2nthrt (problem 3.4.6)

Percentage Accurate: 53.9% → 85.9%

Time: 39.8s

Alternatives: 21

Speedup: 1.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -3.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -3.2e-79)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-16)
       (/ (log (+ 1.0 (/ 1.0 x))) n)
       (pow (sqrt (- (exp (/ (log1p x) n)) t_0)) 2.0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -3.2e-79) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-16) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else {
		tmp = pow(sqrt((exp((log1p(x) / n)) - t_0)), 2.0);
	}
	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) <= -3.2e-79) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-16) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else {
		tmp = Math.pow(Math.sqrt((Math.exp((Math.log1p(x) / n)) - t_0)), 2.0);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -3.2e-79:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-16:
		tmp = math.log((1.0 + (1.0 / x))) / n
	else:
		tmp = math.pow(math.sqrt((math.exp((math.log1p(x) / n)) - t_0)), 2.0)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -3.2e-79)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-16)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	else
		tmp = sqrt(Float64(exp(Float64(log1p(x) / n)) - t_0)) ^ 2.0;
	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], -3.2e-79], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-16], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[Power[N[Sqrt[N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.19999999999999988e-79

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf 25.3%

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

   5. Simplified 25.3%

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

   6. Taylor expanded in x around inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. log-rec 96.4%

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

      3. distribute-neg-frac 96.4%

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

      4. remove-double-neg 96.4%

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

   8. Simplified 96.4%

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

   9. Taylor expanded in x around inf 96.4%

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

       1. mul-1-neg 96.4%

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

       2. log-rec 96.4%

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

       3. distribute-frac-neg 96.4%

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

       4. remove-double-neg 96.4%

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

       5. *-rgt-identity 96.4%

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

       6. associate-*r/ 96.4%

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

       7. exp-to-pow 96.4%

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

   11. Simplified 96.4%

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

   if -3.19999999999999988e-79 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-17

   1. Initial program 31.8%

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

   3. Taylor expanded in n around inf 80.0%

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

      1. log1p-define 80.0%

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

   5. Simplified 80.0%

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

      1. expm1-log1p-u 79.4%

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

   7. Applied egg-rr 79.4%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) - \log x\right)\right)}}{n} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 80.0%

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

      2. log1p-undefine 80.0%

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

      3. diff-log 80.3%

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

   9. Applied egg-rr 80.3%

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

       1. +-commutative 80.3%

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

   11. Simplified 80.3%

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

   12. Taylor expanded in x around inf 80.3%

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

   if 9.9999999999999998e-17 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 64.4%

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

      1. add-sqr-sqrt 64.4%

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

      2. pow2 64.4%

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

      3. pow-to-exp 64.2%

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

      4. un-div-inv 64.3%

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

      5. +-commutative 64.3%

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

      6. log1p-define 91.4%

         \[\leadsto {\left(\sqrt{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{2} \]

   4. Applied egg-rr 91.4%

      \[\leadsto \color{blue}{{\left(\sqrt{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 87.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2200:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{e}^{\left(\frac{\log x}{n}\right)}}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 2200.0)
   (/
    (-
     (+
      (log1p x)
      (/
       (+
        (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0)))
        (*
         0.16666666666666666
         (/ (- (pow (log1p x) 3.0) (pow (log x) 3.0)) n)))
       n))
     (log x))
    n)
   (/ (/ (pow E (/ (log x) n)) n) x)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 2200.0) {
		tmp = ((log1p(x) + (((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) + (0.16666666666666666 * ((pow(log1p(x), 3.0) - pow(log(x), 3.0)) / n))) / n)) - log(x)) / n;
	} else {
		tmp = (pow(((double) M_E), (log(x) / n)) / n) / x;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 2200.0) {
		tmp = ((Math.log1p(x) + (((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) + (0.16666666666666666 * ((Math.pow(Math.log1p(x), 3.0) - Math.pow(Math.log(x), 3.0)) / n))) / n)) - Math.log(x)) / n;
	} else {
		tmp = (Math.pow(Math.E, (Math.log(x) / n)) / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 2200.0:
		tmp = ((math.log1p(x) + (((0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0))) + (0.16666666666666666 * ((math.pow(math.log1p(x), 3.0) - math.pow(math.log(x), 3.0)) / n))) / n)) - math.log(x)) / n
	else:
		tmp = (math.pow(math.e, (math.log(x) / n)) / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 2200.0)
		tmp = Float64(Float64(Float64(log1p(x) + Float64(Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) + Float64(0.16666666666666666 * Float64(Float64((log1p(x) ^ 3.0) - (log(x) ^ 3.0)) / n))) / n)) - log(x)) / n);
	else
		tmp = Float64(Float64((exp(1) ^ Float64(log(x) / n)) / n) / x);
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 2200.0], N[(N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[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[(N[Power[E, N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2200

   1. Initial program 47.8%

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

   3. Taylor expanded in n around -inf 76.8%

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

   5. Simplified 76.8%

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

   if 2200 < x

   1. Initial program 67.2%

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

   3. Taylor expanded in x around inf 81.8%

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

   5. Simplified 81.8%

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

   6. Taylor expanded in x around inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. log-rec 98.7%

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

      3. distribute-neg-frac 98.7%

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

      4. remove-double-neg 98.7%

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

   8. Simplified 98.7%

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

      1. *-un-lft-identity 98.7%

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

      2. exp-prod 98.7%

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

   10. Applied egg-rr 98.7%

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

       1. *-un-lft-identity 98.7%

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

       2. exp-1-e 98.7%

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

   12. Applied egg-rr 98.7%

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

       1. *-lft-identity 98.7%

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

   14. Simplified 98.7%

       \[\leadsto \frac{\frac{{\color{blue}{e}}^{\left(\frac{\log x}{n}\right)}}{n}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2200:\\ \;\;\;\;\frac{\left(\mathsf{log1p}\left(x\right) + \frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right) + 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}}{n}\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{e}^{\left(\frac{\log x}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -3.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-16}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -3.2e-79)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-16)
       (/ (log (+ 1.0 (/ 1.0 x))) n)
       (- (exp (/ (log1p x) n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -3.2e-79) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-16) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else {
		tmp = exp((log1p(x) / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -3.2e-79) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-16) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else {
		tmp = Math.exp((Math.log1p(x) / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -3.2e-79:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-16:
		tmp = math.log((1.0 + (1.0 / x))) / n
	else:
		tmp = math.exp((math.log1p(x) / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -3.2e-79)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-16)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	else
		tmp = Float64(exp(Float64(log1p(x) / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -3.2e-79], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-16], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.19999999999999988e-79

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf 25.3%

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

   5. Simplified 25.3%

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

   6. Taylor expanded in x around inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. log-rec 96.4%

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

      3. distribute-neg-frac 96.4%

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

      4. remove-double-neg 96.4%

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

   8. Simplified 96.4%

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

   9. Taylor expanded in x around inf 96.4%

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

       1. mul-1-neg 96.4%

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

       2. log-rec 96.4%

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

       3. distribute-frac-neg 96.4%

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

       4. remove-double-neg 96.4%

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

       5. *-rgt-identity 96.4%

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

       6. associate-*r/ 96.4%

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

       7. exp-to-pow 96.4%

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

   11. Simplified 96.4%

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

   if -3.19999999999999988e-79 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e-17

   1. Initial program 31.8%

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

   3. Taylor expanded in n around inf 80.0%

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

      1. log1p-define 80.0%

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

   5. Simplified 80.0%

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

      1. expm1-log1p-u 79.4%

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

   7. Applied egg-rr 79.4%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) - \log x\right)\right)}}{n} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 80.0%

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

      2. log1p-undefine 80.0%

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

      3. diff-log 80.3%

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

   9. Applied egg-rr 80.3%

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

       1. +-commutative 80.3%

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

   11. Simplified 80.3%

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

   12. Taylor expanded in x around inf 80.3%

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

   if 9.9999999999999998e-17 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 64.4%

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

   3. Taylor expanded in n around 0 64.4%

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

      1. log1p-define 91.4%

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

      2. *-rgt-identity 91.4%

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

      3. associate-/l* 91.3%

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

      4. exp-to-pow 91.3%

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

   5. Simplified 91.3%

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

Alternative 4: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -3.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -3.2e-79)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-11)
       (/ (log (+ 1.0 (/ 1.0 x))) n)
       (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -3.2e-79) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-11) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-3.2d-79)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-11) then
        tmp = log((1.0d0 + (1.0d0 / x))) / n
    else
        tmp = exp((x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -3.2e-79) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-11) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -3.2e-79:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-11:
		tmp = math.log((1.0 + (1.0 / x))) / n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -3.2e-79)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-11)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -3.2e-79)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-11)
		tmp = log((1.0 + (1.0 / x))) / n;
	else
		tmp = exp((x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -3.2e-79], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-11], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.19999999999999988e-79

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf 25.3%

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

   5. Simplified 25.3%

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

   6. Taylor expanded in x around inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. log-rec 96.4%

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

      3. distribute-neg-frac 96.4%

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

      4. remove-double-neg 96.4%

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

   8. Simplified 96.4%

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

   9. Taylor expanded in x around inf 96.4%

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

       1. mul-1-neg 96.4%

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

       2. log-rec 96.4%

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

       3. distribute-frac-neg 96.4%

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

       4. remove-double-neg 96.4%

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

       5. *-rgt-identity 96.4%

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

       6. associate-*r/ 96.4%

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

       7. exp-to-pow 96.4%

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

   11. Simplified 96.4%

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

   if -3.19999999999999988e-79 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000018e-11

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf 79.4%

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

      1. log1p-define 79.4%

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

   5. Simplified 79.4%

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

      1. expm1-log1p-u 78.8%

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

   7. Applied egg-rr 78.8%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) - \log x\right)\right)}}{n} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 79.4%

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

      2. log1p-undefine 79.4%

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

      3. diff-log 79.7%

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

   9. Applied egg-rr 79.7%

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

       1. +-commutative 79.7%

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

   11. Simplified 79.7%

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

   12. Taylor expanded in x around inf 79.7%

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

   if 5.00000000000000018e-11 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 66.0%

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

   3. Taylor expanded in n around 0 66.0%

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

      1. log1p-define 93.8%

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

      2. *-rgt-identity 93.8%

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

      3. associate-/l* 93.8%

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

      4. exp-to-pow 93.8%

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

   5. Simplified 93.8%

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

   6. Taylor expanded in x around 0 93.7%

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

Alternative 5: 66.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+206}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+142}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.25}{x \cdot n}}{x} - \frac{0.5}{n}}{x} + \frac{-1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= (/ 1.0 n) -2e+206)
     (/ (log (/ x x)) n)
     (if (<= (/ 1.0 n) -1e+24)
       t_0
       (if (<= (/ 1.0 n) 5e-11)
         (/ (log (+ 1.0 (/ 1.0 x))) n)
         (if (<= (/ 1.0 n) 2e+142)
           t_0
           (/
            (+ (/ (- (/ (/ 0.25 (* x n)) x) (/ 0.5 n)) x) (/ -1.0 n))
            x)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+206) {
		tmp = log((x / x)) / n;
	} else if ((1.0 / n) <= -1e+24) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-11) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e+142) {
		tmp = t_0;
	} else {
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if ((1.0d0 / n) <= (-2d+206)) then
        tmp = log((x / x)) / n
    else if ((1.0d0 / n) <= (-1d+24)) then
        tmp = t_0
    else if ((1.0d0 / n) <= 5d-11) then
        tmp = log((1.0d0 + (1.0d0 / x))) / n
    else if ((1.0d0 / n) <= 2d+142) then
        tmp = t_0
    else
        tmp = (((((0.25d0 / (x * n)) / x) - (0.5d0 / n)) / x) + ((-1.0d0) / n)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -2e+206) {
		tmp = Math.log((x / x)) / n;
	} else if ((1.0 / n) <= -1e+24) {
		tmp = t_0;
	} else if ((1.0 / n) <= 5e-11) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e+142) {
		tmp = t_0;
	} else {
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -2e+206:
		tmp = math.log((x / x)) / n
	elif (1.0 / n) <= -1e+24:
		tmp = t_0
	elif (1.0 / n) <= 5e-11:
		tmp = math.log((1.0 + (1.0 / x))) / n
	elif (1.0 / n) <= 2e+142:
		tmp = t_0
	else:
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e+206)
		tmp = Float64(log(Float64(x / x)) / n);
	elseif (Float64(1.0 / n) <= -1e+24)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= 5e-11)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	elseif (Float64(1.0 / n) <= 2e+142)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.25 / Float64(x * n)) / x) - Float64(0.5 / n)) / x) + Float64(-1.0 / n)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if ((1.0 / n) <= -2e+206)
		tmp = log((x / x)) / n;
	elseif ((1.0 / n) <= -1e+24)
		tmp = t_0;
	elseif ((1.0 / n) <= 5e-11)
		tmp = log((1.0 + (1.0 / x))) / n;
	elseif ((1.0 / n) <= 2e+142)
		tmp = t_0;
	else
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e+206], N[(N[Log[N[(x / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -1e+24], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-11], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+142], t$95$0, N[(N[(N[(N[(N[(N[(0.25 / N[(x * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2.0000000000000001e206

   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 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. Step-by-step derivation

      1. expm1-log1p-u 55.8%

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

   7. Applied egg-rr 55.8%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) - \log x\right)\right)}}{n} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 55.8%

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

      2. log1p-undefine 55.8%

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

      3. diff-log 55.8%

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

   9. Applied egg-rr 55.8%

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

       1. +-commutative 55.8%

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

   11. Simplified 55.8%

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

   12. Taylor expanded in x around inf 61.5%

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

   if -2.0000000000000001e206 < (/.f64 #s(literal 1 binary64) n) < -9.9999999999999998e23 or 5.00000000000000018e-11 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e142

   1. Initial program 93.0%

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

   3. Taylor expanded in x around 0 69.1%

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

      1. *-rgt-identity 69.1%

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

      2. associate-/l* 69.1%

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

      3. exp-to-pow 69.1%

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

   5. Simplified 69.1%

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

   if -9.9999999999999998e23 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000018e-11

   1. Initial program 33.6%

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

   3. Taylor expanded in n around inf 77.0%

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

      1. log1p-define 77.0%

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

   5. Simplified 77.0%

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

      1. expm1-log1p-u 76.4%

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

   7. Applied egg-rr 76.4%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) - \log x\right)\right)}}{n} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 77.0%

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

      2. log1p-undefine 77.0%

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

      3. diff-log 77.3%

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

   9. Applied egg-rr 77.3%

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

       1. +-commutative 77.3%

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

   11. Simplified 77.3%

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

   12. Taylor expanded in x around inf 77.3%

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

   if 2.0000000000000001e142 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf 5.6%

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

      1. log1p-define 5.6%

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

   5. Simplified 5.6%

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

   6. Taylor expanded in x around -inf 0.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.1%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 0.1%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \color{blue}{\sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      2. sqrt-unprod 78.5%

         \[\leadsto \color{blue}{\sqrt{\left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right) \cdot \left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right)}} \]

      3. sqr-neg 78.5%

         \[\leadsto \sqrt{\color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \cdot \frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      4. sqrt-unprod 78.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      5. add-sqr-sqrt 78.5%

         \[\leadsto \color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]

   10. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x}} \]

   11. Taylor expanded in x around 0 78.5%

       \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{0.25}{n \cdot x}}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{+206}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{+24}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+142}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.25}{x \cdot n}}{x} - \frac{0.5}{n}}{x} + \frac{-1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -3.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.25}{x \cdot n}}{x} - \frac{0.5}{n}}{x} + \frac{-1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -3.2e-79)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-11)
       (/ (log (+ 1.0 (/ 1.0 x))) n)
       (if (<= (/ 1.0 n) 2e+142)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (+ (/ (- (/ (/ 0.25 (* x n)) x) (/ 0.5 n)) x) (/ -1.0 n)) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -3.2e-79) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-11) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e+142) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-3.2d-79)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-11) then
        tmp = log((1.0d0 + (1.0d0 / x))) / n
    else if ((1.0d0 / n) <= 2d+142) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (((((0.25d0 / (x * n)) / x) - (0.5d0 / n)) / x) + ((-1.0d0) / n)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -3.2e-79) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-11) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e+142) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -3.2e-79:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-11:
		tmp = math.log((1.0 + (1.0 / x))) / n
	elif (1.0 / n) <= 2e+142:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -3.2e-79)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-11)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	elseif (Float64(1.0 / n) <= 2e+142)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.25 / Float64(x * n)) / x) - Float64(0.5 / n)) / x) + Float64(-1.0 / n)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -3.2e-79)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-11)
		tmp = log((1.0 + (1.0 / x))) / n;
	elseif ((1.0 / n) <= 2e+142)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -3.2e-79], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-11], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+142], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.25 / N[(x * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.19999999999999988e-79

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf 25.3%

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

   5. Simplified 25.3%

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

   6. Taylor expanded in x around inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. log-rec 96.4%

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

      3. distribute-neg-frac 96.4%

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

      4. remove-double-neg 96.4%

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

   8. Simplified 96.4%

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

   9. Taylor expanded in x around inf 96.4%

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

       1. mul-1-neg 96.4%

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

       2. log-rec 96.4%

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

       3. distribute-frac-neg 96.4%

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

       4. remove-double-neg 96.4%

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

       5. *-rgt-identity 96.4%

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

       6. associate-*r/ 96.4%

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

       7. exp-to-pow 96.4%

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

   11. Simplified 96.4%

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

   if -3.19999999999999988e-79 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000018e-11

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf 79.4%

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

      1. log1p-define 79.4%

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

   5. Simplified 79.4%

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

      1. expm1-log1p-u 78.8%

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

   7. Applied egg-rr 78.8%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) - \log x\right)\right)}}{n} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 79.4%

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

      2. log1p-undefine 79.4%

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

      3. diff-log 79.7%

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

   9. Applied egg-rr 79.7%

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

       1. +-commutative 79.7%

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

   11. Simplified 79.7%

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

   12. Taylor expanded in x around inf 79.7%

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

   if 5.00000000000000018e-11 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e142

   1. Initial program 80.3%

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

   3. Taylor expanded in x around 0 73.5%

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

   if 2.0000000000000001e142 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf 5.6%

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

      1. log1p-define 5.6%

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

   5. Simplified 5.6%

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

   6. Taylor expanded in x around -inf 0.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.1%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 0.1%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \color{blue}{\sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      2. sqrt-unprod 78.5%

         \[\leadsto \color{blue}{\sqrt{\left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right) \cdot \left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right)}} \]

      3. sqr-neg 78.5%

         \[\leadsto \sqrt{\color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \cdot \frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      4. sqrt-unprod 78.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      5. add-sqr-sqrt 78.5%

         \[\leadsto \color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]

   10. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x}} \]

   11. Taylor expanded in x around 0 78.5%

       \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{0.25}{n \cdot x}}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -3.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.25}{x \cdot n}}{x} - \frac{0.5}{n}}{x} + \frac{-1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -3.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+142}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.25}{x \cdot n}}{x} - \frac{0.5}{n}}{x} + \frac{-1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -3.2e-79)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 5e-11)
       (/ (log (+ 1.0 (/ 1.0 x))) n)
       (if (<= (/ 1.0 n) 2e+142)
         (- 1.0 t_0)
         (/ (+ (/ (- (/ (/ 0.25 (* x n)) x) (/ 0.5 n)) x) (/ -1.0 n)) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -3.2e-79) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-11) {
		tmp = log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e+142) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-3.2d-79)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 5d-11) then
        tmp = log((1.0d0 + (1.0d0 / x))) / n
    else if ((1.0d0 / n) <= 2d+142) then
        tmp = 1.0d0 - t_0
    else
        tmp = (((((0.25d0 / (x * n)) / x) - (0.5d0 / n)) / x) + ((-1.0d0) / n)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -3.2e-79) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 5e-11) {
		tmp = Math.log((1.0 + (1.0 / x))) / n;
	} else if ((1.0 / n) <= 2e+142) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -3.2e-79:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 5e-11:
		tmp = math.log((1.0 + (1.0 / x))) / n
	elif (1.0 / n) <= 2e+142:
		tmp = 1.0 - t_0
	else:
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -3.2e-79)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 5e-11)
		tmp = Float64(log(Float64(1.0 + Float64(1.0 / x))) / n);
	elseif (Float64(1.0 / n) <= 2e+142)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.25 / Float64(x * n)) / x) - Float64(0.5 / n)) / x) + Float64(-1.0 / n)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -3.2e-79)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 5e-11)
		tmp = log((1.0 + (1.0 / x))) / n;
	elseif ((1.0 / n) <= 2e+142)
		tmp = 1.0 - t_0;
	else
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -3.2e-79], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-11], N[(N[Log[N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+142], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.25 / N[(x * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.19999999999999988e-79

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf 25.3%

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

   5. Simplified 25.3%

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

   6. Taylor expanded in x around inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. log-rec 96.4%

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

      3. distribute-neg-frac 96.4%

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

      4. remove-double-neg 96.4%

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

   8. Simplified 96.4%

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

   9. Taylor expanded in x around inf 96.4%

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

       1. mul-1-neg 96.4%

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

       2. log-rec 96.4%

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

       3. distribute-frac-neg 96.4%

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

       4. remove-double-neg 96.4%

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

       5. *-rgt-identity 96.4%

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

       6. associate-*r/ 96.4%

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

       7. exp-to-pow 96.4%

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

   11. Simplified 96.4%

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

   if -3.19999999999999988e-79 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000018e-11

   1. Initial program 31.6%

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

   3. Taylor expanded in n around inf 79.4%

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

      1. log1p-define 79.4%

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

   5. Simplified 79.4%

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

      1. expm1-log1p-u 78.8%

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

   7. Applied egg-rr 78.8%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) - \log x\right)\right)}}{n} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 79.4%

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

      2. log1p-undefine 79.4%

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

      3. diff-log 79.7%

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

   9. Applied egg-rr 79.7%

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

       1. +-commutative 79.7%

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

   11. Simplified 79.7%

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

   12. Taylor expanded in x around inf 79.7%

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

   if 5.00000000000000018e-11 < (/.f64 #s(literal 1 binary64) n) < 2.0000000000000001e142

   1. Initial program 80.3%

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

   3. Taylor expanded in x around 0 72.7%

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

      1. *-rgt-identity 72.7%

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

      2. associate-/l* 72.7%

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

      3. exp-to-pow 72.7%

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

   5. Simplified 72.7%

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

   if 2.0000000000000001e142 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 24.7%

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

   3. Taylor expanded in n around inf 5.6%

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

      1. log1p-define 5.6%

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

   5. Simplified 5.6%

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

   6. Taylor expanded in x around -inf 0.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.1%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 0.1%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \color{blue}{\sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      2. sqrt-unprod 78.5%

         \[\leadsto \color{blue}{\sqrt{\left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right) \cdot \left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right)}} \]

      3. sqr-neg 78.5%

         \[\leadsto \sqrt{\color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \cdot \frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      4. sqrt-unprod 78.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      5. add-sqr-sqrt 78.5%

         \[\leadsto \color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]

   10. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x}} \]

   11. Taylor expanded in x around 0 78.5%

       \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{0.25}{n \cdot x}}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -3.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(1 + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+142}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.25}{x \cdot n}}{x} - \frac{0.5}{n}}{x} + \frac{-1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-94}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.15e-94)
   (/ (log x) (- n))
   (if (<= x 8.6e-7)
     (- 1.0 (pow x (/ 1.0 n)))
     (if (<= x 2.2e+164)
       (/ (/ (+ (/ (+ 0.5 (/ -0.3333333333333333 x)) x) -1.0) (- x)) n)
       (/ (log (/ x x)) n)))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.15e-94) {
		tmp = log(x) / -n;
	} else if (x <= 8.6e-7) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else if (x <= 2.2e+164) {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	} else {
		tmp = log((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 <= 1.15d-94) then
        tmp = log(x) / -n
    else if (x <= 8.6d-7) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else if (x <= 2.2d+164) then
        tmp = ((((0.5d0 + ((-0.3333333333333333d0) / x)) / x) + (-1.0d0)) / -x) / n
    else
        tmp = log((x / x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.15e-94) {
		tmp = Math.log(x) / -n;
	} else if (x <= 8.6e-7) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else if (x <= 2.2e+164) {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	} else {
		tmp = Math.log((x / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.15e-94:
		tmp = math.log(x) / -n
	elif x <= 8.6e-7:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	elif x <= 2.2e+164:
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n
	else:
		tmp = math.log((x / x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.15e-94)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 8.6e-7)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	elseif (x <= 2.2e+164)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x) + -1.0) / Float64(-x)) / n);
	else
		tmp = Float64(log(Float64(x / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.15e-94)
		tmp = log(x) / -n;
	elseif (x <= 8.6e-7)
		tmp = 1.0 - (x ^ (1.0 / n));
	elseif (x <= 2.2e+164)
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	else
		tmp = log((x / x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.15e-94], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 8.6e-7], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e+164], N[(N[(N[(N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision], N[(N[Log[N[(x / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.15e-94

   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 n around inf 56.7%

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

      1. log1p-define 56.7%

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

   5. Simplified 56.7%

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

   6. Taylor expanded in x around 0 56.7%

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

      1. neg-mul-1 56.7%

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

   8. Simplified 56.7%

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

   if 1.15e-94 < x < 8.6000000000000002e-7

   1. Initial program 60.0%

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

   3. Taylor expanded in x around 0 54.9%

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

      1. *-rgt-identity 54.9%

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

      2. associate-/l* 54.9%

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

      3. exp-to-pow 54.9%

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

   5. Simplified 54.9%

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

   if 8.6000000000000002e-7 < x < 2.20000000000000006e164

   1. Initial program 50.4%

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

   3. Taylor expanded in n around inf 53.6%

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

      1. log1p-define 53.6%

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

   5. Simplified 53.6%

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

      1. expm1-log1p-u 53.6%

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

   7. Applied egg-rr 53.6%

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

   8. Taylor expanded in x around -inf 69.3%

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

      1. mul-1-neg 69.3%

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

      2. distribute-neg-frac2 69.3%

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

      3. sub-neg 69.3%

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

      4. associate-*r/ 69.3%

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

      5. sub-neg 69.3%

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

      6. metadata-eval 69.3%

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

      7. distribute-lft-in 69.3%

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

      8. neg-mul-1 69.3%

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

      9. associate-*r/ 69.3%

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

      10. metadata-eval 69.3%

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

      11. distribute-neg-frac 69.3%

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

      12. metadata-eval 69.3%

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

      13. metadata-eval 69.3%

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

      14. metadata-eval 69.3%

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

   10. Simplified 69.3%

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

   if 2.20000000000000006e164 < x

   1. Initial program 83.7%

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

   3. Taylor expanded in n around inf 83.7%

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

      1. log1p-define 83.7%

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

   5. Simplified 83.7%

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

      1. expm1-log1p-u 83.7%

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

   7. Applied egg-rr 83.7%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) - \log x\right)\right)}}{n} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 83.7%

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

      2. log1p-undefine 83.7%

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

      3. diff-log 83.7%

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

   9. Applied egg-rr 83.7%

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

       1. +-commutative 83.7%

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

   11. Simplified 83.7%

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

   12. Taylor expanded in x around inf 83.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-94}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}\\ \mathbf{if}\;x \leq 5 \cdot 10^{-96}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 0.62:\\ \;\;\;\;\frac{\frac{n - \frac{x}{t\_0}}{\frac{x \cdot n}{t\_0}}}{x}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (+ (/ (+ -0.3333333333333333 (/ 0.25 x)) (* x n)) (/ -0.5 n))))
   (if (<= x 5e-96)
     (/ (log x) (- n))
     (if (<= x 0.62)
       (/ (/ (- n (/ x t_0)) (/ (* x n) t_0)) x)
       (if (<= x 9.5e+163)
         (/
          (/
           (+
            1.0
            (/ (- (/ (+ 0.3333333333333333 (* 0.25 (/ -1.0 x))) x) 0.5) x))
           x)
          n)
         (/ (log (/ x x)) n))))))

C

double code(double x, double n) {
	double t_0 = ((-0.3333333333333333 + (0.25 / x)) / (x * n)) + (-0.5 / n);
	double tmp;
	if (x <= 5e-96) {
		tmp = log(x) / -n;
	} else if (x <= 0.62) {
		tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x;
	} else if (x <= 9.5e+163) {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = log((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 = (((-0.3333333333333333d0) + (0.25d0 / x)) / (x * n)) + ((-0.5d0) / n)
    if (x <= 5d-96) then
        tmp = log(x) / -n
    else if (x <= 0.62d0) then
        tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x
    else if (x <= 9.5d+163) then
        tmp = ((1.0d0 + ((((0.3333333333333333d0 + (0.25d0 * ((-1.0d0) / x))) / x) - 0.5d0) / x)) / x) / n
    else
        tmp = log((x / x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = ((-0.3333333333333333 + (0.25 / x)) / (x * n)) + (-0.5 / n);
	double tmp;
	if (x <= 5e-96) {
		tmp = Math.log(x) / -n;
	} else if (x <= 0.62) {
		tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x;
	} else if (x <= 9.5e+163) {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	} else {
		tmp = Math.log((x / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = ((-0.3333333333333333 + (0.25 / x)) / (x * n)) + (-0.5 / n)
	tmp = 0
	if x <= 5e-96:
		tmp = math.log(x) / -n
	elif x <= 0.62:
		tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x
	elif x <= 9.5e+163:
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n
	else:
		tmp = math.log((x / x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / Float64(x * n)) + Float64(-0.5 / n))
	tmp = 0.0
	if (x <= 5e-96)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 0.62)
		tmp = Float64(Float64(Float64(n - Float64(x / t_0)) / Float64(Float64(x * n) / t_0)) / x);
	elseif (x <= 9.5e+163)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(0.25 * Float64(-1.0 / x))) / x) - 0.5) / x)) / x) / n);
	else
		tmp = Float64(log(Float64(x / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((-0.3333333333333333 + (0.25 / x)) / (x * n)) + (-0.5 / n);
	tmp = 0.0;
	if (x <= 5e-96)
		tmp = log(x) / -n;
	elseif (x <= 0.62)
		tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x;
	elseif (x <= 9.5e+163)
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	else
		tmp = log((x / x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e-96], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 0.62], N[(N[(N[(n - N[(x / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(x * n), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 9.5e+163], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(0.25 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], N[(N[Log[N[(x / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.99999999999999995e-96

   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 n around inf 56.7%

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

      1. log1p-define 56.7%

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

   5. Simplified 56.7%

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

   6. Taylor expanded in x around 0 56.7%

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

      1. neg-mul-1 56.7%

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

   8. Simplified 56.7%

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

   if 4.99999999999999995e-96 < x < 0.619999999999999996

   1. Initial program 57.2%

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

   3. Taylor expanded in n around inf 35.8%

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

      1. log1p-define 35.8%

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

   5. Simplified 35.8%

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

   6. Taylor expanded in x around -inf 0.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.9%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 0.9%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.4%

         \[\leadsto \color{blue}{\sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      2. sqrt-unprod 14.1%

         \[\leadsto \color{blue}{\sqrt{\left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right) \cdot \left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right)}} \]

      3. sqr-neg 14.1%

         \[\leadsto \sqrt{\color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \cdot \frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      4. sqrt-unprod 9.0%

         \[\leadsto \color{blue}{\sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      5. add-sqr-sqrt 36.3%

         \[\leadsto \color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]

   10. Applied egg-rr 36.3%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x}} \]
   11. Step-by-step derivation

       1. clear-num 36.3%

          \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}}} - \frac{1}{n}}{x} \]

       2. frac-sub 50.8%

          \[\leadsto \frac{\color{blue}{\frac{1 \cdot n - \frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}}{x} \]

       3. *-un-lft-identity 50.8%

          \[\leadsto \frac{\frac{\color{blue}{n} - \frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}{x} \]

       4. associate-/l/ 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\color{blue}{\frac{\frac{0.25}{x} - 0.3333333333333333}{x \cdot n}} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}{x} \]

       5. sub-neg 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\color{blue}{\frac{0.25}{x} + \left(-0.3333333333333333\right)}}{x \cdot n} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}{x} \]

       6. metadata-eval 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\frac{0.25}{x} + \color{blue}{-0.3333333333333333}}{x \cdot n} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}{x} \]

   12. Applied egg-rr 50.8%

       \[\leadsto \frac{\color{blue}{\frac{n - \frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}}{x} \]
   13. Step-by-step derivation

       1. *-rgt-identity 50.8%

          \[\leadsto \frac{\frac{n - \color{blue}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       2. sub-neg 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\color{blue}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       3. *-lft-identity 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\color{blue}{1 \cdot \frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n}} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       4. *-lft-identity 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\color{blue}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n}} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       5. metadata-eval 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\frac{\color{blue}{0.25 \cdot 1}}{x} + -0.3333333333333333}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       6. associate-*r/ 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\color{blue}{0.25 \cdot \frac{1}{x}} + -0.3333333333333333}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       7. +-commutative 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\color{blue}{-0.3333333333333333 + 0.25 \cdot \frac{1}{x}}}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       8. associate-*r/ 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{x}}}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       9. metadata-eval 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{\color{blue}{0.25}}{x}}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       10. distribute-neg-frac 50.8%

           \[\leadsto \frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \color{blue}{\frac{-0.5}{n}}}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       11. metadata-eval 50.8%

           \[\leadsto \frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{\color{blue}{-0.5}}{n}}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

   14. Simplified 50.8%

       \[\leadsto \frac{\color{blue}{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}}}{\frac{x \cdot n}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}}}}}{x} \]

   if 0.619999999999999996 < x < 9.50000000000000053e163

   1. Initial program 51.9%

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

   3. Taylor expanded in n around inf 52.0%

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

      1. log1p-define 52.0%

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

   5. Simplified 52.0%

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

   6. Taylor expanded in x around -inf 71.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]

   if 9.50000000000000053e163 < x

   1. Initial program 83.7%

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

   3. Taylor expanded in n around inf 83.7%

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

      1. log1p-define 83.7%

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

   5. Simplified 83.7%

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

      1. expm1-log1p-u 83.7%

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

   7. Applied egg-rr 83.7%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x\right) - \log x\right)\right)}}{n} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 83.7%

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

      2. log1p-undefine 83.7%

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

      3. diff-log 83.7%

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

   9. Applied egg-rr 83.7%

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

       1. +-commutative 83.7%

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

   11. Simplified 83.7%

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

   12. Taylor expanded in x around inf 83.7%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-96}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 0.62:\\ \;\;\;\;\frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}}}{\frac{x \cdot n}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}}}}{x}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}\\ \mathbf{if}\;x \leq 1.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 0.62:\\ \;\;\;\;\frac{\frac{n - \frac{x}{t\_0}}{\frac{x \cdot n}{t\_0}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (+ (/ (+ -0.3333333333333333 (/ 0.25 x)) (* x n)) (/ -0.5 n))))
   (if (<= x 1.2e-94)
     (/ (log x) (- n))
     (if (<= x 0.62)
       (/ (/ (- n (/ x t_0)) (/ (* x n) t_0)) x)
       (/
        (/
         (+ 1.0 (/ (- (/ (+ 0.3333333333333333 (* 0.25 (/ -1.0 x))) x) 0.5) x))
         x)
        n)))))

C

double code(double x, double n) {
	double t_0 = ((-0.3333333333333333 + (0.25 / x)) / (x * n)) + (-0.5 / n);
	double tmp;
	if (x <= 1.2e-94) {
		tmp = log(x) / -n;
	} else if (x <= 0.62) {
		tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 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) :: tmp
    t_0 = (((-0.3333333333333333d0) + (0.25d0 / x)) / (x * n)) + ((-0.5d0) / n)
    if (x <= 1.2d-94) then
        tmp = log(x) / -n
    else if (x <= 0.62d0) then
        tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x
    else
        tmp = ((1.0d0 + ((((0.3333333333333333d0 + (0.25d0 * ((-1.0d0) / x))) / x) - 0.5d0) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = ((-0.3333333333333333 + (0.25 / x)) / (x * n)) + (-0.5 / n);
	double tmp;
	if (x <= 1.2e-94) {
		tmp = Math.log(x) / -n;
	} else if (x <= 0.62) {
		tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = ((-0.3333333333333333 + (0.25 / x)) / (x * n)) + (-0.5 / n)
	tmp = 0
	if x <= 1.2e-94:
		tmp = math.log(x) / -n
	elif x <= 0.62:
		tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x
	else:
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / Float64(x * n)) + Float64(-0.5 / n))
	tmp = 0.0
	if (x <= 1.2e-94)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 0.62)
		tmp = Float64(Float64(Float64(n - Float64(x / t_0)) / Float64(Float64(x * n) / t_0)) / x);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(0.25 * Float64(-1.0 / x))) / x) - 0.5) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((-0.3333333333333333 + (0.25 / x)) / (x * n)) + (-0.5 / n);
	tmp = 0.0;
	if (x <= 1.2e-94)
		tmp = log(x) / -n;
	elseif (x <= 0.62)
		tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x;
	else
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.2e-94], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 0.62], N[(N[(N[(n - N[(x / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(x * n), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(0.25 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.2e-94

   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 n around inf 56.7%

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

      1. log1p-define 56.7%

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

   5. Simplified 56.7%

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

   6. Taylor expanded in x around 0 56.7%

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

      1. neg-mul-1 56.7%

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

   8. Simplified 56.7%

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

   if 1.2e-94 < x < 0.619999999999999996

   1. Initial program 57.2%

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

   3. Taylor expanded in n around inf 35.8%

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

      1. log1p-define 35.8%

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

   5. Simplified 35.8%

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

   6. Taylor expanded in x around -inf 0.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.9%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 0.9%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.4%

         \[\leadsto \color{blue}{\sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      2. sqrt-unprod 14.1%

         \[\leadsto \color{blue}{\sqrt{\left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right) \cdot \left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right)}} \]

      3. sqr-neg 14.1%

         \[\leadsto \sqrt{\color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \cdot \frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      4. sqrt-unprod 9.0%

         \[\leadsto \color{blue}{\sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      5. add-sqr-sqrt 36.3%

         \[\leadsto \color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]

   10. Applied egg-rr 36.3%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x}} \]
   11. Step-by-step derivation

       1. clear-num 36.3%

          \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}}} - \frac{1}{n}}{x} \]

       2. frac-sub 50.8%

          \[\leadsto \frac{\color{blue}{\frac{1 \cdot n - \frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}}{x} \]

       3. *-un-lft-identity 50.8%

          \[\leadsto \frac{\frac{\color{blue}{n} - \frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}{x} \]

       4. associate-/l/ 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\color{blue}{\frac{\frac{0.25}{x} - 0.3333333333333333}{x \cdot n}} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}{x} \]

       5. sub-neg 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\color{blue}{\frac{0.25}{x} + \left(-0.3333333333333333\right)}}{x \cdot n} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}{x} \]

       6. metadata-eval 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\frac{0.25}{x} + \color{blue}{-0.3333333333333333}}{x \cdot n} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}{x} \]

   12. Applied egg-rr 50.8%

       \[\leadsto \frac{\color{blue}{\frac{n - \frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}}{x} \]
   13. Step-by-step derivation

       1. *-rgt-identity 50.8%

          \[\leadsto \frac{\frac{n - \color{blue}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       2. sub-neg 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\color{blue}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       3. *-lft-identity 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\color{blue}{1 \cdot \frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n}} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       4. *-lft-identity 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\color{blue}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n}} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       5. metadata-eval 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\frac{\color{blue}{0.25 \cdot 1}}{x} + -0.3333333333333333}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       6. associate-*r/ 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\color{blue}{0.25 \cdot \frac{1}{x}} + -0.3333333333333333}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       7. +-commutative 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\color{blue}{-0.3333333333333333 + 0.25 \cdot \frac{1}{x}}}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       8. associate-*r/ 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{x}}}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       9. metadata-eval 50.8%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{\color{blue}{0.25}}{x}}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       10. distribute-neg-frac 50.8%

           \[\leadsto \frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \color{blue}{\frac{-0.5}{n}}}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       11. metadata-eval 50.8%

           \[\leadsto \frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{\color{blue}{-0.5}}{n}}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

   14. Simplified 50.8%

       \[\leadsto \frac{\color{blue}{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}}}{\frac{x \cdot n}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}}}}}{x} \]

   if 0.619999999999999996 < x

   1. Initial program 66.7%

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

   3. Taylor expanded in n around inf 66.8%

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

      1. log1p-define 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in x around -inf 66.2%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 0.62:\\ \;\;\;\;\frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}}}{\frac{x \cdot n}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}\\ \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;\frac{\frac{n - \frac{x}{t\_0}}{\frac{x \cdot n}{t\_0}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (+ (/ (+ -0.3333333333333333 (/ 0.25 x)) (* x n)) (/ -0.5 n))))
   (if (<= x 0.62)
     (/ (/ (- n (/ x t_0)) (/ (* x n) t_0)) x)
     (/
      (/
       (+ 1.0 (/ (- (/ (+ 0.3333333333333333 (* 0.25 (/ -1.0 x))) x) 0.5) x))
       x)
      n))))

C

double code(double x, double n) {
	double t_0 = ((-0.3333333333333333 + (0.25 / x)) / (x * n)) + (-0.5 / n);
	double tmp;
	if (x <= 0.62) {
		tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 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) :: tmp
    t_0 = (((-0.3333333333333333d0) + (0.25d0 / x)) / (x * n)) + ((-0.5d0) / n)
    if (x <= 0.62d0) then
        tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x
    else
        tmp = ((1.0d0 + ((((0.3333333333333333d0 + (0.25d0 * ((-1.0d0) / x))) / x) - 0.5d0) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = ((-0.3333333333333333 + (0.25 / x)) / (x * n)) + (-0.5 / n);
	double tmp;
	if (x <= 0.62) {
		tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = ((-0.3333333333333333 + (0.25 / x)) / (x * n)) + (-0.5 / n)
	tmp = 0
	if x <= 0.62:
		tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x
	else:
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / Float64(x * n)) + Float64(-0.5 / n))
	tmp = 0.0
	if (x <= 0.62)
		tmp = Float64(Float64(Float64(n - Float64(x / t_0)) / Float64(Float64(x * n) / t_0)) / x);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(0.25 * Float64(-1.0 / x))) / x) - 0.5) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = ((-0.3333333333333333 + (0.25 / x)) / (x * n)) + (-0.5 / n);
	tmp = 0.0;
	if (x <= 0.62)
		tmp = ((n - (x / t_0)) / ((x * n) / t_0)) / x;
	else
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.62], N[(N[(N[(n - N[(x / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(x * n), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(0.25 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.619999999999999996

   1. Initial program 48.1%

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

   3. Taylor expanded in n around inf 50.9%

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

      1. log1p-define 50.9%

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

   5. Simplified 50.9%

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

   6. Taylor expanded in x around -inf 0.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.8%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 0.8%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.4%

         \[\leadsto \color{blue}{\sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      2. sqrt-unprod 8.5%

         \[\leadsto \color{blue}{\sqrt{\left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right) \cdot \left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right)}} \]

      3. sqr-neg 8.5%

         \[\leadsto \sqrt{\color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \cdot \frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      4. sqrt-unprod 6.8%

         \[\leadsto \color{blue}{\sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      5. add-sqr-sqrt 30.8%

         \[\leadsto \color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]

   10. Applied egg-rr 30.8%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x}} \]
   11. Step-by-step derivation

       1. clear-num 30.8%

          \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}}} - \frac{1}{n}}{x} \]

       2. frac-sub 34.9%

          \[\leadsto \frac{\color{blue}{\frac{1 \cdot n - \frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}}{x} \]

       3. *-un-lft-identity 34.9%

          \[\leadsto \frac{\frac{\color{blue}{n} - \frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}{x} \]

       4. associate-/l/ 34.9%

          \[\leadsto \frac{\frac{n - \frac{x}{\color{blue}{\frac{\frac{0.25}{x} - 0.3333333333333333}{x \cdot n}} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}{x} \]

       5. sub-neg 34.9%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\color{blue}{\frac{0.25}{x} + \left(-0.3333333333333333\right)}}{x \cdot n} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}{x} \]

       6. metadata-eval 34.9%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\frac{0.25}{x} + \color{blue}{-0.3333333333333333}}{x \cdot n} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}} \cdot n}}{x} \]

   12. Applied egg-rr 34.9%

       \[\leadsto \frac{\color{blue}{\frac{n - \frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot 1}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}}{x} \]
   13. Step-by-step derivation

       1. *-rgt-identity 34.9%

          \[\leadsto \frac{\frac{n - \color{blue}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}}}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       2. sub-neg 34.9%

          \[\leadsto \frac{\frac{n - \frac{x}{\color{blue}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       3. *-lft-identity 34.9%

          \[\leadsto \frac{\frac{n - \frac{x}{\color{blue}{1 \cdot \frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n}} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       4. *-lft-identity 34.9%

          \[\leadsto \frac{\frac{n - \frac{x}{\color{blue}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n}} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       5. metadata-eval 34.9%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\frac{\color{blue}{0.25 \cdot 1}}{x} + -0.3333333333333333}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       6. associate-*r/ 34.9%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\color{blue}{0.25 \cdot \frac{1}{x}} + -0.3333333333333333}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       7. +-commutative 34.9%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{\color{blue}{-0.3333333333333333 + 0.25 \cdot \frac{1}{x}}}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       8. associate-*r/ 34.9%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{x}}}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       9. metadata-eval 34.9%

          \[\leadsto \frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{\color{blue}{0.25}}{x}}{x \cdot n} + \left(-\frac{0.5}{n}\right)}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       10. distribute-neg-frac 34.9%

           \[\leadsto \frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \color{blue}{\frac{-0.5}{n}}}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

       11. metadata-eval 34.9%

           \[\leadsto \frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{\color{blue}{-0.5}}{n}}}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{x \cdot n} - \frac{0.5}{n}} \cdot n}}{x} \]

   14. Simplified 34.9%

       \[\leadsto \frac{\color{blue}{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}}}{\frac{x \cdot n}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}}}}}{x} \]

   if 0.619999999999999996 < x

   1. Initial program 66.7%

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

   3. Taylor expanded in n around inf 66.8%

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

      1. log1p-define 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in x around -inf 66.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;\frac{\frac{n - \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}}}{\frac{x \cdot n}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x \cdot n} + \frac{-0.5}{n}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{n}}\\ \mathbf{if}\;x \leq 0.45:\\ \;\;\;\;\frac{\frac{\frac{n - 0.5 \cdot t\_0}{n \cdot t\_0}}{x} + \frac{-1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ x (/ (+ -0.3333333333333333 (/ 0.25 x)) n))))
   (if (<= x 0.45)
     (/ (+ (/ (/ (- n (* 0.5 t_0)) (* n t_0)) x) (/ -1.0 n)) x)
     (/
      (/
       (+ 1.0 (/ (- (/ (+ 0.3333333333333333 (* 0.25 (/ -1.0 x))) x) 0.5) x))
       x)
      n))))

C

double code(double x, double n) {
	double t_0 = x / ((-0.3333333333333333 + (0.25 / x)) / n);
	double tmp;
	if (x <= 0.45) {
		tmp = ((((n - (0.5 * t_0)) / (n * t_0)) / x) + (-1.0 / n)) / x;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 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) :: tmp
    t_0 = x / (((-0.3333333333333333d0) + (0.25d0 / x)) / n)
    if (x <= 0.45d0) then
        tmp = ((((n - (0.5d0 * t_0)) / (n * t_0)) / x) + ((-1.0d0) / n)) / x
    else
        tmp = ((1.0d0 + ((((0.3333333333333333d0 + (0.25d0 * ((-1.0d0) / x))) / x) - 0.5d0) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = x / ((-0.3333333333333333 + (0.25 / x)) / n);
	double tmp;
	if (x <= 0.45) {
		tmp = ((((n - (0.5 * t_0)) / (n * t_0)) / x) + (-1.0 / n)) / x;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = x / ((-0.3333333333333333 + (0.25 / x)) / n)
	tmp = 0
	if x <= 0.45:
		tmp = ((((n - (0.5 * t_0)) / (n * t_0)) / x) + (-1.0 / n)) / x
	else:
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(x / Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / n))
	tmp = 0.0
	if (x <= 0.45)
		tmp = Float64(Float64(Float64(Float64(Float64(n - Float64(0.5 * t_0)) / Float64(n * t_0)) / x) + Float64(-1.0 / n)) / x);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(0.25 * Float64(-1.0 / x))) / x) - 0.5) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x / ((-0.3333333333333333 + (0.25 / x)) / n);
	tmp = 0.0;
	if (x <= 0.45)
		tmp = ((((n - (0.5 * t_0)) / (n * t_0)) / x) + (-1.0 / n)) / x;
	else
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(x / N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.45], N[(N[(N[(N[(N[(n - N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(n * t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(0.25 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.450000000000000011

   1. Initial program 48.1%

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

   3. Taylor expanded in n around inf 50.9%

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

      1. log1p-define 50.9%

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

   5. Simplified 50.9%

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

   6. Taylor expanded in x around -inf 0.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.8%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 0.8%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.4%

         \[\leadsto \color{blue}{\sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      2. sqrt-unprod 8.5%

         \[\leadsto \color{blue}{\sqrt{\left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right) \cdot \left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right)}} \]

      3. sqr-neg 8.5%

         \[\leadsto \sqrt{\color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \cdot \frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      4. sqrt-unprod 6.8%

         \[\leadsto \color{blue}{\sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      5. add-sqr-sqrt 30.8%

         \[\leadsto \color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]

   10. Applied egg-rr 30.8%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x}} \]
   11. Step-by-step derivation

       1. clear-num 30.8%

          \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\frac{x}{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}}} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x} \]

       2. frac-sub 34.1%

          \[\leadsto \frac{\frac{\color{blue}{\frac{1 \cdot n - \frac{x}{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}} \cdot 0.5}{\frac{x}{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}} \cdot n}}}{x} - \frac{1}{n}}{x} \]

       3. *-un-lft-identity 34.1%

          \[\leadsto \frac{\frac{\frac{\color{blue}{n} - \frac{x}{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}} \cdot 0.5}{\frac{x}{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}} \cdot n}}{x} - \frac{1}{n}}{x} \]

       4. sub-neg 34.1%

          \[\leadsto \frac{\frac{\frac{n - \frac{x}{\frac{\color{blue}{\frac{0.25}{x} + \left(-0.3333333333333333\right)}}{n}} \cdot 0.5}{\frac{x}{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}} \cdot n}}{x} - \frac{1}{n}}{x} \]

       5. metadata-eval 34.1%

          \[\leadsto \frac{\frac{\frac{n - \frac{x}{\frac{\frac{0.25}{x} + \color{blue}{-0.3333333333333333}}{n}} \cdot 0.5}{\frac{x}{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}} \cdot n}}{x} - \frac{1}{n}}{x} \]

       6. sub-neg 34.1%

          \[\leadsto \frac{\frac{\frac{n - \frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{n}} \cdot 0.5}{\frac{x}{\frac{\color{blue}{\frac{0.25}{x} + \left(-0.3333333333333333\right)}}{n}} \cdot n}}{x} - \frac{1}{n}}{x} \]

       7. metadata-eval 34.1%

          \[\leadsto \frac{\frac{\frac{n - \frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{n}} \cdot 0.5}{\frac{x}{\frac{\frac{0.25}{x} + \color{blue}{-0.3333333333333333}}{n}} \cdot n}}{x} - \frac{1}{n}}{x} \]

   12. Applied egg-rr 34.1%

       \[\leadsto \frac{\frac{\color{blue}{\frac{n - \frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{n}} \cdot 0.5}{\frac{x}{\frac{\frac{0.25}{x} + -0.3333333333333333}{n}} \cdot n}}}{x} - \frac{1}{n}}{x} \]

   if 0.450000000000000011 < x

   1. Initial program 66.7%

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

   3. Taylor expanded in n around inf 66.8%

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

      1. log1p-define 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in x around -inf 66.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.45:\\ \;\;\;\;\frac{\frac{\frac{n - 0.5 \cdot \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{n}}}{n \cdot \frac{x}{\frac{-0.3333333333333333 + \frac{0.25}{x}}{n}}}}{x} + \frac{-1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.45:\\ \;\;\;\;\frac{\frac{\frac{n \cdot \left(-0.3333333333333333 + \frac{0.25}{x}\right) - 0.5 \cdot \left(x \cdot n\right)}{n \cdot \left(x \cdot n\right)}}{x} + \frac{-1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.45)
   (/
    (+
     (/
      (/
       (- (* n (+ -0.3333333333333333 (/ 0.25 x))) (* 0.5 (* x n)))
       (* n (* x n)))
      x)
     (/ -1.0 n))
    x)
   (/
    (/
     (+ 1.0 (/ (- (/ (+ 0.3333333333333333 (* 0.25 (/ -1.0 x))) x) 0.5) x))
     x)
    n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.45) {
		tmp = (((((n * (-0.3333333333333333 + (0.25 / x))) - (0.5 * (x * n))) / (n * (x * n))) / x) + (-1.0 / n)) / x;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 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) :: tmp
    if (x <= 0.45d0) then
        tmp = (((((n * ((-0.3333333333333333d0) + (0.25d0 / x))) - (0.5d0 * (x * n))) / (n * (x * n))) / x) + ((-1.0d0) / n)) / x
    else
        tmp = ((1.0d0 + ((((0.3333333333333333d0 + (0.25d0 * ((-1.0d0) / x))) / x) - 0.5d0) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.45) {
		tmp = (((((n * (-0.3333333333333333 + (0.25 / x))) - (0.5 * (x * n))) / (n * (x * n))) / x) + (-1.0 / n)) / x;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.45:
		tmp = (((((n * (-0.3333333333333333 + (0.25 / x))) - (0.5 * (x * n))) / (n * (x * n))) / x) + (-1.0 / n)) / x
	else:
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.45)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(n * Float64(-0.3333333333333333 + Float64(0.25 / x))) - Float64(0.5 * Float64(x * n))) / Float64(n * Float64(x * n))) / x) + Float64(-1.0 / n)) / x);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(0.25 * Float64(-1.0 / x))) / x) - 0.5) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.45)
		tmp = (((((n * (-0.3333333333333333 + (0.25 / x))) - (0.5 * (x * n))) / (n * (x * n))) / x) + (-1.0 / n)) / x;
	else
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.45], N[(N[(N[(N[(N[(N[(n * N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(n * N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(0.25 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.450000000000000011

   1. Initial program 48.1%

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

   3. Taylor expanded in n around inf 50.9%

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

      1. log1p-define 50.9%

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

   5. Simplified 50.9%

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

   6. Taylor expanded in x around -inf 0.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.8%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 0.8%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.4%

         \[\leadsto \color{blue}{\sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      2. sqrt-unprod 8.5%

         \[\leadsto \color{blue}{\sqrt{\left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right) \cdot \left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right)}} \]

      3. sqr-neg 8.5%

         \[\leadsto \sqrt{\color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \cdot \frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      4. sqrt-unprod 6.8%

         \[\leadsto \color{blue}{\sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      5. add-sqr-sqrt 30.8%

         \[\leadsto \color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]

   10. Applied egg-rr 30.8%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x}} \]
   11. Step-by-step derivation

       1. associate-/l/ 30.8%

          \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{0.25}{x} - 0.3333333333333333}{x \cdot n}} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x} \]

       2. frac-sub 32.5%

          \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\frac{0.25}{x} - 0.3333333333333333\right) \cdot n - \left(x \cdot n\right) \cdot 0.5}{\left(x \cdot n\right) \cdot n}}}{x} - \frac{1}{n}}{x} \]

       3. sub-neg 32.5%

          \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\frac{0.25}{x} + \left(-0.3333333333333333\right)\right)} \cdot n - \left(x \cdot n\right) \cdot 0.5}{\left(x \cdot n\right) \cdot n}}{x} - \frac{1}{n}}{x} \]

       4. metadata-eval 32.5%

          \[\leadsto \frac{\frac{\frac{\left(\frac{0.25}{x} + \color{blue}{-0.3333333333333333}\right) \cdot n - \left(x \cdot n\right) \cdot 0.5}{\left(x \cdot n\right) \cdot n}}{x} - \frac{1}{n}}{x} \]

   12. Applied egg-rr 32.5%

       \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\frac{0.25}{x} + -0.3333333333333333\right) \cdot n - \left(x \cdot n\right) \cdot 0.5}{\left(x \cdot n\right) \cdot n}}}{x} - \frac{1}{n}}{x} \]

   if 0.450000000000000011 < x

   1. Initial program 66.7%

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

   3. Taylor expanded in n around inf 66.8%

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

      1. log1p-define 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in x around -inf 66.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.45:\\ \;\;\;\;\frac{\frac{\frac{n \cdot \left(-0.3333333333333333 + \frac{0.25}{x}\right) - 0.5 \cdot \left(x \cdot n\right)}{n \cdot \left(x \cdot n\right)}}{x} + \frac{-1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.46:\\ \;\;\;\;\frac{\frac{\frac{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x} - 0.5}{n}}{x} + \frac{-1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.46)
   (/
    (+ (/ (/ (- (/ (+ -0.3333333333333333 (/ 0.25 x)) x) 0.5) n) x) (/ -1.0 n))
    x)
   (/
    (/
     (+ 1.0 (/ (- (/ (+ 0.3333333333333333 (* 0.25 (/ -1.0 x))) x) 0.5) x))
     x)
    n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.46) {
		tmp = ((((((-0.3333333333333333 + (0.25 / x)) / x) - 0.5) / n) / x) + (-1.0 / n)) / x;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 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) :: tmp
    if (x <= 0.46d0) then
        tmp = (((((((-0.3333333333333333d0) + (0.25d0 / x)) / x) - 0.5d0) / n) / x) + ((-1.0d0) / n)) / x
    else
        tmp = ((1.0d0 + ((((0.3333333333333333d0 + (0.25d0 * ((-1.0d0) / x))) / x) - 0.5d0) / x)) / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.46) {
		tmp = ((((((-0.3333333333333333 + (0.25 / x)) / x) - 0.5) / n) / x) + (-1.0 / n)) / x;
	} else {
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.46:
		tmp = ((((((-0.3333333333333333 + (0.25 / x)) / x) - 0.5) / n) / x) + (-1.0 / n)) / x
	else:
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.46)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.3333333333333333 + Float64(0.25 / x)) / x) - 0.5) / n) / x) + Float64(-1.0 / n)) / x);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(0.25 * Float64(-1.0 / x))) / x) - 0.5) / x)) / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.46)
		tmp = ((((((-0.3333333333333333 + (0.25 / x)) / x) - 0.5) / n) / x) + (-1.0 / n)) / x;
	else
		tmp = ((1.0 + ((((0.3333333333333333 + (0.25 * (-1.0 / x))) / x) - 0.5) / x)) / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.46], N[(N[(N[(N[(N[(N[(N[(-0.3333333333333333 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(0.25 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.46000000000000002

   1. Initial program 48.1%

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

   3. Taylor expanded in n around inf 50.9%

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

      1. log1p-define 50.9%

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

   5. Simplified 50.9%

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

   6. Taylor expanded in x around -inf 0.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.8%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 0.8%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.4%

         \[\leadsto \color{blue}{\sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      2. sqrt-unprod 8.5%

         \[\leadsto \color{blue}{\sqrt{\left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right) \cdot \left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right)}} \]

      3. sqr-neg 8.5%

         \[\leadsto \sqrt{\color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \cdot \frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      4. sqrt-unprod 6.8%

         \[\leadsto \color{blue}{\sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      5. add-sqr-sqrt 30.8%

         \[\leadsto \color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]

   10. Applied egg-rr 30.8%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x}} \]

   11. Taylor expanded in n around -inf 30.8%

       \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \frac{0.5 + -1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x}}{n}}}{x} - \frac{1}{n}}{x} \]
   12. Step-by-step derivation

       1. mul-1-neg 30.8%

          \[\leadsto \frac{\frac{\color{blue}{-\frac{0.5 + -1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x}}{n}}}{x} - \frac{1}{n}}{x} \]

       2. distribute-neg-frac2 30.8%

          \[\leadsto \frac{\frac{\color{blue}{\frac{0.5 + -1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x}}{-n}}}{x} - \frac{1}{n}}{x} \]

       3. mul-1-neg 30.8%

          \[\leadsto \frac{\frac{\frac{0.5 + \color{blue}{\left(-\frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x}\right)}}{-n}}{x} - \frac{1}{n}}{x} \]

       4. unsub-neg 30.8%

          \[\leadsto \frac{\frac{\frac{\color{blue}{0.5 - \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x}}}{-n}}{x} - \frac{1}{n}}{x} \]

       5. sub-neg 30.8%

          \[\leadsto \frac{\frac{\frac{0.5 - \frac{\color{blue}{0.25 \cdot \frac{1}{x} + \left(-0.3333333333333333\right)}}{x}}{-n}}{x} - \frac{1}{n}}{x} \]

       6. metadata-eval 30.8%

          \[\leadsto \frac{\frac{\frac{0.5 - \frac{0.25 \cdot \frac{1}{x} + \color{blue}{-0.3333333333333333}}{x}}{-n}}{x} - \frac{1}{n}}{x} \]

       7. +-commutative 30.8%

          \[\leadsto \frac{\frac{\frac{0.5 - \frac{\color{blue}{-0.3333333333333333 + 0.25 \cdot \frac{1}{x}}}{x}}{-n}}{x} - \frac{1}{n}}{x} \]

       8. associate-*r/ 30.8%

          \[\leadsto \frac{\frac{\frac{0.5 - \frac{-0.3333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{x}}}{x}}{-n}}{x} - \frac{1}{n}}{x} \]

       9. metadata-eval 30.8%

          \[\leadsto \frac{\frac{\frac{0.5 - \frac{-0.3333333333333333 + \frac{\color{blue}{0.25}}{x}}{x}}{-n}}{x} - \frac{1}{n}}{x} \]

   13. Simplified 30.8%

       \[\leadsto \frac{\frac{\color{blue}{\frac{0.5 - \frac{-0.3333333333333333 + \frac{0.25}{x}}{x}}{-n}}}{x} - \frac{1}{n}}{x} \]

   if 0.46000000000000002 < x

   1. Initial program 66.7%

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

   3. Taylor expanded in n around inf 66.8%

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

      1. log1p-define 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in x around -inf 66.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{x} - 0.3333333333333333}{x} - 0.5}{x} - 1}{x}}}{n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.46:\\ \;\;\;\;\frac{\frac{\frac{\frac{-0.3333333333333333 + \frac{0.25}{x}}{x} - 0.5}{n}}{x} + \frac{-1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{\frac{0.3333333333333333 + 0.25 \cdot \frac{-1}{x}}{x} - 0.5}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.25}{x \cdot n}}{x} - \frac{0.5}{n}}{x} + \frac{-1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.4e-24)
   (/ (+ (/ (- (/ (/ 0.25 (* x n)) x) (/ 0.5 n)) x) (/ -1.0 n)) x)
   (/ (/ (+ (/ (+ 0.5 (/ -0.3333333333333333 x)) x) -1.0) (- x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.4e-24) {
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x;
	} else {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.4d-24) then
        tmp = (((((0.25d0 / (x * n)) / x) - (0.5d0 / n)) / x) + ((-1.0d0) / n)) / x
    else
        tmp = ((((0.5d0 + ((-0.3333333333333333d0) / x)) / x) + (-1.0d0)) / -x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.4e-24) {
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x;
	} else {
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.4e-24:
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x
	else:
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.4e-24)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.25 / Float64(x * n)) / x) - Float64(0.5 / n)) / x) + Float64(-1.0 / n)) / x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x) + -1.0) / Float64(-x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.4e-24)
		tmp = (((((0.25 / (x * n)) / x) - (0.5 / n)) / x) + (-1.0 / n)) / x;
	else
		tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.4e-24], N[(N[(N[(N[(N[(N[(0.25 / N[(x * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-1.0 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4000000000000001e-24

   1. Initial program 47.3%

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

   3. Taylor expanded in n around inf 51.2%

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

      1. log1p-define 51.2%

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

   5. Simplified 51.2%

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

   6. Taylor expanded in x around -inf 0.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.8%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   8. Simplified 0.8%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.3%

         \[\leadsto \color{blue}{\sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      2. sqrt-unprod 8.2%

         \[\leadsto \color{blue}{\sqrt{\left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right) \cdot \left(-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}\right)}} \]

      3. sqr-neg 8.2%

         \[\leadsto \sqrt{\color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x} \cdot \frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      4. sqrt-unprod 7.1%

         \[\leadsto \color{blue}{\sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \cdot \sqrt{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}}} \]

      5. add-sqr-sqrt 32.9%

         \[\leadsto \color{blue}{\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]

   10. Applied egg-rr 32.9%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x}} \]

   11. Taylor expanded in x around 0 32.9%

       \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{0.25}{n \cdot x}}}{x} - \frac{0.5}{n}}{x} - \frac{1}{n}}{x} \]

   if 1.4000000000000001e-24 < x

   1. Initial program 65.8%

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

   3. Taylor expanded in n around inf 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. Step-by-step derivation

      1. expm1-log1p-u 65.0%

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

   7. Applied egg-rr 65.0%

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

   8. Taylor expanded in x around -inf 60.9%

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

      1. mul-1-neg 60.9%

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

      2. distribute-neg-frac2 60.9%

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

      3. sub-neg 60.9%

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

      4. associate-*r/ 60.9%

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

      5. sub-neg 60.9%

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

      6. metadata-eval 60.9%

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

      7. distribute-lft-in 60.9%

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

      8. neg-mul-1 60.9%

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

      9. associate-*r/ 60.9%

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

      10. metadata-eval 60.9%

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

      11. distribute-neg-frac 60.9%

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

      12. metadata-eval 60.9%

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

      13. metadata-eval 60.9%

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

      14. metadata-eval 60.9%

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

   10. Simplified 60.9%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{\frac{\frac{0.25}{x \cdot n}}{x} - \frac{0.5}{n}}{x} + \frac{-1}{n}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.4% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x + n \cdot \left(\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}\right)}{x \cdot n}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/
  (/
   (+ x (* n (- (/ (/ (- (/ 0.25 x) 0.3333333333333333) n) x) (/ 0.5 n))))
   (* x n))
  x))

C

double code(double x, double n) {
	return ((x + (n * (((((0.25 / x) - 0.3333333333333333) / n) / x) - (0.5 / n)))) / (x * n)) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((x + (n * (((((0.25d0 / x) - 0.3333333333333333d0) / n) / x) - (0.5d0 / n)))) / (x * n)) / x
end function

Java

public static double code(double x, double n) {
	return ((x + (n * (((((0.25 / x) - 0.3333333333333333) / n) / x) - (0.5 / n)))) / (x * n)) / x;
}

Python

def code(x, n):
	return ((x + (n * (((((0.25 / x) - 0.3333333333333333) / n) / x) - (0.5 / n)))) / (x * n)) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(x + Float64(n * Float64(Float64(Float64(Float64(Float64(0.25 / x) - 0.3333333333333333) / n) / x) - Float64(0.5 / n)))) / Float64(x * n)) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((x + (n * (((((0.25 / x) - 0.3333333333333333) / n) / x) - (0.5 / n)))) / (x * n)) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(x + N[(n * N[(N[(N[(N[(N[(0.25 / x), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in n around inf 57.8%

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

   1. log1p-define 57.8%

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

5. Simplified 57.8%

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

6. Taylor expanded in x around -inf 29.4%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
7. Step-by-step derivation

   1. mul-1-neg 29.4%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

8. Simplified 29.4%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]
9. Step-by-step derivation

   1. distribute-neg-frac 29.4%

      \[\leadsto -\frac{\color{blue}{\frac{-\left(\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}\right)}{x}} - \frac{1}{n}}{x} \]

   2. frac-sub 28.3%

      \[\leadsto -\frac{\color{blue}{\frac{\left(-\left(\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}\right)\right) \cdot n - x \cdot 1}{x \cdot n}}}{x} \]

10. Applied egg-rr 45.2%

    \[\leadsto -\frac{\color{blue}{\frac{\left(-\left(\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}\right)\right) \cdot n - x \cdot 1}{x \cdot n}}}{x} \]

11. Final simplification 45.2%

    \[\leadsto \frac{\frac{x + n \cdot \left(\frac{\frac{\frac{0.25}{x} - 0.3333333333333333}{n}}{x} - \frac{0.5}{n}\right)}{x \cdot n}}{x} \]
12. Add Preprocessing

Alternative 17: 47.2% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n} - \frac{\frac{0.5}{n} - \frac{0.3333333333333333}{x \cdot n}}{x}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (- (/ 1.0 n) (/ (- (/ 0.5 n) (/ 0.3333333333333333 (* x n))) x)) x))

C

double code(double x, double n) {
	return ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (x * n))) / x)) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 / n) - (((0.5d0 / n) - (0.3333333333333333d0 / (x * n))) / x)) / x
end function

Java

public static double code(double x, double n) {
	return ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (x * n))) / x)) / x;
}

Python

def code(x, n):
	return ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (x * n))) / x)) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 / n) - Float64(Float64(Float64(0.5 / n) - Float64(0.3333333333333333 / Float64(x * n))) / x)) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 / n) - (((0.5 / n) - (0.3333333333333333 / (x * n))) / x)) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 / n), $MachinePrecision] - N[(N[(N[(0.5 / n), $MachinePrecision] - N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in n around inf 57.8%

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

   1. log1p-define 57.8%

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

5. Simplified 57.8%

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

6. Taylor expanded in x around -inf 29.4%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
7. Step-by-step derivation

   1. mul-1-neg 29.4%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{n \cdot x} - 0.3333333333333333 \cdot \frac{1}{n}}{x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

8. Simplified 29.4%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{0.25}{x \cdot n} - \frac{0.3333333333333333}{n}}{x}\right) - \frac{0.5}{n}}{x}\right) - \frac{1}{n}}{x}} \]

9. Taylor expanded in x around inf 44.2%

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

    1. associate-*r/ 44.2%

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

    2. metadata-eval 44.2%

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

    3. associate-*r/ 44.2%

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

    4. metadata-eval 44.2%

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

    5. *-commutative 44.2%

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

11. Simplified 44.2%

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

12. Final simplification 44.2%

    \[\leadsto \frac{\frac{1}{n} - \frac{\frac{0.5}{n} - \frac{0.3333333333333333}{x \cdot n}}{x}}{x} \]
13. Add Preprocessing

Alternative 18: 47.2% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (/ (+ (/ (+ 0.5 (/ -0.3333333333333333 x)) x) -1.0) (- x)) n))

C

double code(double x, double n) {
	return ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((((0.5d0 + ((-0.3333333333333333d0) / x)) / x) + (-1.0d0)) / -x) / n
end function

Java

public static double code(double x, double n) {
	return ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
}

Python

def code(x, n):
	return ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n

Julia

function code(x, n)
	return Float64(Float64(Float64(Float64(Float64(0.5 + Float64(-0.3333333333333333 / x)) / x) + -1.0) / Float64(-x)) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = ((((0.5 + (-0.3333333333333333 / x)) / x) + -1.0) / -x) / n;
end

Wolfram

code[x_, n_] := N[(N[(N[(N[(N[(0.5 + N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + -1.0), $MachinePrecision] / (-x)), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in n around inf 57.8%

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

   1. log1p-define 57.8%

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

5. Simplified 57.8%

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

   1. expm1-log1p-u 57.5%

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

7. Applied egg-rr 57.5%

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

8. Taylor expanded in x around -inf 44.2%

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

   1. mul-1-neg 44.2%

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

   2. distribute-neg-frac2 44.2%

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

   3. sub-neg 44.2%

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

   4. associate-*r/ 44.2%

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

   5. sub-neg 44.2%

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

   6. metadata-eval 44.2%

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

   7. distribute-lft-in 44.2%

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

   8. neg-mul-1 44.2%

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

   9. associate-*r/ 44.2%

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

   10. metadata-eval 44.2%

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

   11. distribute-neg-frac 44.2%

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

   12. metadata-eval 44.2%

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

   13. metadata-eval 44.2%

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

   14. metadata-eval 44.2%

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

10. Simplified 44.2%

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

11. Final simplification 44.2%

    \[\leadsto \frac{\frac{\frac{0.5 + \frac{-0.3333333333333333}{x}}{x} + -1}{-x}}{n} \]
12. Add Preprocessing

Alternative 19: 41.0% 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 56.2%

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

3. Taylor expanded in n around inf 57.8%

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

   1. log1p-define 57.8%

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

5. Simplified 57.8%

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

6. Taylor expanded in x around inf 40.2%

   \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
7. Add Preprocessing

Alternative 20: 41.0% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in x around inf 36.3%

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

5. Simplified 36.2%

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

6. Taylor expanded in x around inf 61.0%

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

   1. mul-1-neg 61.0%

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

   2. log-rec 61.0%

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

   3. distribute-neg-frac 61.0%

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

   4. remove-double-neg 61.0%

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

8. Simplified 61.0%

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

9. Taylor expanded in n around inf 40.1%

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

Alternative 21: 40.4% 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 56.2%

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

3. Taylor expanded in n around inf 57.8%

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

   1. log1p-define 57.8%

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

5. Simplified 57.8%

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

6. Taylor expanded in x around inf 39.1%

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

   1. *-commutative 39.1%

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

8. Simplified 39.1%

   \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))