simplified 95.7%

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

      1. clear-num 96.1%

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

      2. inv-pow 96.1%

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

   7. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 96.1%

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

   9. Simplified 96.1%

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

   if -4.99999999999999979e-104 < (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145

   1. Initial program 20.9%

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

   3. Taylor expanded in x around inf 92.4%

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

      1. mul-1-neg 92.4%

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

      2. log-rec 92.4%

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

      3. mul-1-neg 92.4%

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

      4. distribute-neg-frac 92.4%

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

      5. mul-1-neg 92.4%

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

      6. remove-double-neg 92.4%

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

      7. *-commutative 92.4%

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

   5. Simplified 92.4%

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

   6. Taylor expanded in n around inf 92.4%

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

      1. *-commutative 92.4%

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

   8. Simplified 92.4%

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

   if -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-13

   1. Initial program 32.4%

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

   3. Taylor expanded in n around inf 79.2%

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

      1. log1p-define 79.2%

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

   5. Simplified 79.2%

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

      1. log1p-undefine 79.2%

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

      2. diff-log 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. +-commutative 79.2%

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

   9. Simplified 79.2%

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

   if 2.0000000000000001e-13 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 56.3%

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

   3. Taylor expanded in x around 0 88.3%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.5 \cdot \frac{1}{{n}^{2}} - 0.5 \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x - \mathsf{log1p}\left(x\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\log x}{x \cdot n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} - x \cdot \left(0.5 \cdot \frac{1}{n} + 0.5 \cdot \frac{-1}{{n}^{2}}\right)\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x - \mathsf{log1p}\left(x\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\log x}{x \cdot n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-42)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -5e-104)
       (/ -1.0 (/ n (- (log x) (log1p x))))
       (if (<= (/ 1.0 n) -5e-145)
         (/ (+ (/ 1.0 x) (/ (log x) (* x n))) n)
         (if (<= (/ 1.0 n) 2e-13)
           (/ (log (/ (+ x 1.0) x)) n)
           (if (<= (/ 1.0 n) 5e+152)
             (- (pow (+ x 1.0) (/ 1.0 n)) t_0)
             (log1p (expm1 (/ x n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = -1.0 / (n / (log(x) - log1p(x)));
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 / x) + (log(x) / (x * n))) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+152) {
		tmp = pow((x + 1.0), (1.0 / n)) - t_0;
	} else {
		tmp = log1p(expm1((x / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = -1.0 / (n / (Math.log(x) - Math.log1p(x)));
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 / x) + (Math.log(x) / (x * n))) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+152) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - t_0;
	} else {
		tmp = Math.log1p(Math.expm1((x / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-42:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -5e-104:
		tmp = -1.0 / (n / (math.log(x) - math.log1p(x)))
	elif (1.0 / n) <= -5e-145:
		tmp = ((1.0 / x) + (math.log(x) / (x * n))) / n
	elif (1.0 / n) <= 2e-13:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+152:
		tmp = math.pow((x + 1.0), (1.0 / n)) - t_0
	else:
		tmp = math.log1p(math.expm1((x / n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-42)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e-104)
		tmp = Float64(-1.0 / Float64(n / Float64(log(x) - log1p(x))));
	elseif (Float64(1.0 / n) <= -5e-145)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(log(x) / Float64(x * n))) / n);
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+152)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0);
	else
		tmp = log1p(expm1(Float64(x / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-42], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-104], N[(-1.0 / N[(n / N[(N[Log[x], $MachinePrecision] - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+152], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[N[(x / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000015e-42

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. associate-*r/ 98.9%

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

      3. exp-to-pow 98.9%

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

      4. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   if -4.00000000000000015e-42 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999979e-104

   1. Initial program 17.4%

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

   3. Taylor expanded in n around inf 95.7%

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

      1. log1p-define 95.7%

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

   5. Simplified 95.7%

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

      1. clear-num 96.1%

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

      2. inv-pow 96.1%

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

   7. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 96.1%

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

   9. Simplified 96.1%

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

   if -4.99999999999999979e-104 < (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145

   1. Initial program 20.9%

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

   3. Taylor expanded in x around inf 92.4%

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

      1. mul-1-neg 92.4%

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

      2. log-rec 92.4%

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

      3. mul-1-neg 92.4%

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

      4. distribute-neg-frac 92.4%

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

      5. mul-1-neg 92.4%

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

      6. remove-double-neg 92.4%

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

      7. *-commutative 92.4%

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

   5. Simplified 92.4%

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

   6. Taylor expanded in n around inf 92.4%

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

      1. *-commutative 92.4%

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

   8. Simplified 92.4%

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

   if -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-13

   1. Initial program 32.4%

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

   3. Taylor expanded in n around inf 79.2%

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

      1. log1p-define 79.2%

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

   5. Simplified 79.2%

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

      1. log1p-undefine 79.2%

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

      2. diff-log 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. +-commutative 79.2%

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

   9. Simplified 79.2%

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

   if 2.0000000000000001e-13 < (/.f64 #s(literal 1 binary64) n) < 5e152

   1. Initial program 90.8%

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

   if 5e152 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 15.2%

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

   3. Taylor expanded in n around inf 6.6%

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

      1. log1p-define 6.6%

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

   5. Simplified 6.6%

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

   6. Taylor expanded in x around inf 71.5%

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

      1. *-commutative 71.5%

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

   8. Simplified 71.5%

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

      1. associate-/r* 71.5%

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

      2. rem-exp-log 71.5%

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

      3. neg-log 71.5%

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

      4. add-sqr-sqrt 71.5%

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

      5. sqrt-unprod 71.5%

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

      6. sqr-neg 71.5%

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

      7. unpow2 71.5%

         \[\leadsto \frac{e^{\sqrt{\color{blue}{{\log x}^{2}}}}}{n} \]

      8. sqrt-pow1 4.3%

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

      9. metadata-eval 4.3%

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

      10. pow1 4.3%

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

      11. add-exp-log 4.3%

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

      12. log1p-expm1-u 94.1%

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

   10. Applied egg-rr 94.1%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x - \mathsf{log1p}\left(x\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\log x}{x \cdot n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x - \mathsf{log1p}\left(x\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\log x}{x \cdot n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -4e-42)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -5e-104)
       (/ -1.0 (/ n (- (log x) (log1p x))))
       (if (<= (/ 1.0 n) -5e-145)
         (/ (+ (/ 1.0 x) (/ (log x) (* x n))) n)
         (if (<= (/ 1.0 n) 2e-13)
           (/ (log (/ (+ x 1.0) x)) n)
           (if (<= (/ 1.0 n) 5e+134)
             (- (+ 1.0 (/ x n)) t_0)
             (log1p (expm1 (/ x n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = -1.0 / (n / (log(x) - log1p(x)));
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 / x) + (log(x) / (x * n))) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+134) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log1p(expm1((x / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = -1.0 / (n / (Math.log(x) - Math.log1p(x)));
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 / x) + (Math.log(x) / (x * n))) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+134) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log1p(Math.expm1((x / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -4e-42:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -5e-104:
		tmp = -1.0 / (n / (math.log(x) - math.log1p(x)))
	elif (1.0 / n) <= -5e-145:
		tmp = ((1.0 / x) + (math.log(x) / (x * n))) / n
	elif (1.0 / n) <= 2e-13:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+134:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log1p(math.expm1((x / n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-42)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e-104)
		tmp = Float64(-1.0 / Float64(n / Float64(log(x) - log1p(x))));
	elseif (Float64(1.0 / n) <= -5e-145)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(log(x) / Float64(x * n))) / n);
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+134)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = log1p(expm1(Float64(x / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-42], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-104], N[(-1.0 / N[(n / N[(N[Log[x], $MachinePrecision] - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+134], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[N[(x / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000015e-42

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. associate-*r/ 98.9%

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

      3. exp-to-pow 98.9%

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

      4. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   if -4.00000000000000015e-42 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999979e-104

   1. Initial program 17.4%

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

   3. Taylor expanded in n around inf 95.7%

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

      1. log1p-define 95.7%

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

   5. Simplified 95.7%

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

      1. clear-num 96.1%

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

      2. inv-pow 96.1%

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

   7. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{{\left(\frac{n}{\mathsf{log1p}\left(x\right) - \log x}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 96.1%

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

   9. Simplified 96.1%

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

   if -4.99999999999999979e-104 < (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145

   1. Initial program 20.9%

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

   3. Taylor expanded in x around inf 92.4%

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

      1. mul-1-neg 92.4%

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

      2. log-rec 92.4%

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

      3. mul-1-neg 92.4%

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

      4. distribute-neg-frac 92.4%

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

      5. mul-1-neg 92.4%

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

      6. remove-double-neg 92.4%

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

      7. *-commutative 92.4%

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

   5. Simplified 92.4%

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

   6. Taylor expanded in n around inf 92.4%

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

      1. *-commutative 92.4%

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

   8. Simplified 92.4%

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

   if -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-13

   1. Initial program 32.4%

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

   3. Taylor expanded in n around inf 79.2%

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

      1. log1p-define 79.2%

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

   5. Simplified 79.2%

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

      1. log1p-undefine 79.2%

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

      2. diff-log 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. +-commutative 79.2%

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

   9. Simplified 79.2%

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

   if 2.0000000000000001e-13 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999981e134

   1. Initial program 89.4%

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

   3. Taylor expanded in x around 0 91.7%

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

   if 4.99999999999999981e134 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 28.3%

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

   3. Taylor expanded in n around inf 6.3%

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

      1. log1p-define 6.3%

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

   5. Simplified 6.3%

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

   6. Taylor expanded in x around inf 60.9%

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

      1. *-commutative 60.9%

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

   8. Simplified 60.9%

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

      1. associate-/r* 60.9%

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

      2. rem-exp-log 60.9%

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

      3. neg-log 60.9%

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

      4. add-sqr-sqrt 60.9%

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

      5. sqrt-unprod 60.9%

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

      6. sqr-neg 60.9%

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

      7. unpow2 60.9%

         \[\leadsto \frac{e^{\sqrt{\color{blue}{{\log x}^{2}}}}}{n} \]

      8. sqrt-pow1 4.6%

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

      9. metadata-eval 4.6%

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

      10. pow1 4.6%

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

      11. add-exp-log 4.6%

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

      12. log1p-expm1-u 90.3%

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

   10. Applied egg-rr 90.3%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\frac{-1}{\frac{n}{\log x - \mathsf{log1p}\left(x\right)}}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\log x}{x \cdot n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\log x}{x \cdot n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -4e-42)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -5e-104)
       t_1
       (if (<= (/ 1.0 n) -5e-145)
         (/ (+ (/ 1.0 x) (/ (log x) (* x n))) n)
         (if (<= (/ 1.0 n) 2e-13)
           t_1
           (if (<= (/ 1.0 n) 5e+134)
             (- (+ 1.0 (/ x n)) t_0)
             (log1p (expm1 (/ x n))))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 / x) + (log(x) / (x * n))) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+134) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log1p(expm1((x / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 / x) + (Math.log(x) / (x * n))) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+134) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log1p(Math.expm1((x / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -4e-42:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -5e-104:
		tmp = t_1
	elif (1.0 / n) <= -5e-145:
		tmp = ((1.0 / x) + (math.log(x) / (x * n))) / n
	elif (1.0 / n) <= 2e-13:
		tmp = t_1
	elif (1.0 / n) <= 5e+134:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log1p(math.expm1((x / n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-42)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e-104)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-145)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(log(x) / Float64(x * n))) / n);
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+134)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = log1p(expm1(Float64(x / n)));
	end
	return 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[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-42], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-104], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+134], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[N[(x / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000015e-42

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. associate-*r/ 98.9%

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

      3. exp-to-pow 98.9%

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

      4. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   if -4.00000000000000015e-42 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999979e-104 or -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-13

   1. Initial program 31.5%

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

   3. Taylor expanded in n around inf 80.1%

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

      1. log1p-define 80.1%

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

   5. Simplified 80.1%

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

      1. log1p-undefine 80.1%

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

      2. diff-log 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. +-commutative 80.2%

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

   9. Simplified 80.2%

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

   if -4.99999999999999979e-104 < (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145

   1. Initial program 20.9%

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

   3. Taylor expanded in x around inf 92.4%

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

      1. mul-1-neg 92.4%

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

      2. log-rec 92.4%

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

      3. mul-1-neg 92.4%

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

      4. distribute-neg-frac 92.4%

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

      5. mul-1-neg 92.4%

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

      6. remove-double-neg 92.4%

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

      7. *-commutative 92.4%

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

   5. Simplified 92.4%

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

   6. Taylor expanded in n around inf 92.4%

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

      1. *-commutative 92.4%

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

   8. Simplified 92.4%

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

   if 2.0000000000000001e-13 < (/.f64 #s(literal 1 binary64) n) < 4.99999999999999981e134

   1. Initial program 89.4%

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

   3. Taylor expanded in x around 0 91.7%

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

   if 4.99999999999999981e134 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 28.3%

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

   3. Taylor expanded in n around inf 6.3%

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

      1. log1p-define 6.3%

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

   5. Simplified 6.3%

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

   6. Taylor expanded in x around inf 60.9%

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

      1. *-commutative 60.9%

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

   8. Simplified 60.9%

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

      1. associate-/r* 60.9%

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

      2. rem-exp-log 60.9%

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

      3. neg-log 60.9%

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

      4. add-sqr-sqrt 60.9%

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

      5. sqrt-unprod 60.9%

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

      6. sqr-neg 60.9%

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

      7. unpow2 60.9%

         \[\leadsto \frac{e^{\sqrt{\color{blue}{{\log x}^{2}}}}}{n} \]

      8. sqrt-pow1 4.6%

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

      9. metadata-eval 4.6%

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

      10. pow1 4.6%

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

      11. add-exp-log 4.6%

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

      12. log1p-expm1-u 90.3%

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

   10. Applied egg-rr 90.3%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\log x}{x \cdot n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(\frac{0.3333333333333333}{{x}^{2}} - \frac{0.5}{x}\right)}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -4e-42)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -5e-104)
       t_1
       (if (<= (/ 1.0 n) -5e-145)
         (/ (+ (/ 1.0 x) (/ (log x) (* x n))) n)
         (if (<= (/ 1.0 n) 2e-13)
           t_1
           (if (<= (/ 1.0 n) 5e+152)
             (-
              (+ 1.0 (* x (+ (/ 1.0 n) (/ (+ (* x -0.5) (* 0.5 (/ x n))) n))))
              t_0)
             (/
              (/ (+ 1.0 (- (/ 0.3333333333333333 (pow x 2.0)) (/ 0.5 x))) x)
              n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 / x) + (log(x) / (x * n))) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+152) {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	} else {
		tmp = ((1.0 + ((0.3333333333333333 / pow(x, 2.0)) - (0.5 / x))) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(((x + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= (-4d-42)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= (-5d-104)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-145)) then
        tmp = ((1.0d0 / x) + (log(x) / (x * n))) / n
    else if ((1.0d0 / n) <= 2d-13) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d+152) then
        tmp = (1.0d0 + (x * ((1.0d0 / n) + (((x * (-0.5d0)) + (0.5d0 * (x / n))) / n)))) - t_0
    else
        tmp = ((1.0d0 + ((0.3333333333333333d0 / (x ** 2.0d0)) - (0.5d0 / x))) / x) / n
    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 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 / x) + (Math.log(x) / (x * n))) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+152) {
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	} else {
		tmp = ((1.0 + ((0.3333333333333333 / Math.pow(x, 2.0)) - (0.5 / x))) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -4e-42:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -5e-104:
		tmp = t_1
	elif (1.0 / n) <= -5e-145:
		tmp = ((1.0 / x) + (math.log(x) / (x * n))) / n
	elif (1.0 / n) <= 2e-13:
		tmp = t_1
	elif (1.0 / n) <= 5e+152:
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0
	else:
		tmp = ((1.0 + ((0.3333333333333333 / math.pow(x, 2.0)) - (0.5 / x))) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-42)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e-104)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-145)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(log(x) / Float64(x * n))) / n);
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+152)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(Float64(x * -0.5) + Float64(0.5 * Float64(x / n))) / n)))) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(0.3333333333333333 / (x ^ 2.0)) - Float64(0.5 / x))) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -4e-42)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= -5e-104)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-145)
		tmp = ((1.0 / x) + (log(x) / (x * n))) / n;
	elseif ((1.0 / n) <= 2e-13)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e+152)
		tmp = (1.0 + (x * ((1.0 / n) + (((x * -0.5) + (0.5 * (x / n))) / n)))) - t_0;
	else
		tmp = ((1.0 + ((0.3333333333333333 / (x ^ 2.0)) - (0.5 / x))) / x) / n;
	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[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-42], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-104], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+152], N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(x * -0.5), $MachinePrecision] + N[(0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(0.3333333333333333 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000015e-42

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. associate-*r/ 98.9%

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

      3. exp-to-pow 98.9%

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

      4. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   if -4.00000000000000015e-42 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999979e-104 or -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-13

   1. Initial program 31.5%

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

   3. Taylor expanded in n around inf 80.1%

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

      1. log1p-define 80.1%

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

   5. Simplified 80.1%

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

      1. log1p-undefine 80.1%

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

      2. diff-log 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. +-commutative 80.2%

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

   9. Simplified 80.2%

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

   if -4.99999999999999979e-104 < (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145

   1. Initial program 20.9%

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

   3. Taylor expanded in x around inf 92.4%

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

      1. mul-1-neg 92.4%

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

      2. log-rec 92.4%

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

      3. mul-1-neg 92.4%

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

      4. distribute-neg-frac 92.4%

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

      5. mul-1-neg 92.4%

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

      6. remove-double-neg 92.4%

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

      7. *-commutative 92.4%

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

   5. Simplified 92.4%

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

   6. Taylor expanded in n around inf 92.4%

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

      1. *-commutative 92.4%

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

   8. Simplified 92.4%

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

   if 2.0000000000000001e-13 < (/.f64 #s(literal 1 binary64) n) < 5e152

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0 83.5%

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

   4. Taylor expanded in n around inf 83.5%

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

   if 5e152 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 15.2%

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

   3. Taylor expanded in n around inf 6.6%

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

      1. log1p-define 6.6%

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

   5. Simplified 6.6%

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

   6. Taylor expanded in x around inf 88.3%

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

      1. associate--l+ 88.3%

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

      2. associate-*r/ 88.3%

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

      3. metadata-eval 88.3%

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

   8. Simplified 88.3%

      \[\leadsto \frac{\color{blue}{\frac{1 + \left(\frac{0.3333333333333333}{{x}^{2}} - \frac{0.5}{x}\right)}{x}}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\log x}{x \cdot n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot -0.5 + 0.5 \cdot \frac{x}{n}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(\frac{0.3333333333333333}{{x}^{2}} - \frac{0.5}{x}\right)}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\log x}{x \cdot n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \left(\frac{0.3333333333333333}{{x}^{2}} - \frac{0.5}{x}\right)}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -4e-42)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -5e-104)
       t_1
       (if (<= (/ 1.0 n) -5e-145)
         (/ (+ (/ 1.0 x) (/ (log x) (* x n))) n)
         (if (<= (/ 1.0 n) 2e-13)
           t_1
           (if (<= (/ 1.0 n) 5e+152)
             (- (+ 1.0 (/ x n)) t_0)
             (/
              (/ (+ 1.0 (- (/ 0.3333333333333333 (pow x 2.0)) (/ 0.5 x))) x)
              n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 / x) + (log(x) / (x * n))) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+152) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((1.0 + ((0.3333333333333333 / pow(x, 2.0)) - (0.5 / x))) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(((x + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= (-4d-42)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= (-5d-104)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-145)) then
        tmp = ((1.0d0 / x) + (log(x) / (x * n))) / n
    else if ((1.0d0 / n) <= 2d-13) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d+152) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = ((1.0d0 + ((0.3333333333333333d0 / (x ** 2.0d0)) - (0.5d0 / x))) / x) / n
    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 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 / x) + (Math.log(x) / (x * n))) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+152) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((1.0 + ((0.3333333333333333 / Math.pow(x, 2.0)) - (0.5 / x))) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -4e-42:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -5e-104:
		tmp = t_1
	elif (1.0 / n) <= -5e-145:
		tmp = ((1.0 / x) + (math.log(x) / (x * n))) / n
	elif (1.0 / n) <= 2e-13:
		tmp = t_1
	elif (1.0 / n) <= 5e+152:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = ((1.0 + ((0.3333333333333333 / math.pow(x, 2.0)) - (0.5 / x))) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-42)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e-104)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-145)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(log(x) / Float64(x * n))) / n);
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+152)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(0.3333333333333333 / (x ^ 2.0)) - Float64(0.5 / x))) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -4e-42)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= -5e-104)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-145)
		tmp = ((1.0 / x) + (log(x) / (x * n))) / n;
	elseif ((1.0 / n) <= 2e-13)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e+152)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = ((1.0 + ((0.3333333333333333 / (x ^ 2.0)) - (0.5 / x))) / x) / n;
	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[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-42], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-104], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+152], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(0.3333333333333333 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000015e-42

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. associate-*r/ 98.9%

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

      3. exp-to-pow 98.9%

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

      4. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   if -4.00000000000000015e-42 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999979e-104 or -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-13

   1. Initial program 31.5%

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

   3. Taylor expanded in n around inf 80.1%

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

      1. log1p-define 80.1%

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

   5. Simplified 80.1%

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

      1. log1p-undefine 80.1%

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

      2. diff-log 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. +-commutative 80.2%

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

   9. Simplified 80.2%

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

   if -4.99999999999999979e-104 < (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145

   1. Initial program 20.9%

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

   3. Taylor expanded in x around inf 92.4%

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

      1. mul-1-neg 92.4%

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

      2. log-rec 92.4%

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

      3. mul-1-neg 92.4%

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

      4. distribute-neg-frac 92.4%

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

      5. mul-1-neg 92.4%

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

      6. remove-double-neg 92.4%

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

      7. *-commutative 92.4%

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

   5. Simplified 92.4%

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

   6. Taylor expanded in n around inf 92.4%

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

      1. *-commutative 92.4%

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

   8. Simplified 92.4%

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

   if 2.0000000000000001e-13 < (/.f64 #s(literal 1 binary64) n) < 5e152

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0 83.0%

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

   if 5e152 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 15.2%

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

   3. Taylor expanded in n around inf 6.6%

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

      1. log1p-define 6.6%

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

   5. Simplified 6.6%

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

   6. Taylor expanded in x around inf 88.3%

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

      1. associate--l+ 88.3%

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

      2. associate-*r/ 88.3%

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

      3. metadata-eval 88.3%

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

   8. Simplified 88.3%

      \[\leadsto \frac{\color{blue}{\frac{1 + \left(\frac{0.3333333333333333}{{x}^{2}} - \frac{0.5}{x}\right)}{x}}}{n} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 9: 81.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\log x}{x \cdot n}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -4e-42)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -5e-104)
       t_1
       (if (<= (/ 1.0 n) -5e-145)
         (/ (+ (/ 1.0 x) (/ (log x) (* x n))) n)
         (if (<= (/ 1.0 n) 2e-13)
           t_1
           (if (<= (/ 1.0 n) 5e+152)
             (- (+ 1.0 (/ x n)) t_0)
             (/ (/ 0.3333333333333333 n) (pow x 3.0)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 / x) + (log(x) / (x * n))) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+152) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (0.3333333333333333 / n) / pow(x, 3.0);
	}
	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 = x ** (1.0d0 / n)
    t_1 = log(((x + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= (-4d-42)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= (-5d-104)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-145)) then
        tmp = ((1.0d0 / x) + (log(x) / (x * n))) / n
    else if ((1.0d0 / n) <= 2d-13) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d+152) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (0.3333333333333333d0 / n) / (x ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 / x) + (Math.log(x) / (x * n))) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+152) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (0.3333333333333333 / n) / Math.pow(x, 3.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -4e-42:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -5e-104:
		tmp = t_1
	elif (1.0 / n) <= -5e-145:
		tmp = ((1.0 / x) + (math.log(x) / (x * n))) / n
	elif (1.0 / n) <= 2e-13:
		tmp = t_1
	elif (1.0 / n) <= 5e+152:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (0.3333333333333333 / n) / math.pow(x, 3.0)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-42)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e-104)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-145)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(log(x) / Float64(x * n))) / n);
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+152)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(0.3333333333333333 / n) / (x ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -4e-42)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= -5e-104)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-145)
		tmp = ((1.0 / x) + (log(x) / (x * n))) / n;
	elseif ((1.0 / n) <= 2e-13)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e+152)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = (0.3333333333333333 / n) / (x ^ 3.0);
	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[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-42], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-104], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+152], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(0.3333333333333333 / n), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000015e-42

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. associate-*r/ 98.9%

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

      3. exp-to-pow 98.9%

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

      4. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   if -4.00000000000000015e-42 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999979e-104 or -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-13

   1. Initial program 31.5%

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

   3. Taylor expanded in n around inf 80.1%

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

      1. log1p-define 80.1%

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

   5. Simplified 80.1%

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

      1. log1p-undefine 80.1%

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

      2. diff-log 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. +-commutative 80.2%

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

   9. Simplified 80.2%

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

   if -4.99999999999999979e-104 < (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145

   1. Initial program 20.9%

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

   3. Taylor expanded in x around inf 92.4%

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

      1. mul-1-neg 92.4%

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

      2. log-rec 92.4%

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

      3. mul-1-neg 92.4%

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

      4. distribute-neg-frac 92.4%

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

      5. mul-1-neg 92.4%

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

      6. remove-double-neg 92.4%

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

      7. *-commutative 92.4%

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

   5. Simplified 92.4%

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

   6. Taylor expanded in n around inf 92.4%

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

      1. *-commutative 92.4%

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

   8. Simplified 92.4%

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

   if 2.0000000000000001e-13 < (/.f64 #s(literal 1 binary64) n) < 5e152

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0 83.0%

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

   if 5e152 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 15.2%

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

   3. Taylor expanded in n around inf 6.6%

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

      1. log1p-define 6.6%

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

   5. Simplified 6.6%

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

   6. Taylor expanded in x around -inf 88.3%

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

   7. Taylor expanded in x around 0 88.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 88.3%

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

   9. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 10: 81.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -4e-42)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -5e-104)
       t_1
       (if (<= (/ 1.0 n) -5e-145)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 2e-13)
           t_1
           (if (<= (/ 1.0 n) 5e+152)
             (- (+ 1.0 (/ x n)) t_0)
             (/ (/ 0.3333333333333333 n) (pow x 3.0)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-145) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+152) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (0.3333333333333333 / n) / pow(x, 3.0);
	}
	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 = x ** (1.0d0 / n)
    t_1 = log(((x + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= (-4d-42)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= (-5d-104)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-145)) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 2d-13) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d+152) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (0.3333333333333333d0 / n) / (x ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-145) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+152) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (0.3333333333333333 / n) / Math.pow(x, 3.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -4e-42:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -5e-104:
		tmp = t_1
	elif (1.0 / n) <= -5e-145:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e-13:
		tmp = t_1
	elif (1.0 / n) <= 5e+152:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (0.3333333333333333 / n) / math.pow(x, 3.0)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-42)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e-104)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-145)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+152)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(0.3333333333333333 / n) / (x ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -4e-42)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= -5e-104)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-145)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 2e-13)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e+152)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = (0.3333333333333333 / n) / (x ^ 3.0);
	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[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-42], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-104], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+152], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(0.3333333333333333 / n), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000015e-42

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. associate-*r/ 98.9%

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

      3. exp-to-pow 98.9%

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

      4. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   if -4.00000000000000015e-42 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999979e-104 or -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-13

   1. Initial program 31.5%

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

   3. Taylor expanded in n around inf 80.1%

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

      1. log1p-define 80.1%

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

   5. Simplified 80.1%

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

      1. log1p-undefine 80.1%

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

      2. diff-log 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. +-commutative 80.2%

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

   9. Simplified 80.2%

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

   if -4.99999999999999979e-104 < (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145

   1. Initial program 20.9%

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

   3. Taylor expanded in n around inf 28.2%

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

      1. log1p-define 28.2%

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

   5. Simplified 28.2%

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

   6. Taylor expanded in x around inf 92.4%

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

   if 2.0000000000000001e-13 < (/.f64 #s(literal 1 binary64) n) < 5e152

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0 83.0%

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

   if 5e152 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 15.2%

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

   3. Taylor expanded in n around inf 6.6%

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

      1. log1p-define 6.6%

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

   5. Simplified 6.6%

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

   6. Taylor expanded in x around -inf 88.3%

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

   7. Taylor expanded in x around 0 88.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 88.3%

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

   9. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 11: 81.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -4e-42)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -5e-104)
       t_1
       (if (<= (/ 1.0 n) -5e-145)
         (/ (/ 1.0 x) n)
         (if (<= (/ 1.0 n) 2e-13)
           t_1
           (if (<= (/ 1.0 n) 5e+152)
             (- 1.0 t_0)
             (/ (/ 0.3333333333333333 n) (pow x 3.0)))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-145) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+152) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (0.3333333333333333 / n) / pow(x, 3.0);
	}
	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 = x ** (1.0d0 / n)
    t_1 = log(((x + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= (-4d-42)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= (-5d-104)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-145)) then
        tmp = (1.0d0 / x) / n
    else if ((1.0d0 / n) <= 2d-13) then
        tmp = t_1
    else if ((1.0d0 / n) <= 5d+152) then
        tmp = 1.0d0 - t_0
    else
        tmp = (0.3333333333333333d0 / n) / (x ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -4e-42) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -5e-104) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-145) {
		tmp = (1.0 / x) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = t_1;
	} else if ((1.0 / n) <= 5e+152) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (0.3333333333333333 / n) / Math.pow(x, 3.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -4e-42:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -5e-104:
		tmp = t_1
	elif (1.0 / n) <= -5e-145:
		tmp = (1.0 / x) / n
	elif (1.0 / n) <= 2e-13:
		tmp = t_1
	elif (1.0 / n) <= 5e+152:
		tmp = 1.0 - t_0
	else:
		tmp = (0.3333333333333333 / n) / math.pow(x, 3.0)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-42)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -5e-104)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-145)
		tmp = Float64(Float64(1.0 / x) / n);
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= 5e+152)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(0.3333333333333333 / n) / (x ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -4e-42)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= -5e-104)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-145)
		tmp = (1.0 / x) / n;
	elseif ((1.0 / n) <= 2e-13)
		tmp = t_1;
	elseif ((1.0 / n) <= 5e+152)
		tmp = 1.0 - t_0;
	else
		tmp = (0.3333333333333333 / n) / (x ^ 3.0);
	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[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-42], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-104], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+152], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(0.3333333333333333 / n), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4.00000000000000015e-42

   1. Initial program 92.7%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. *-rgt-identity 98.9%

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

      2. associate-*r/ 98.9%

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

      3. exp-to-pow 98.9%

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

      4. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   if -4.00000000000000015e-42 < (/.f64 #s(literal 1 binary64) n) < -4.99999999999999979e-104 or -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-13

   1. Initial program 31.5%

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

   3. Taylor expanded in n around inf 80.1%

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

      1. log1p-define 80.1%

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

   5. Simplified 80.1%

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

      1. log1p-undefine 80.1%

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

      2. diff-log 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. +-commutative 80.2%

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

   9. Simplified 80.2%

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

   if -4.99999999999999979e-104 < (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145

   1. Initial program 20.9%

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

   3. Taylor expanded in n around inf 28.2%

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

      1. log1p-define 28.2%

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

   5. Simplified 28.2%

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

   6. Taylor expanded in x around inf 92.4%

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

   if 2.0000000000000001e-13 < (/.f64 #s(literal 1 binary64) n) < 5e152

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0 80.6%

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

   if 5e152 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 15.2%

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

   3. Taylor expanded in n around inf 6.6%

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

      1. log1p-define 6.6%

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

   5. Simplified 6.6%

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

   6. Taylor expanded in x around -inf 88.3%

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

   7. Taylor expanded in x around 0 88.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 88.3%

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

   9. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 12: 73.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5000000:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + 0.3333333333333333 \cdot \frac{-1}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -5000000.0)
   (/ 0.3333333333333333 (* n (pow x 3.0)))
   (if (<= (/ 1.0 n) -5e-145)
     (/ (/ (- 1.0 (/ (+ 0.5 (* 0.3333333333333333 (/ -1.0 x))) x)) x) n)
     (if (<= (/ 1.0 n) 2e-13)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+152)
         (- 1.0 (pow x (/ 1.0 n)))
         (/ (/ 0.3333333333333333 n) (pow x 3.0)))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5000000.0) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 - ((0.5 + (0.3333333333333333 * (-1.0 / x))) / x)) / x) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+152) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (0.3333333333333333 / n) / pow(x, 3.0);
	}
	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) <= (-5000000.0d0)) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if ((1.0d0 / n) <= (-5d-145)) then
        tmp = ((1.0d0 - ((0.5d0 + (0.3333333333333333d0 * ((-1.0d0) / x))) / x)) / x) / n
    else if ((1.0d0 / n) <= 2d-13) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 5d+152) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (0.3333333333333333d0 / n) / (x ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -5000000.0) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if ((1.0 / n) <= -5e-145) {
		tmp = ((1.0 - ((0.5 + (0.3333333333333333 * (-1.0 / x))) / x)) / x) / n;
	} else if ((1.0 / n) <= 2e-13) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+152) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (0.3333333333333333 / n) / Math.pow(x, 3.0);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -5000000.0:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif (1.0 / n) <= -5e-145:
		tmp = ((1.0 - ((0.5 + (0.3333333333333333 * (-1.0 / x))) / x)) / x) / n
	elif (1.0 / n) <= 2e-13:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+152:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (0.3333333333333333 / n) / math.pow(x, 3.0)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -5000000.0)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (Float64(1.0 / n) <= -5e-145)
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(0.3333333333333333 * Float64(-1.0 / x))) / x)) / x) / n);
	elseif (Float64(1.0 / n) <= 2e-13)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+152)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(0.3333333333333333 / n) / (x ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -5000000.0)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif ((1.0 / n) <= -5e-145)
		tmp = ((1.0 - ((0.5 + (0.3333333333333333 * (-1.0 / x))) / x)) / x) / n;
	elseif ((1.0 / n) <= 2e-13)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 5e+152)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (0.3333333333333333 / n) / (x ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -5000000.0], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-145], N[(N[(N[(1.0 - N[(N[(0.5 + N[(0.3333333333333333 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-13], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+152], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / n), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -5e6

   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 61.8%

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

      1. log1p-define 61.8%

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

   5. Simplified 61.8%

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

   6. Taylor expanded in x around -inf 35.9%

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

   7. Taylor expanded in x around 0 76.1%

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

   if -5e6 < (/.f64 #s(literal 1 binary64) n) < -4.9999999999999998e-145

   1. Initial program 16.0%

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

   3. Taylor expanded in n around inf 43.4%

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

      1. log1p-define 43.4%

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

   5. Simplified 43.4%

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

   6. Taylor expanded in x around -inf 71.3%

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

   if -4.9999999999999998e-145 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e-13

   1. Initial program 32.4%

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

   3. Taylor expanded in n around inf 79.2%

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

      1. log1p-define 79.2%

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

   5. Simplified 79.2%

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

      1. log1p-undefine 79.2%

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

      2. diff-log 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. +-commutative 79.2%

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

   9. Simplified 79.2%

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

   if 2.0000000000000001e-13 < (/.f64 #s(literal 1 binary64) n) < 5e152

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0 80.6%

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

   if 5e152 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 15.2%

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

   3. Taylor expanded in n around inf 6.6%

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

      1. log1p-define 6.6%

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

   5. Simplified 6.6%

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

   6. Taylor expanded in x around -inf 88.3%

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

   7. Taylor expanded in x around 0 88.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 88.3%

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

   9. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5000000:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 + 0.3333333333333333 \cdot \frac{-1}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - \log x}{n}\\ \mathbf{if}\;x \leq 4.4 \cdot 10^{-186}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-46}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- x (log x)) n)))
   (if (<= x 4.4e-186)
     (/ (log x) (- n))
     (if (<= x 4.2e-146)
       (/ (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* x n)) (/ 0.5 n)) x)) x)
       (if (<= x 3.25e-72)
         t_0
         (if (<= x 4e-46)
           (- 1.0 (pow x (/ 1.0 n)))
           (if (<= x 0.9)
             t_0
             (/
              (/
               (-
                1.0
                (/ (- 0.5 (/ (+ 0.3333333333333333 (* 0.25 (/ -1.0 x))) x)) x))
               x)
              n))))))))

C

double code(double x, double n) {
	double t_0 = (x - log(x)) / n;
	double tmp;
	if (x <= 4.4e-186) {
		tmp = log(x) / -n;
	} else if (x <= 4.2e-146) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else if (x <= 3.25e-72) {
		tmp = t_0;
	} else if (x <= 4e-46) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 0.9) {
		tmp = t_0;
	} else {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (0.25 * (-1.0 / x))) / x)) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - log(x)) / n
    if (x <= 4.4d-186) then
        tmp = log(x) / -n
    else if (x <= 4.2d-146) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) - (0.5d0 / n)) / x)) / x
    else if (x <= 3.25d-72) then
        tmp = t_0
    else if (x <= 4d-46) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 0.9d0) then
        tmp = t_0
    else
        tmp = ((1.0d0 - ((0.5d0 - ((0.3333333333333333d0 + (0.25d0 * ((-1.0d0) / x))) / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (x - Math.log(x)) / n;
	double tmp;
	if (x <= 4.4e-186) {
		tmp = Math.log(x) / -n;
	} else if (x <= 4.2e-146) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else if (x <= 3.25e-72) {
		tmp = t_0;
	} else if (x <= 4e-46) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 0.9) {
		tmp = t_0;
	} else {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (0.25 * (-1.0 / x))) / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (x - math.log(x)) / n
	tmp = 0
	if x <= 4.4e-186:
		tmp = math.log(x) / -n
	elif x <= 4.2e-146:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x
	elif x <= 3.25e-72:
		tmp = t_0
	elif x <= 4e-46:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 0.9:
		tmp = t_0
	else:
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (0.25 * (-1.0 / x))) / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(x - log(x)) / n)
	tmp = 0.0
	if (x <= 4.4e-186)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 4.2e-146)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) - Float64(0.5 / n)) / x)) / x);
	elseif (x <= 3.25e-72)
		tmp = t_0;
	elseif (x <= 4e-46)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 0.9)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 + Float64(0.25 * Float64(-1.0 / x))) / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (x - log(x)) / n;
	tmp = 0.0;
	if (x <= 4.4e-186)
		tmp = log(x) / -n;
	elseif (x <= 4.2e-146)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	elseif (x <= 3.25e-72)
		tmp = t_0;
	elseif (x <= 4e-46)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 0.9)
		tmp = t_0;
	else
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (0.25 * (-1.0 / x))) / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[x, 4.4e-186], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 4.2e-146], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 3.25e-72], t$95$0, If[LessEqual[x, 4e-46], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.9], t$95$0, N[(N[(N[(1.0 - N[(N[(0.5 - N[(N[(0.3333333333333333 + N[(0.25 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 4.40000000000000026e-186

   1. Initial program 49.2%

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

   3. Taylor expanded in n around inf 54.0%

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

      1. log1p-define 54.0%

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

   5. Simplified 54.0%

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

   6. Taylor expanded in x around 0 54.0%

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

      1. neg-mul-1 54.0%

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

   8. Simplified 54.0%

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

   if 4.40000000000000026e-186 < x < 4.1999999999999998e-146

   1. Initial program 43.0%

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

   3. Taylor expanded in n around inf 36.8%

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

      1. log1p-define 36.8%

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

   5. Simplified 36.8%

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

   6. Taylor expanded in x around -inf 56.7%

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

      1. associate-*r/ 56.7%

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

      2. mul-1-neg 56.7%

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

      3. associate-*r/ 56.7%

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

      4. mul-1-neg 56.7%

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

      5. associate-*r/ 56.7%

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

      6. metadata-eval 56.7%

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

      7. *-commutative 56.7%

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

      8. associate-*r/ 56.7%

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

      9. metadata-eval 56.7%

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

   8. Simplified 56.7%

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

   if 4.1999999999999998e-146 < x < 3.2499999999999998e-72 or 4.00000000000000009e-46 < x < 0.900000000000000022

   1. Initial program 32.1%

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

   3. Taylor expanded in n around inf 65.1%

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

      1. log1p-define 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in x around 0 65.1%

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

   if 3.2499999999999998e-72 < x < 4.00000000000000009e-46

   1. Initial program 76.2%

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

   3. Taylor expanded in x around 0 76.2%

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

   if 0.900000000000000022 < x

   1. Initial program 68.2%

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

   3. Taylor expanded in n around inf 68.9%

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

      1. log1p-define 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in x around -inf 59.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{-186}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{-72}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-46}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -12.2:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 2.15 \cdot 10^{+268}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -12.2)
   (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) (* x n))
   (if (<= n 1.3e-157)
     (/ (/ 0.3333333333333333 n) (pow x 3.0))
     (if (<= n 2.4e+14)
       (- 1.0 (pow x (/ 1.0 n)))
       (if (<= n 2.15e+268) (/ (/ 1.0 x) n) (/ (log x) (- n)))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -12.2) {
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
	} else if (n <= 1.3e-157) {
		tmp = (0.3333333333333333 / n) / pow(x, 3.0);
	} else if (n <= 2.4e+14) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (n <= 2.15e+268) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = log(x) / -n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-12.2d0)) then
        tmp = (1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / (x * n)
    else if (n <= 1.3d-157) then
        tmp = (0.3333333333333333d0 / n) / (x ** 3.0d0)
    else if (n <= 2.4d+14) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (n <= 2.15d+268) then
        tmp = (1.0d0 / x) / n
    else
        tmp = log(x) / -n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -12.2) {
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
	} else if (n <= 1.3e-157) {
		tmp = (0.3333333333333333 / n) / Math.pow(x, 3.0);
	} else if (n <= 2.4e+14) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (n <= 2.15e+268) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = Math.log(x) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -12.2:
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n)
	elif n <= 1.3e-157:
		tmp = (0.3333333333333333 / n) / math.pow(x, 3.0)
	elif n <= 2.4e+14:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif n <= 2.15e+268:
		tmp = (1.0 / x) / n
	else:
		tmp = math.log(x) / -n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -12.2)
		tmp = Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / Float64(x * n));
	elseif (n <= 1.3e-157)
		tmp = Float64(Float64(0.3333333333333333 / n) / (x ^ 3.0));
	elseif (n <= 2.4e+14)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (n <= 2.15e+268)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(log(x) / Float64(-n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -12.2)
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
	elseif (n <= 1.3e-157)
		tmp = (0.3333333333333333 / n) / (x ^ 3.0);
	elseif (n <= 2.4e+14)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (n <= 2.15e+268)
		tmp = (1.0 / x) / n;
	else
		tmp = log(x) / -n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -12.2], N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.3e-157], N[(N[(0.3333333333333333 / n), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.4e+14], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.15e+268], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -12.199999999999999

   1. Initial program 30.1%

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

   3. Taylor expanded in n around inf 68.7%

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

      1. log1p-define 68.7%

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

   5. Simplified 68.7%

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

   6. Taylor expanded in x around -inf 61.1%

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

   7. Taylor expanded in n around 0 61.1%

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

   9. Simplified 61.1%

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

   if -12.199999999999999 < n < 1.29999999999999994e-157

   1. Initial program 86.4%

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

   3. Taylor expanded in n around inf 52.9%

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

      1. log1p-define 52.9%

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

   5. Simplified 52.9%

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

   6. Taylor expanded in x around -inf 44.3%

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

   7. Taylor expanded in x around 0 78.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 78.0%

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

   9. Simplified 78.0%

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

   if 1.29999999999999994e-157 < n < 2.4e14

   1. Initial program 85.3%

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

   3. Taylor expanded in x around 0 76.4%

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

   if 2.4e14 < n < 2.14999999999999985e268

   1. Initial program 25.8%

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

   3. Taylor expanded in n around inf 68.3%

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

      1. log1p-define 68.3%

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

   5. Simplified 68.3%

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

   6. Taylor expanded in x around inf 58.0%

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

   if 2.14999999999999985e268 < n

   1. Initial program 33.5%

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

   3. Taylor expanded in n around inf 99.7%

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

      1. log1p-define 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 75.3%

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

      1. neg-mul-1 75.3%

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

   8. Simplified 75.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -12.2:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;n \leq 1.3 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 2.15 \cdot 10^{+268}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-156}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{+262}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -5.0)
   (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) (* x n))
   (if (<= n 7e-156)
     (/ 0.3333333333333333 (* n (pow x 3.0)))
     (if (<= n 2.4e+14)
       (- 1.0 (pow x (/ 1.0 n)))
       (if (<= n 1.8e+262) (/ (/ 1.0 x) n) (/ (log x) (- n)))))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -5.0) {
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
	} else if (n <= 7e-156) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (n <= 2.4e+14) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (n <= 1.8e+262) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = log(x) / -n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-5.0d0)) then
        tmp = (1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / (x * n)
    else if (n <= 7d-156) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (n <= 2.4d+14) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (n <= 1.8d+262) then
        tmp = (1.0d0 / x) / n
    else
        tmp = log(x) / -n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -5.0) {
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
	} else if (n <= 7e-156) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (n <= 2.4e+14) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (n <= 1.8e+262) {
		tmp = (1.0 / x) / n;
	} else {
		tmp = Math.log(x) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -5.0:
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n)
	elif n <= 7e-156:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif n <= 2.4e+14:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif n <= 1.8e+262:
		tmp = (1.0 / x) / n
	else:
		tmp = math.log(x) / -n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -5.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / Float64(x * n));
	elseif (n <= 7e-156)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (n <= 2.4e+14)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (n <= 1.8e+262)
		tmp = Float64(Float64(1.0 / x) / n);
	else
		tmp = Float64(log(x) / Float64(-n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -5.0)
		tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
	elseif (n <= 7e-156)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (n <= 2.4e+14)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (n <= 1.8e+262)
		tmp = (1.0 / x) / n;
	else
		tmp = log(x) / -n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -5.0], N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7e-156], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.4e+14], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.8e+262], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -5

   1. Initial program 30.1%

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

   3. Taylor expanded in n around inf 68.7%

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

      1. log1p-define 68.7%

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

   5. Simplified 68.7%

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

   6. Taylor expanded in x around -inf 61.1%

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

   7. Taylor expanded in n around 0 61.1%

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

   9. Simplified 61.1%

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

   if -5 < n < 6.9999999999999999e-156

   1. Initial program 86.4%

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

   3. Taylor expanded in n around inf 52.9%

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

      1. log1p-define 52.9%

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

   5. Simplified 52.9%

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

   6. Taylor expanded in x around -inf 44.3%

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

   7. Taylor expanded in x around 0 78.0%

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

   if 6.9999999999999999e-156 < n < 2.4e14

   1. Initial program 85.3%

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

   3. Taylor expanded in x around 0 76.4%

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

   if 2.4e14 < n < 1.79999999999999996e262

   1. Initial program 25.8%

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

   3. Taylor expanded in n around inf 68.3%

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

      1. log1p-define 68.3%

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

   5. Simplified 68.3%

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

   6. Taylor expanded in x around inf 58.0%

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

   if 1.79999999999999996e262 < n

   1. Initial program 33.5%

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

   3. Taylor expanded in n around inf 99.7%

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

      1. log1p-define 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 75.3%

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

      1. neg-mul-1 75.3%

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

   8. Simplified 75.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-156}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 2.4 \cdot 10^{+14}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{+262}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{-186}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.4e-186)
   (/ (log x) (- n))
   (if (<= x 5.7e-145)
     (/ (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* x n)) (/ 0.5 n)) x)) x)
     (if (<= x 0.88)
       (/ (- x (log x)) n)
       (/
        (/
         (- 1.0 (/ (- 0.5 (/ (+ 0.3333333333333333 (* 0.25 (/ -1.0 x))) x)) x))
         x)
        n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.4e-186) {
		tmp = log(x) / -n;
	} else if (x <= 5.7e-145) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.88) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (0.25 * (-1.0 / x))) / x)) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 4.4d-186) then
        tmp = log(x) / -n
    else if (x <= 5.7d-145) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) - (0.5d0 / n)) / x)) / x
    else if (x <= 0.88d0) then
        tmp = (x - log(x)) / n
    else
        tmp = ((1.0d0 - ((0.5d0 - ((0.3333333333333333d0 + (0.25d0 * ((-1.0d0) / x))) / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.4e-186) {
		tmp = Math.log(x) / -n;
	} else if (x <= 5.7e-145) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.88) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (0.25 * (-1.0 / x))) / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.4e-186:
		tmp = math.log(x) / -n
	elif x <= 5.7e-145:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x
	elif x <= 0.88:
		tmp = (x - math.log(x)) / n
	else:
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (0.25 * (-1.0 / x))) / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.4e-186)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 5.7e-145)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) - Float64(0.5 / n)) / x)) / x);
	elseif (x <= 0.88)
		tmp = Float64(Float64(x - log(x)) / n);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 + Float64(0.25 * Float64(-1.0 / x))) / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.4e-186)
		tmp = log(x) / -n;
	elseif (x <= 5.7e-145)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	elseif (x <= 0.88)
		tmp = (x - log(x)) / n;
	else
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (0.25 * (-1.0 / x))) / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.4e-186], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 5.7e-145], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.88], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(0.5 - N[(N[(0.3333333333333333 + N[(0.25 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.40000000000000026e-186

   1. Initial program 49.2%

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

   3. Taylor expanded in n around inf 54.0%

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

      1. log1p-define 54.0%

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

   5. Simplified 54.0%

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

   6. Taylor expanded in x around 0 54.0%

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

      1. neg-mul-1 54.0%

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

   8. Simplified 54.0%

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

   if 4.40000000000000026e-186 < x < 5.70000000000000032e-145

   1. Initial program 43.0%

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

   3. Taylor expanded in n around inf 36.8%

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

      1. log1p-define 36.8%

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

   5. Simplified 36.8%

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

   6. Taylor expanded in x around -inf 56.7%

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

      1. associate-*r/ 56.7%

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

      2. mul-1-neg 56.7%

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

      3. associate-*r/ 56.7%

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

      4. mul-1-neg 56.7%

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

      5. associate-*r/ 56.7%

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

      6. metadata-eval 56.7%

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

      7. *-commutative 56.7%

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

      8. associate-*r/ 56.7%

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

      9. metadata-eval 56.7%

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

   8. Simplified 56.7%

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

   if 5.70000000000000032e-145 < x < 0.880000000000000004

   1. Initial program 40.9%

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

   3. Taylor expanded in n around inf 55.8%

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

      1. log1p-define 55.8%

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

   5. Simplified 55.8%

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

   6. Taylor expanded in x around 0 55.8%

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

   if 0.880000000000000004 < x

   1. Initial program 68.2%

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

   3. Taylor expanded in n around inf 68.9%

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

      1. log1p-define 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in x around -inf 59.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{-186}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 4 \cdot 10^{-186}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.72:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x}}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log x) (- n))))
   (if (<= x 4e-186)
     t_0
     (if (<= x 3.3e-146)
       (/ (+ (/ 1.0 n) (/ (- (/ 0.3333333333333333 (* x n)) (/ 0.5 n)) x)) x)
       (if (<= x 0.72)
         t_0
         (/
          (/
           (-
            1.0
            (/ (- 0.5 (/ (+ 0.3333333333333333 (* 0.25 (/ -1.0 x))) x)) x))
           x)
          n))))))

C

double code(double x, double n) {
	double t_0 = log(x) / -n;
	double tmp;
	if (x <= 4e-186) {
		tmp = t_0;
	} else if (x <= 3.3e-146) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.72) {
		tmp = t_0;
	} else {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (0.25 * (-1.0 / x))) / x)) / x)) / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(x) / -n
    if (x <= 4d-186) then
        tmp = t_0
    else if (x <= 3.3d-146) then
        tmp = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) - (0.5d0 / n)) / x)) / x
    else if (x <= 0.72d0) then
        tmp = t_0
    else
        tmp = ((1.0d0 - ((0.5d0 - ((0.3333333333333333d0 + (0.25d0 * ((-1.0d0) / x))) / x)) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(x) / -n;
	double tmp;
	if (x <= 4e-186) {
		tmp = t_0;
	} else if (x <= 3.3e-146) {
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	} else if (x <= 0.72) {
		tmp = t_0;
	} else {
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (0.25 * (-1.0 / x))) / x)) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(x) / -n
	tmp = 0
	if x <= 4e-186:
		tmp = t_0
	elif x <= 3.3e-146:
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x
	elif x <= 0.72:
		tmp = t_0
	else:
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (0.25 * (-1.0 / x))) / x)) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 4e-186)
		tmp = t_0;
	elseif (x <= 3.3e-146)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) - Float64(0.5 / n)) / x)) / x);
	elseif (x <= 0.72)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 + Float64(0.25 * Float64(-1.0 / x))) / x)) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(x) / -n;
	tmp = 0.0;
	if (x <= 4e-186)
		tmp = t_0;
	elseif (x <= 3.3e-146)
		tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) - (0.5 / n)) / x)) / x;
	elseif (x <= 0.72)
		tmp = t_0;
	else
		tmp = ((1.0 - ((0.5 - ((0.3333333333333333 + (0.25 * (-1.0 / x))) / x)) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 4e-186], t$95$0, If[LessEqual[x, 3.3e-146], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 0.72], t$95$0, N[(N[(N[(1.0 - N[(N[(0.5 - N[(N[(0.3333333333333333 + N[(0.25 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.9999999999999996e-186 or 3.3e-146 < x < 0.71999999999999997

   1. Initial program 44.7%

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

   3. Taylor expanded in n around inf 55.0%

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

      1. log1p-define 55.0%

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

   5. Simplified 55.0%

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

   6. Taylor expanded in x around 0 54.0%

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

      1. neg-mul-1 54.0%

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

   8. Simplified 54.0%

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

   if 3.9999999999999996e-186 < x < 3.3e-146

   1. Initial program 43.0%

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

   3. Taylor expanded in n around inf 36.8%

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

      1. log1p-define 36.8%

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

   5. Simplified 36.8%

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

   6. Taylor expanded in x around -inf 56.7%

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

      1. associate-*r/ 56.7%

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

      2. mul-1-neg 56.7%

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

      3. associate-*r/ 56.7%

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

      4. mul-1-neg 56.7%

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

      5. associate-*r/ 56.7%

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

      6. metadata-eval 56.7%

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

      7. *-commutative 56.7%

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

      8. associate-*r/ 56.7%

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

      9. metadata-eval 56.7%

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

   8. Simplified 56.7%

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

   if 0.71999999999999997 < x

   1. Initial program 68.2%

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

   3. Taylor expanded in n around inf 68.9%

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

      1. log1p-define 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in x around -inf 59.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-186}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}{x}}{x}\\ \mathbf{elif}\;x \leq 0.72:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 47.2% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x \cdot n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) (* x n)))

C

double code(double x, double n) {
	return (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = (1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / (x * n)
end function

Java

public static double code(double x, double n) {
	return (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
}

Python

def code(x, n):
	return (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n)

Julia

function code(x, n)
	return Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / Float64(x * n))
end

MATLAB

function tmp = code(x, n)
	tmp = (1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / (x * n);
end

Wolfram

code[x_, n_] := N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in n around inf 59.9%

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

   1. log1p-define 59.9%

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

5. Simplified 59.9%

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

6. Taylor expanded in x around -inf 46.8%

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

7. Taylor expanded in n around 0 46.8%

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

9. Simplified 46.8%

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

Alternative 19: 41.7% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in n around inf 59.9%

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

   1. log1p-define 59.9%

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

5. Simplified 59.9%

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

6. Taylor expanded in x around inf 39.9%

   \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
7. Add Preprocessing

Alternative 20: 41.2% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in n around inf 59.9%

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

   1. log1p-define 59.9%

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

5. Simplified 59.9%

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

6. Taylor expanded in x around inf 39.9%

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

   1. *-commutative 39.9%

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

8. Simplified 39.9%

   \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
9. Add Preprocessing

Alternative 21: 4.5% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in x around 0 28.9%

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

4. Taylor expanded in x around inf 4.4%

   \[\leadsto \color{blue}{\frac{x}{n}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))