2nthrt (problem 3.4.6)

Percentage Accurate: 53.4% → 90.9%

Time: 39.9s

Alternatives: 24

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: 53.4% 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: 90.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 330:\\ \;\;\;\;\frac{\log \left(\frac{x}{e^{\mathsf{log1p}\left(x\right) + \frac{\mathsf{fma}\left(0.5, {\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}, 0.16666666666666666 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{3} - {\log x}^{3}}{n}\right)}{n}}}\right)}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 330.0)
   (/
    (log
     (/
      x
      (exp
       (+
        (log1p x)
        (/
         (fma
          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)))))
    (- n))
   (/ (pow x (/ 1.0 n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 330.0) {
		tmp = log((x / exp((log1p(x) + (fma(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))))) / -n;
	} else {
		tmp = pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 330.0)
		tmp = Float64(log(Float64(x / exp(Float64(log1p(x) + Float64(fma(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))))) / Float64(-n));
	else
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 330.0], N[(N[Log[N[(x / N[Exp[N[(N[Log[1 + x], $MachinePrecision] + 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] + 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]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 330

   1. Initial program 48.4%

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

   3. Taylor expanded in n around -inf 74.9%

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

   5. Simplified 74.9%

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

      1. add-log-exp 83.2%

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

      2. diff-log 83.2%

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

      3. fma-define 83.2%

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

   7. Applied egg-rr 83.2%

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

   if 330 < x

   1. Initial program 68.6%

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

   3. Taylor expanded in x around inf 98.5%

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

      1. mul-1-neg 98.5%

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

      2. log-rec 98.5%

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

      3. mul-1-neg 98.5%

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

      4. distribute-neg-frac 98.5%

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

      5. mul-1-neg 98.5%

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

      6. remove-double-neg 98.5%

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

      7. *-rgt-identity 98.5%

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

      8. associate-/l* 98.5%

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

      9. exp-to-pow 98.5%

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

      10. *-commutative 98.5%

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

   5. Simplified 98.5%

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

Alternative 2: 85.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(x + 1\right)}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\frac{1}{n} + -0.5 \cdot \frac{x}{n}\right)} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (- (pow (+ x 1.0) (/ 1.0 n)) t_0) 2e-8)
     (/
      (log
       (/
        (*
         (sqrt (exp (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n)))
         (+ x 1.0))
        x))
      n)
     (- (exp (* x (+ (/ 1.0 n) (* -0.5 (/ x n))))) t_0))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((pow((x + 1.0), (1.0 / n)) - t_0) <= 2e-8) {
		tmp = log(((sqrt(exp(((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n))) * (x + 1.0)) / x)) / n;
	} else {
		tmp = exp((x * ((1.0 / n) + (-0.5 * (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 ((Math.pow((x + 1.0), (1.0 / n)) - t_0) <= 2e-8) {
		tmp = Math.log(((Math.sqrt(Math.exp(((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) / n))) * (x + 1.0)) / x)) / n;
	} else {
		tmp = Math.exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (math.pow((x + 1.0), (1.0 / n)) - t_0) <= 2e-8:
		tmp = math.log(((math.sqrt(math.exp(((math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)) / n))) * (x + 1.0)) / x)) / n
	else:
		tmp = math.exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - t_0) <= 2e-8)
		tmp = Float64(log(Float64(Float64(sqrt(exp(Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n))) * Float64(x + 1.0)) / x)) / n);
	else
		tmp = Float64(exp(Float64(x * Float64(Float64(1.0 / n) + Float64(-0.5 * Float64(x / n))))) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], 2e-8], N[(N[Log[N[(N[(N[Sqrt[N[Exp[N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(-0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n))) < 2e-8

   1. Initial program 53.3%

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

   3. Taylor expanded in n around inf 77.1%

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

      1. associate--l+ 62.2%

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

      2. log1p-define 62.2%

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

      3. +-commutative 62.2%

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

      4. associate--r+ 77.1%

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

      5. distribute-lft-out-- 77.1%

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

      6. div-sub 78.1%

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

      7. log1p-define 78.1%

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

   5. Simplified 78.1%

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

      1. add-log-exp 85.5%

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

      2. associate-+r- 85.5%

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

      3. exp-diff 85.5%

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

      4. add-exp-log 57.3%

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

   7. Applied egg-rr 57.3%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(x\right) + 0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}}{x}\right)}}{n} \]
   8. Step-by-step derivation

      1. +-commutative 57.3%

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

      2. exp-sum 57.3%

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

      3. *-commutative 57.3%

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

      4. exp-prod 57.3%

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

      5. unpow1/2 57.3%

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

      6. log1p-define 57.3%

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

      7. rem-exp-log 85.6%

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

   9. Simplified 85.6%

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

   if 2e-8 < (-.f64 (pow.f64 (+.f64 x #s(literal 1 binary64)) (/.f64 #s(literal 1 binary64) n)) (pow.f64 x (/.f64 #s(literal 1 binary64) n)))

   1. Initial program 71.2%

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

   3. Taylor expanded in n around 0 71.2%

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

      1. log1p-define 97.5%

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

      2. *-rgt-identity 97.5%

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

      3. associate-/l* 97.5%

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

      4. exp-to-pow 97.5%

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

   5. Simplified 97.5%

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

   6. Taylor expanded in x around 0 97.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\log \left(\frac{\sqrt{e^{\frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}} \cdot \left(x + 1\right)}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\frac{1}{n} + -0.5 \cdot \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\frac{1}{n} + -0.5 \cdot \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -40.0)
   (-
    (pow (+ x 1.0) (/ 1.0 n))
    (* (pow (pow (cbrt x) 2.0) (/ 1.0 n)) (pow (cbrt x) (/ 1.0 n))))
   (if (<= (/ 1.0 n) 5e-11)
     (/
      (+
       (log1p x)
       (- (* 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n)) (log x)))
      n)
     (- (exp (* x (+ (/ 1.0 n) (* -0.5 (/ x n))))) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -40.0) {
		tmp = pow((x + 1.0), (1.0 / n)) - (pow(pow(cbrt(x), 2.0), (1.0 / n)) * pow(cbrt(x), (1.0 / n)));
	} else if ((1.0 / n) <= 5e-11) {
		tmp = (log1p(x) + ((0.5 * ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n)) - log(x))) / n;
	} else {
		tmp = exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -40.0) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - (Math.pow(Math.pow(Math.cbrt(x), 2.0), (1.0 / n)) * Math.pow(Math.cbrt(x), (1.0 / n)));
	} else if ((1.0 / n) <= 5e-11) {
		tmp = (Math.log1p(x) + ((0.5 * ((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) / n)) - Math.log(x))) / n;
	} else {
		tmp = Math.exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40.0)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - Float64(((cbrt(x) ^ 2.0) ^ Float64(1.0 / n)) * (cbrt(x) ^ Float64(1.0 / n))));
	elseif (Float64(1.0 / n) <= 5e-11)
		tmp = Float64(Float64(log1p(x) + Float64(Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n)) - log(x))) / n);
	else
		tmp = Float64(exp(Float64(x * Float64(Float64(1.0 / n) + Float64(-0.5 * Float64(x / n))))) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -40.0], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[(N[Power[N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[x, 1/3], $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-11], N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(-0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -40

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.9%

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

      2. unpow-prod-down 100.0%

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

      3. pow2 100.0%

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

   4. Applied egg-rr 100.0%

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

   if -40 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000018e-11

   1. Initial program 27.5%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. associate--l+ 78.9%

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

      2. log1p-define 78.9%

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

      3. +-commutative 78.9%

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

      4. associate--r+ 78.9%

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

      5. distribute-lft-out-- 78.9%

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

      6. div-sub 78.9%

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

      7. log1p-define 78.9%

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

   5. Simplified 78.9%

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

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

   1. Initial program 71.2%

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

   3. Taylor expanded in n around 0 71.2%

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

      1. log1p-define 97.5%

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

      2. *-rgt-identity 97.5%

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

      3. associate-/l* 97.5%

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

      4. exp-to-pow 97.5%

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

   5. Simplified 97.5%

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

   6. Taylor expanded in x around 0 97.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\frac{1}{n} + -0.5 \cdot \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -40:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - t\_0 \cdot t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\frac{1}{n} + -0.5 \cdot \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow (sqrt x) (/ 1.0 n))))
   (if (<= (/ 1.0 n) -40.0)
     (- (pow (+ x 1.0) (/ 1.0 n)) (* t_0 t_0))
     (if (<= (/ 1.0 n) 5e-11)
       (/
        (+
         (log1p x)
         (- (* 0.5 (/ (- (pow (log1p x) 2.0) (pow (log x) 2.0)) n)) (log x)))
        n)
       (- (exp (* x (+ (/ 1.0 n) (* -0.5 (/ x n))))) (pow x (/ 1.0 n)))))))

C

double code(double x, double n) {
	double t_0 = pow(sqrt(x), (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -40.0) {
		tmp = pow((x + 1.0), (1.0 / n)) - (t_0 * t_0);
	} else if ((1.0 / n) <= 5e-11) {
		tmp = (log1p(x) + ((0.5 * ((pow(log1p(x), 2.0) - pow(log(x), 2.0)) / n)) - log(x))) / n;
	} else {
		tmp = exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(Math.sqrt(x), (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -40.0) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - (t_0 * t_0);
	} else if ((1.0 / n) <= 5e-11) {
		tmp = (Math.log1p(x) + ((0.5 * ((Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0)) / n)) - Math.log(x))) / n;
	} else {
		tmp = Math.exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(math.sqrt(x), (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -40.0:
		tmp = math.pow((x + 1.0), (1.0 / n)) - (t_0 * t_0)
	elif (1.0 / n) <= 5e-11:
		tmp = (math.log1p(x) + ((0.5 * ((math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0)) / n)) - math.log(x))) / n
	else:
		tmp = math.exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = sqrt(x) ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40.0)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - Float64(t_0 * t_0));
	elseif (Float64(1.0 / n) <= 5e-11)
		tmp = Float64(Float64(log1p(x) + Float64(Float64(0.5 * Float64(Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0)) / n)) - log(x))) / n);
	else
		tmp = Float64(exp(Float64(x * Float64(Float64(1.0 / n) + Float64(-0.5 * Float64(x / n))))) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[N[Sqrt[x], $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -40.0], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-11], N[(N[(N[Log[1 + x], $MachinePrecision] + N[(N[(0.5 * N[(N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(-0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -40

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. unpow-prod-down 99.9%

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

   4. Applied egg-rr 99.9%

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

   if -40 < (/.f64 #s(literal 1 binary64) n) < 5.00000000000000018e-11

   1. Initial program 27.5%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. associate--l+ 78.9%

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

      2. log1p-define 78.9%

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

      3. +-commutative 78.9%

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

      4. associate--r+ 78.9%

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

      5. distribute-lft-out-- 78.9%

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

      6. div-sub 78.9%

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

      7. log1p-define 78.9%

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

   5. Simplified 78.9%

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

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

   1. Initial program 71.2%

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

   3. Taylor expanded in n around 0 71.2%

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

      1. log1p-define 97.5%

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

      2. *-rgt-identity 97.5%

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

      3. associate-/l* 97.5%

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

      4. exp-to-pow 97.5%

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

   5. Simplified 97.5%

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

   6. Taylor expanded in x around 0 97.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) + \left(0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n} - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\frac{1}{n} + -0.5 \cdot \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-12}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{2}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\left(1 + \frac{\frac{\log x}{n}}{x}\right) + \frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{x \cdot \left(\frac{1}{n} + -0.5 \cdot \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-12)
   (- (pow (+ x 1.0) (/ 1.0 n)) (pow (sqrt x) (/ 2.0 n)))
   (if (<= (/ 1.0 n) 5e-12)
     (/ (log (+ (+ 1.0 (/ (/ (log x) n) x)) (/ 1.0 x))) n)
     (log
      (exp
       (- (exp (* x (+ (/ 1.0 n) (* -0.5 (/ x n))))) (pow x (/ 1.0 n))))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-12) {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(sqrt(x), (2.0 / n));
	} else if ((1.0 / n) <= 5e-12) {
		tmp = log(((1.0 + ((log(x) / n) / x)) + (1.0 / x))) / n;
	} else {
		tmp = log(exp((exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - pow(x, (1.0 / n)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d-12)) then
        tmp = ((x + 1.0d0) ** (1.0d0 / n)) - (sqrt(x) ** (2.0d0 / n))
    else if ((1.0d0 / n) <= 5d-12) then
        tmp = log(((1.0d0 + ((log(x) / n) / x)) + (1.0d0 / x))) / n
    else
        tmp = log(exp((exp((x * ((1.0d0 / n) + ((-0.5d0) * (x / n))))) - (x ** (1.0d0 / n)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-12) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(Math.sqrt(x), (2.0 / n));
	} else if ((1.0 / n) <= 5e-12) {
		tmp = Math.log(((1.0 + ((Math.log(x) / n) / x)) + (1.0 / x))) / n;
	} else {
		tmp = Math.log(Math.exp((Math.exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - Math.pow(x, (1.0 / n)))));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-12:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(math.sqrt(x), (2.0 / n))
	elif (1.0 / n) <= 5e-12:
		tmp = math.log(((1.0 + ((math.log(x) / n) / x)) + (1.0 / x))) / n
	else:
		tmp = math.log(math.exp((math.exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - math.pow(x, (1.0 / n)))))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-12)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (sqrt(x) ^ Float64(2.0 / n)));
	elseif (Float64(1.0 / n) <= 5e-12)
		tmp = Float64(log(Float64(Float64(1.0 + Float64(Float64(log(x) / n) / x)) + Float64(1.0 / x))) / n);
	else
		tmp = log(exp(Float64(exp(Float64(x * Float64(Float64(1.0 / n) + Float64(-0.5 * Float64(x / n))))) - (x ^ Float64(1.0 / n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-12)
		tmp = ((x + 1.0) ^ (1.0 / n)) - (sqrt(x) ^ (2.0 / n));
	elseif ((1.0 / n) <= 5e-12)
		tmp = log(((1.0 + ((log(x) / n) / x)) + (1.0 / x))) / n;
	else
		tmp = log(exp((exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - (x ^ (1.0 / n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-12], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[N[Sqrt[x], $MachinePrecision], N[(2.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-12], N[(N[Log[N[(N[(1.0 + N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[Log[N[Exp[N[(N[Exp[N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(-0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999996e-12

   1. Initial program 99.2%

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

      1. add-sqr-sqrt 99.3%

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

      2. unpow-prod-down 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. pow-sqr 99.3%

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

      2. associate-*r/ 99.3%

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

      3. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

   if -1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 26.5%

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

   3. Taylor expanded in n around inf 78.6%

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

      1. associate--l+ 78.6%

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

      2. log1p-define 78.6%

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

      3. +-commutative 78.6%

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

      4. associate--r+ 78.6%

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

      5. distribute-lft-out-- 78.6%

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

      6. div-sub 78.6%

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

      7. log1p-define 78.6%

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

   5. Simplified 78.6%

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

      1. add-log-exp 78.6%

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

      2. associate-+r- 78.6%

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

      3. exp-diff 78.5%

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

      4. add-exp-log 58.8%

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

   7. Applied egg-rr 58.8%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(x\right) + 0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}}{x}\right)}}{n} \]
   8. Step-by-step derivation

      1. +-commutative 58.8%

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

      2. exp-sum 58.8%

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

      3. *-commutative 58.8%

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

      4. exp-prod 58.8%

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

      5. unpow1/2 58.8%

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

      6. log1p-define 58.8%

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

      7. rem-exp-log 78.6%

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

   9. Simplified 78.6%

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

   10. Taylor expanded in x around inf 78.4%

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

       1. associate-+r+ 78.4%

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

       2. mul-1-neg 78.4%

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

       3. associate-/r* 78.4%

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

       4. distribute-neg-frac 78.4%

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

       5. log-rec 78.4%

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

       6. distribute-neg-frac 78.4%

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

       7. remove-double-neg 78.4%

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

   12. Simplified 78.4%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 71.0%

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

      1. add-log-exp 71.2%

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

      2. pow-to-exp 71.2%

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

      3. un-div-inv 71.2%

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

      4. +-commutative 71.2%

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

      5. log1p-define 96.8%

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

   4. Applied egg-rr 96.8%

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

   5. Taylor expanded in x around 0 96.8%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-12}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{2}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\left(1 + \frac{\frac{\log x}{n}}{x}\right) + \frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{x \cdot \left(\frac{1}{n} + -0.5 \cdot \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-12}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{2}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\left(1 + \frac{\frac{\log x}{n}}{x}\right) + \frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\frac{1}{n} + -0.5 \cdot \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -2e-12)
   (- (pow (+ x 1.0) (/ 1.0 n)) (pow (sqrt x) (/ 2.0 n)))
   (if (<= (/ 1.0 n) 5e-12)
     (/ (log (+ (+ 1.0 (/ (/ (log x) n) x)) (/ 1.0 x))) n)
     (- (exp (* x (+ (/ 1.0 n) (* -0.5 (/ x n))))) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-12) {
		tmp = pow((x + 1.0), (1.0 / n)) - pow(sqrt(x), (2.0 / n));
	} else if ((1.0 / n) <= 5e-12) {
		tmp = log(((1.0 + ((log(x) / n) / x)) + (1.0 / x))) / n;
	} else {
		tmp = exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - pow(x, (1.0 / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-2d-12)) then
        tmp = ((x + 1.0d0) ** (1.0d0 / n)) - (sqrt(x) ** (2.0d0 / n))
    else if ((1.0d0 / n) <= 5d-12) then
        tmp = log(((1.0d0 + ((log(x) / n) / x)) + (1.0d0 / x))) / n
    else
        tmp = exp((x * ((1.0d0 / n) + ((-0.5d0) * (x / n))))) - (x ** (1.0d0 / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -2e-12) {
		tmp = Math.pow((x + 1.0), (1.0 / n)) - Math.pow(Math.sqrt(x), (2.0 / n));
	} else if ((1.0 / n) <= 5e-12) {
		tmp = Math.log(((1.0 + ((Math.log(x) / n) / x)) + (1.0 / x))) / n;
	} else {
		tmp = Math.exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -2e-12:
		tmp = math.pow((x + 1.0), (1.0 / n)) - math.pow(math.sqrt(x), (2.0 / n))
	elif (1.0 / n) <= 5e-12:
		tmp = math.log(((1.0 + ((math.log(x) / n) / x)) + (1.0 / x))) / n
	else:
		tmp = math.exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -2e-12)
		tmp = Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (sqrt(x) ^ Float64(2.0 / n)));
	elseif (Float64(1.0 / n) <= 5e-12)
		tmp = Float64(log(Float64(Float64(1.0 + Float64(Float64(log(x) / n) / x)) + Float64(1.0 / x))) / n);
	else
		tmp = Float64(exp(Float64(x * Float64(Float64(1.0 / n) + Float64(-0.5 * Float64(x / n))))) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -2e-12)
		tmp = ((x + 1.0) ^ (1.0 / n)) - (sqrt(x) ^ (2.0 / n));
	elseif ((1.0 / n) <= 5e-12)
		tmp = log(((1.0 + ((log(x) / n) / x)) + (1.0 / x))) / n;
	else
		tmp = exp((x * ((1.0 / n) + (-0.5 * (x / n))))) - (x ^ (1.0 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-12], N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[N[Sqrt[x], $MachinePrecision], N[(2.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-12], N[(N[Log[N[(N[(1.0 + N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(-0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999996e-12

   1. Initial program 99.2%

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

      1. add-sqr-sqrt 99.3%

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

      2. unpow-prod-down 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. pow-sqr 99.3%

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

      2. associate-*r/ 99.3%

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

      3. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

   if -1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 26.5%

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

   3. Taylor expanded in n around inf 78.6%

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

      1. associate--l+ 78.6%

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

      2. log1p-define 78.6%

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

      3. +-commutative 78.6%

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

      4. associate--r+ 78.6%

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

      5. distribute-lft-out-- 78.6%

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

      6. div-sub 78.6%

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

      7. log1p-define 78.6%

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

   5. Simplified 78.6%

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

      1. add-log-exp 78.6%

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

      2. associate-+r- 78.6%

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

      3. exp-diff 78.5%

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

      4. add-exp-log 58.8%

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

   7. Applied egg-rr 58.8%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(x\right) + 0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}}{x}\right)}}{n} \]
   8. Step-by-step derivation

      1. +-commutative 58.8%

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

      2. exp-sum 58.8%

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

      3. *-commutative 58.8%

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

      4. exp-prod 58.8%

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

      5. unpow1/2 58.8%

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

      6. log1p-define 58.8%

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

      7. rem-exp-log 78.6%

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

   9. Simplified 78.6%

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

   10. Taylor expanded in x around inf 78.4%

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

       1. associate-+r+ 78.4%

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

       2. mul-1-neg 78.4%

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

       3. associate-/r* 78.4%

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

       4. distribute-neg-frac 78.4%

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

       5. log-rec 78.4%

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

       6. distribute-neg-frac 78.4%

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

       7. remove-double-neg 78.4%

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

   12. Simplified 78.4%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 71.0%

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

   3. Taylor expanded in n around 0 71.0%

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

      1. log1p-define 96.6%

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

      2. *-rgt-identity 96.6%

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

      3. associate-/l* 96.6%

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

      4. exp-to-pow 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in x around 0 96.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-12}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{2}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\left(1 + \frac{\frac{\log x}{n}}{x}\right) + \frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\frac{1}{n} + -0.5 \cdot \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-12}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\left(1 + \frac{\frac{\log x}{n}}{x}\right) + \frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\frac{1}{n} + -0.5 \cdot \frac{x}{n}\right)} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -2e-12)
     (- (exp (/ x n)) t_0)
     (if (<= (/ 1.0 n) 5e-12)
       (/ (log (+ (+ 1.0 (/ (/ (log x) n) x)) (/ 1.0 x))) n)
       (- (exp (* x (+ (/ 1.0 n) (* -0.5 (/ 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) <= -2e-12) {
		tmp = exp((x / n)) - t_0;
	} else if ((1.0 / n) <= 5e-12) {
		tmp = log(((1.0 + ((log(x) / n) / x)) + (1.0 / x))) / n;
	} else {
		tmp = exp((x * ((1.0 / n) + (-0.5 * (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) <= (-2d-12)) then
        tmp = exp((x / n)) - t_0
    else if ((1.0d0 / n) <= 5d-12) then
        tmp = log(((1.0d0 + ((log(x) / n) / x)) + (1.0d0 / x))) / n
    else
        tmp = exp((x * ((1.0d0 / n) + ((-0.5d0) * (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) <= -2e-12) {
		tmp = Math.exp((x / n)) - t_0;
	} else if ((1.0 / n) <= 5e-12) {
		tmp = Math.log(((1.0 + ((Math.log(x) / n) / x)) + (1.0 / x))) / n;
	} else {
		tmp = Math.exp((x * ((1.0 / n) + (-0.5 * (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) <= -2e-12:
		tmp = math.exp((x / n)) - t_0
	elif (1.0 / n) <= 5e-12:
		tmp = math.log(((1.0 + ((math.log(x) / n) / x)) + (1.0 / x))) / n
	else:
		tmp = math.exp((x * ((1.0 / n) + (-0.5 * (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) <= -2e-12)
		tmp = Float64(exp(Float64(x / n)) - t_0);
	elseif (Float64(1.0 / n) <= 5e-12)
		tmp = Float64(log(Float64(Float64(1.0 + Float64(Float64(log(x) / n) / x)) + Float64(1.0 / x))) / n);
	else
		tmp = Float64(exp(Float64(x * Float64(Float64(1.0 / n) + Float64(-0.5 * 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) <= -2e-12)
		tmp = exp((x / n)) - t_0;
	elseif ((1.0 / n) <= 5e-12)
		tmp = log(((1.0 + ((log(x) / n) / x)) + (1.0 / x))) / n;
	else
		tmp = exp((x * ((1.0 / n) + (-0.5 * (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], -2e-12], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-12], N[(N[Log[N[(N[(1.0 + N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(-0.5 * N[(x / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -1.99999999999999996e-12

   1. Initial program 99.2%

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

   3. Taylor expanded in n around 0 99.3%

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

      1. log1p-define 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. associate-/l* 99.3%

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

      4. exp-to-pow 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in x around 0 99.2%

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

   if -1.99999999999999996e-12 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 26.5%

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

   3. Taylor expanded in n around inf 78.6%

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

      1. associate--l+ 78.6%

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

      2. log1p-define 78.6%

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

      3. +-commutative 78.6%

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

      4. associate--r+ 78.6%

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

      5. distribute-lft-out-- 78.6%

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

      6. div-sub 78.6%

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

      7. log1p-define 78.6%

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

   5. Simplified 78.6%

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

      1. add-log-exp 78.6%

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

      2. associate-+r- 78.6%

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

      3. exp-diff 78.5%

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

      4. add-exp-log 58.8%

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

   7. Applied egg-rr 58.8%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(x\right) + 0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}}{x}\right)}}{n} \]
   8. Step-by-step derivation

      1. +-commutative 58.8%

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

      2. exp-sum 58.8%

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

      3. *-commutative 58.8%

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

      4. exp-prod 58.8%

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

      5. unpow1/2 58.8%

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

      6. log1p-define 58.8%

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

      7. rem-exp-log 78.6%

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

   9. Simplified 78.6%

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

   10. Taylor expanded in x around inf 78.4%

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

       1. associate-+r+ 78.4%

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

       2. mul-1-neg 78.4%

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

       3. associate-/r* 78.4%

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

       4. distribute-neg-frac 78.4%

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

       5. log-rec 78.4%

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

       6. distribute-neg-frac 78.4%

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

       7. remove-double-neg 78.4%

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

   12. Simplified 78.4%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 71.0%

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

   3. Taylor expanded in n around 0 71.0%

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

      1. log1p-define 96.6%

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

      2. *-rgt-identity 96.6%

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

      3. associate-/l* 96.6%

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

      4. exp-to-pow 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in x around 0 96.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -2 \cdot 10^{-12}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\left(1 + \frac{\frac{\log x}{n}}{x}\right) + \frac{1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\frac{1}{n} + -0.5 \cdot \frac{x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -40:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+178}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -40.0)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-14)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+178)
         (- (+ 1.0 (/ x n)) t_0)
         (log1p (expm1 (/ x n))))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -40.0) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-14) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+178) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = log1p(expm1((x / n)));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -40.0) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-14) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+178) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = Math.log1p(Math.expm1((x / n)));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -40.0:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 2e-14:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+178:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = math.log1p(math.expm1((x / n)))
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40.0)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-14)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+178)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = log1p(expm1(Float64(x / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -40.0], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-14], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+178], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[Log[1 + N[(Exp[N[(x / n), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -40

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-rgt-identity 98.9%

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

      8. associate-/l* 98.9%

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

      9. exp-to-pow 98.9%

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

      10. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   if -40 < (/.f64 #s(literal 1 binary64) n) < 2e-14

   1. Initial program 27.4%

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

   3. Taylor expanded in n around inf 78.6%

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

      1. log1p-define 78.6%

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

   5. Simplified 78.6%

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

      1. log1p-undefine 78.6%

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

      2. diff-log 78.6%

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

   7. Applied egg-rr 78.6%

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

   if 2e-14 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e178

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0 77.0%

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

   if 4.9999999999999999e178 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.1%

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

   3. Taylor expanded in n around inf 6.7%

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

      1. log1p-define 6.7%

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

   5. Simplified 6.7%

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

   6. Taylor expanded in x around inf 48.4%

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

      1. *-commutative 48.4%

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

   8. Simplified 48.4%

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

      1. log1p-expm1-u 73.6%

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

      2. associate-/r* 73.6%

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

      3. add-exp-log 73.6%

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

      4. rec-exp 73.6%

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

      5. add-sqr-sqrt 73.6%

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

      6. sqrt-unprod 73.6%

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

      7. sqr-neg 73.6%

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

      8. sqrt-prod 0.0%

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

      9. add-sqr-sqrt 75.7%

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

      10. add-exp-log 75.7%

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

   10. Applied egg-rr 75.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+178}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{n}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -380000000000 \lor \neg \left(n \leq 23000000000\right):\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (or (<= n -380000000000.0) (not (<= n 23000000000.0)))
   (/ (log (/ (+ x 1.0) x)) n)
   (- (exp (/ x n)) (pow x (/ 1.0 n)))))

C

double code(double x, double n) {
	double tmp;
	if ((n <= -380000000000.0) || !(n <= 23000000000.0)) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-380000000000.0d0)) .or. (.not. (n <= 23000000000.0d0))) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = exp((x / n)) - (x ** (1.0d0 / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((n <= -380000000000.0) || !(n <= 23000000000.0)) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (n <= -380000000000.0) or not (n <= 23000000000.0):
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if ((n <= -380000000000.0) || !(n <= 23000000000.0))
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((n <= -380000000000.0) || ~((n <= 23000000000.0)))
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = exp((x / n)) - (x ^ (1.0 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[Or[LessEqual[n, -380000000000.0], N[Not[LessEqual[n, 23000000000.0]], $MachinePrecision]], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -3.8e11 or 2.3e10 < n

   1. Initial program 26.5%

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

   3. Taylor expanded in n around inf 78.3%

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

      1. log1p-define 78.3%

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

   5. Simplified 78.3%

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

      1. log1p-undefine 78.3%

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

      2. diff-log 78.4%

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

   7. Applied egg-rr 78.4%

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

   if -3.8e11 < n < 2.3e10

   1. Initial program 89.6%

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

   3. Taylor expanded in n around 0 89.6%

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

      1. log1p-define 98.4%

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

      2. *-rgt-identity 98.4%

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

      3. associate-/l* 98.4%

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

      4. exp-to-pow 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in x around 0 98.3%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -380000000000 \lor \neg \left(n \leq 23000000000\right):\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -380000000000:\\ \;\;\;\;\frac{\log \left(\left(1 + \frac{\frac{\log x}{n}}{x}\right) + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq 26000000000:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= n -380000000000.0)
   (/ (log (+ (+ 1.0 (/ (/ (log x) n) x)) (/ 1.0 x))) n)
   (if (<= n 26000000000.0)
     (- (exp (/ x n)) (pow x (/ 1.0 n)))
     (/ (log (/ (+ x 1.0) x)) n))))

C

double code(double x, double n) {
	double tmp;
	if (n <= -380000000000.0) {
		tmp = log(((1.0 + ((log(x) / n) / x)) + (1.0 / x))) / n;
	} else if (n <= 26000000000.0) {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = log(((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 (n <= (-380000000000.0d0)) then
        tmp = log(((1.0d0 + ((log(x) / n) / x)) + (1.0d0 / x))) / n
    else if (n <= 26000000000.0d0) then
        tmp = exp((x / n)) - (x ** (1.0d0 / n))
    else
        tmp = log(((x + 1.0d0) / x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (n <= -380000000000.0) {
		tmp = Math.log(((1.0 + ((Math.log(x) / n) / x)) + (1.0 / x))) / n;
	} else if (n <= 26000000000.0) {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log(((x + 1.0) / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if n <= -380000000000.0:
		tmp = math.log(((1.0 + ((math.log(x) / n) / x)) + (1.0 / x))) / n
	elif n <= 26000000000.0:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.log(((x + 1.0) / x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (n <= -380000000000.0)
		tmp = Float64(log(Float64(Float64(1.0 + Float64(Float64(log(x) / n) / x)) + Float64(1.0 / x))) / n);
	elseif (n <= 26000000000.0)
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (n <= -380000000000.0)
		tmp = log(((1.0 + ((log(x) / n) / x)) + (1.0 / x))) / n;
	elseif (n <= 26000000000.0)
		tmp = exp((x / n)) - (x ^ (1.0 / n));
	else
		tmp = log(((x + 1.0) / x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[n, -380000000000.0], N[(N[Log[N[(N[(1.0 + N[(N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[n, 26000000000.0], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.8e11

   1. Initial program 34.4%

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

   3. Taylor expanded in n around inf 76.7%

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

      1. associate--l+ 76.7%

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

      2. log1p-define 76.7%

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

      3. +-commutative 76.7%

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

      4. associate--r+ 76.7%

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

      5. distribute-lft-out-- 76.7%

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

      6. div-sub 76.7%

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

      7. log1p-define 76.7%

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

   5. Simplified 76.7%

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

      1. add-log-exp 76.7%

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

      2. associate-+r- 76.7%

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

      3. exp-diff 76.7%

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

      4. add-exp-log 49.2%

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

   7. Applied egg-rr 49.2%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(x\right) + 0.5 \cdot \frac{{\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}}{n}}}{x}\right)}}{n} \]
   8. Step-by-step derivation

      1. +-commutative 49.2%

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

      2. exp-sum 49.2%

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

      3. *-commutative 49.2%

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

      4. exp-prod 49.2%

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

      5. unpow1/2 49.2%

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

      6. log1p-define 49.2%

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

      7. rem-exp-log 76.9%

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

   9. Simplified 76.9%

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

   10. Taylor expanded in x around inf 76.4%

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

       1. associate-+r+ 76.4%

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

       2. mul-1-neg 76.4%

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

       3. associate-/r* 76.4%

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

       4. distribute-neg-frac 76.4%

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

       5. log-rec 76.4%

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

       6. distribute-neg-frac 76.4%

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

       7. remove-double-neg 76.4%

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

   12. Simplified 76.4%

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

   if -3.8e11 < n < 2.6e10

   1. Initial program 89.6%

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

   3. Taylor expanded in n around 0 89.6%

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

      1. log1p-define 98.4%

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

      2. *-rgt-identity 98.4%

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

      3. associate-/l* 98.4%

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

      4. exp-to-pow 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in x around 0 98.3%

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

   if 2.6e10 < n

   1. Initial program 20.2%

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

   3. Taylor expanded in n around inf 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. log1p-undefine 80.0%

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

      2. diff-log 80.0%

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

   7. Applied egg-rr 80.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -380000000000:\\ \;\;\;\;\frac{\log \left(\left(1 + \frac{\frac{\log x}{n}}{x}\right) + \frac{1}{x}\right)}{n}\\ \mathbf{elif}\;n \leq 26000000000:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -40:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -40.0)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 2e-14)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 2e+214) (- (+ 1.0 (/ x n)) t_0) (/ (/ 1.0 n) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -40.0) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-14) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+214) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-40.0d0)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= 2d-14) then
        tmp = log(((x + 1.0d0) / x)) / n
    else if ((1.0d0 / n) <= 2d+214) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -40.0) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 2e-14) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 2e+214) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -40.0:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 2e-14:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 2e+214:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40.0)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 2e-14)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+214)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(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) <= -40.0)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= 2e-14)
		tmp = log(((x + 1.0) / x)) / n;
	elseif ((1.0 / n) <= 2e+214)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -40.0], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-14], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+214], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -40

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-rgt-identity 98.9%

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

      8. associate-/l* 98.9%

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

      9. exp-to-pow 98.9%

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

      10. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   if -40 < (/.f64 #s(literal 1 binary64) n) < 2e-14

   1. Initial program 27.4%

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

   3. Taylor expanded in n around inf 78.6%

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

      1. log1p-define 78.6%

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

   5. Simplified 78.6%

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

      1. log1p-undefine 78.6%

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

      2. diff-log 78.6%

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

   7. Applied egg-rr 78.6%

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

   if 2e-14 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999999e214

   1. Initial program 79.9%

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

   3. Taylor expanded in x around 0 72.3%

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

   if 1.9999999999999999e214 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 6.0%

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

   3. Taylor expanded in n around inf 8.2%

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

      1. log1p-define 8.2%

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

   5. Simplified 8.2%

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

   6. Taylor expanded in x around -inf 83.9%

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

   7. Taylor expanded in x around inf 83.9%

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

      1. associate-/r* 83.9%

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

   9. Simplified 83.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+214}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -40:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+178}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -40.0)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) 5e-12)
       (/ (log (/ (+ x 1.0) x)) n)
       (if (<= (/ 1.0 n) 5e+178) (- 1.0 t_0) (/ (log1p (+ x -1.0)) (- n)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -40.0) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-12) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+178) {
		tmp = 1.0 - t_0;
	} else {
		tmp = log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -40.0) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= 5e-12) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+178) {
		tmp = 1.0 - t_0;
	} else {
		tmp = Math.log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -40.0:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= 5e-12:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+178:
		tmp = 1.0 - t_0
	else:
		tmp = math.log1p((x + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40.0)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= 5e-12)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+178)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(log1p(Float64(x + -1.0)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -40.0], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-12], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+178], N[(1.0 - t$95$0), $MachinePrecision], N[(N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -40

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-neg-frac 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

      7. *-rgt-identity 98.9%

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

      8. associate-/l* 98.9%

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

      9. exp-to-pow 98.9%

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

      10. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   if -40 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 27.2%

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

   3. Taylor expanded in n around inf 78.0%

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

      1. log1p-define 78.0%

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

   5. Simplified 78.0%

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

      1. log1p-undefine 78.0%

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

      2. diff-log 78.1%

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

   7. Applied egg-rr 78.1%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e178

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0 79.1%

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

      1. *-rgt-identity 79.1%

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

      2. associate-/l* 79.1%

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

      3. exp-to-pow 79.1%

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

   5. Simplified 79.1%

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

   if 4.9999999999999999e178 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.1%

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

   3. Taylor expanded in n around inf 6.7%

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

      1. log1p-define 6.7%

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

   5. Simplified 6.7%

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

   6. Taylor expanded in x around 0 6.7%

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

      1. neg-mul-1 6.7%

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

   8. Simplified 6.7%

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

      1. log1p-expm1-u 65.0%

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

      2. expm1-undefine 65.0%

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

      3. add-exp-log 65.0%

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

   10. Applied egg-rr 65.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+178}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4000000000000:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+178}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4000000000000.0)
   (/ 0.3333333333333333 (* n (pow x 3.0)))
   (if (<= (/ 1.0 n) 5e-12)
     (/ (log (/ (+ x 1.0) x)) n)
     (if (<= (/ 1.0 n) 5e+178)
       (- 1.0 (pow x (/ 1.0 n)))
       (/ (log1p (+ x -1.0)) (- n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4000000000000.0) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if ((1.0 / n) <= 5e-12) {
		tmp = log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+178) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4000000000000.0) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if ((1.0 / n) <= 5e-12) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else if ((1.0 / n) <= 5e+178) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.log1p((x + -1.0)) / -n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -4000000000000.0:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif (1.0 / n) <= 5e-12:
		tmp = math.log(((x + 1.0) / x)) / n
	elif (1.0 / n) <= 5e+178:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = math.log1p((x + -1.0)) / -n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4000000000000.0)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (Float64(1.0 / n) <= 5e-12)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	elseif (Float64(1.0 / n) <= 5e+178)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(log1p(Float64(x + -1.0)) / Float64(-n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -4000000000000.0], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e-12], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 5e+178], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(x + -1.0), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e12

   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 49.0%

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

      1. log1p-define 49.0%

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

   5. Simplified 49.0%

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

   6. Taylor expanded in x around -inf 44.3%

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

   7. Taylor expanded in x around 0 79.6%

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

   if -4e12 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999997e-12

   1. Initial program 28.7%

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

   3. Taylor expanded in n around inf 77.1%

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

      1. log1p-define 77.1%

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

   5. Simplified 77.1%

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

      1. log1p-undefine 77.1%

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

      2. diff-log 77.2%

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

   7. Applied egg-rr 77.2%

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

   if 4.9999999999999997e-12 < (/.f64 #s(literal 1 binary64) n) < 4.9999999999999999e178

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0 79.1%

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

      1. *-rgt-identity 79.1%

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

      2. associate-/l* 79.1%

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

      3. exp-to-pow 79.1%

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

   5. Simplified 79.1%

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

   if 4.9999999999999999e178 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 31.1%

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

   3. Taylor expanded in n around inf 6.7%

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

      1. log1p-define 6.7%

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

   5. Simplified 6.7%

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

   6. Taylor expanded in x around 0 6.7%

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

      1. neg-mul-1 6.7%

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

   8. Simplified 6.7%

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

      1. log1p-expm1-u 65.0%

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

      2. expm1-undefine 65.0%

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

      3. add-exp-log 65.0%

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

   10. Applied egg-rr 65.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4000000000000:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 5 \cdot 10^{+178}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x + -1\right)}{-n}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - \log x}{n}\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x \cdot n}}{x}\\ \mathbf{elif}\;n \leq -1.58:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 7.6 \cdot 10^{-215}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 35000000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (- x (log x)) n)))
   (if (<= n -1.15e+89)
     (/ (+ (/ 1.0 n) (/ (+ -0.5 (/ 0.3333333333333333 x)) (* x n))) x)
     (if (<= n -1.58)
       t_0
       (if (<= n 7.6e-215)
         (/ 0.3333333333333333 (* n (pow x 3.0)))
         (if (<= n 35000000000.0) (- 1.0 (pow x (/ 1.0 n))) t_0))))))

C

double code(double x, double n) {
	double t_0 = (x - log(x)) / n;
	double tmp;
	if (n <= -1.15e+89) {
		tmp = ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (x * n))) / x;
	} else if (n <= -1.58) {
		tmp = t_0;
	} else if (n <= 7.6e-215) {
		tmp = 0.3333333333333333 / (n * pow(x, 3.0));
	} else if (n <= 35000000000.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - log(x)) / n
    if (n <= (-1.15d+89)) then
        tmp = ((1.0d0 / n) + (((-0.5d0) + (0.3333333333333333d0 / x)) / (x * n))) / x
    else if (n <= (-1.58d0)) then
        tmp = t_0
    else if (n <= 7.6d-215) then
        tmp = 0.3333333333333333d0 / (n * (x ** 3.0d0))
    else if (n <= 35000000000.0d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (x - Math.log(x)) / n;
	double tmp;
	if (n <= -1.15e+89) {
		tmp = ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (x * n))) / x;
	} else if (n <= -1.58) {
		tmp = t_0;
	} else if (n <= 7.6e-215) {
		tmp = 0.3333333333333333 / (n * Math.pow(x, 3.0));
	} else if (n <= 35000000000.0) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (x - math.log(x)) / n
	tmp = 0
	if n <= -1.15e+89:
		tmp = ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (x * n))) / x
	elif n <= -1.58:
		tmp = t_0
	elif n <= 7.6e-215:
		tmp = 0.3333333333333333 / (n * math.pow(x, 3.0))
	elif n <= 35000000000.0:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(x - log(x)) / n)
	tmp = 0.0
	if (n <= -1.15e+89)
		tmp = Float64(Float64(Float64(1.0 / n) + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / Float64(x * n))) / x);
	elseif (n <= -1.58)
		tmp = t_0;
	elseif (n <= 7.6e-215)
		tmp = Float64(0.3333333333333333 / Float64(n * (x ^ 3.0)));
	elseif (n <= 35000000000.0)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (x - log(x)) / n;
	tmp = 0.0;
	if (n <= -1.15e+89)
		tmp = ((1.0 / n) + ((-0.5 + (0.3333333333333333 / x)) / (x * n))) / x;
	elseif (n <= -1.58)
		tmp = t_0;
	elseif (n <= 7.6e-215)
		tmp = 0.3333333333333333 / (n * (x ^ 3.0));
	elseif (n <= 35000000000.0)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[n, -1.15e+89], N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[n, -1.58], t$95$0, If[LessEqual[n, 7.6e-215], N[(0.3333333333333333 / N[(n * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 35000000000.0], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.1499999999999999e89

   1. Initial program 44.4%

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

   3. Taylor expanded in n around inf 78.9%

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

      1. log1p-define 78.9%

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

   5. Simplified 78.9%

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

   6. Taylor expanded in x around -inf 65.4%

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

   7. Taylor expanded in x around inf 65.4%

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

   9. Simplified 65.4%

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

   if -1.1499999999999999e89 < n < -1.5800000000000001 or 3.5e10 < n

   1. Initial program 19.7%

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

   3. Taylor expanded in n around inf 77.6%

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

      1. log1p-define 77.6%

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

   5. Simplified 77.6%

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

   6. Taylor expanded in x around 0 63.1%

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

   if -1.5800000000000001 < n < 7.59999999999999954e-215

   1. Initial program 93.1%

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

   3. Taylor expanded in n around inf 45.6%

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

      1. log1p-define 45.6%

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

   5. Simplified 45.6%

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

   6. Taylor expanded in x around -inf 45.8%

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

   7. Taylor expanded in x around 0 77.2%

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

   if 7.59999999999999954e-215 < n < 3.5e10

   1. Initial program 82.1%

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

   3. Taylor expanded in x around 0 73.8%

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

      1. *-rgt-identity 73.8%

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

      2. associate-/l* 73.8%

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

      3. exp-to-pow 73.8%

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

   5. Simplified 73.8%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{1}{n} + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x \cdot n}}{x}\\ \mathbf{elif}\;n \leq -1.58:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;n \leq 7.6 \cdot 10^{-215}:\\ \;\;\;\;\frac{0.3333333333333333}{n \cdot {x}^{3}}\\ \mathbf{elif}\;n \leq 35000000000:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 6.5 \cdot 10^{-276}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.35 \cdot 10^{-240}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.74:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333}{x} - 0.5}{x}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x 6.5e-276)
     t_0
     (if (<= x 4.35e-240)
       (/ (log x) (- n))
       (if (<= x 1.8e-154)
         t_0
         (if (<= x 0.74)
           (/ (- x (log x)) n)
           (if (<= x 3.1e+165)
             (/ (+ 1.0 (/ (- (/ 0.3333333333333333 x) 0.5) x)) (* x n))
             0.0)))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= 6.5e-276) {
		tmp = t_0;
	} else if (x <= 4.35e-240) {
		tmp = log(x) / -n;
	} else if (x <= 1.8e-154) {
		tmp = t_0;
	} else if (x <= 0.74) {
		tmp = (x - log(x)) / n;
	} else if (x <= 3.1e+165) {
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= 6.5d-276) then
        tmp = t_0
    else if (x <= 4.35d-240) then
        tmp = log(x) / -n
    else if (x <= 1.8d-154) then
        tmp = t_0
    else if (x <= 0.74d0) then
        tmp = (x - log(x)) / n
    else if (x <= 3.1d+165) then
        tmp = (1.0d0 + (((0.3333333333333333d0 / x) - 0.5d0) / x)) / (x * n)
    else
        tmp = 0.0d0
    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 (x <= 6.5e-276) {
		tmp = t_0;
	} else if (x <= 4.35e-240) {
		tmp = Math.log(x) / -n;
	} else if (x <= 1.8e-154) {
		tmp = t_0;
	} else if (x <= 0.74) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 3.1e+165) {
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 6.5e-276:
		tmp = t_0
	elif x <= 4.35e-240:
		tmp = math.log(x) / -n
	elif x <= 1.8e-154:
		tmp = t_0
	elif x <= 0.74:
		tmp = (x - math.log(x)) / n
	elif x <= 3.1e+165:
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= 6.5e-276)
		tmp = t_0;
	elseif (x <= 4.35e-240)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 1.8e-154)
		tmp = t_0;
	elseif (x <= 0.74)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 3.1e+165)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) / Float64(x * n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= 6.5e-276)
		tmp = t_0;
	elseif (x <= 4.35e-240)
		tmp = log(x) / -n;
	elseif (x <= 1.8e-154)
		tmp = t_0;
	elseif (x <= 0.74)
		tmp = (x - log(x)) / n;
	elseif (x <= 3.1e+165)
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n);
	else
		tmp = 0.0;
	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[x, 6.5e-276], t$95$0, If[LessEqual[x, 4.35e-240], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 1.8e-154], t$95$0, If[LessEqual[x, 0.74], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 3.1e+165], N[(N[(1.0 + N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 6.49999999999999981e-276 or 4.3500000000000003e-240 < x < 1.8000000000000001e-154

   1. Initial program 63.9%

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

   3. Taylor expanded in x around 0 63.9%

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

      1. *-rgt-identity 63.9%

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

      2. associate-/l* 63.9%

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

      3. exp-to-pow 63.9%

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

   5. Simplified 63.9%

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

   if 6.49999999999999981e-276 < x < 4.3500000000000003e-240

   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 75.3%

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

      1. log1p-define 75.3%

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

   5. Simplified 75.3%

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

   6. Taylor expanded in x around 0 75.3%

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

      1. neg-mul-1 75.3%

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

   8. Simplified 75.3%

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

   if 1.8000000000000001e-154 < x < 0.73999999999999999

   1. Initial program 38.7%

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

   3. Taylor expanded in n around inf 57.7%

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

      1. log1p-define 57.7%

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

   5. Simplified 57.7%

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

   6. Taylor expanded in x around 0 55.9%

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

   if 0.73999999999999999 < x < 3.1000000000000002e165

   1. Initial program 52.1%

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

   3. Taylor expanded in n around inf 50.5%

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

      1. log1p-define 50.5%

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

   5. Simplified 50.5%

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

   6. Taylor expanded in x around -inf 70.7%

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

   7. Taylor expanded in n around -inf 70.7%

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

      1. mul-1-neg 70.7%

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

      2. unsub-neg 70.7%

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

      3. associate-*r/ 70.7%

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

      4. metadata-eval 70.7%

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

      5. *-commutative 70.7%

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

   9. Simplified 70.7%

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

   if 3.1000000000000002e165 < x

   1. Initial program 95.9%

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

      1. add-log-exp 95.9%

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

      2. pow-to-exp 95.9%

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

      3. un-div-inv 95.9%

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

      4. +-commutative 95.9%

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

      5. log1p-define 95.9%

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

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in x around inf 95.9%

      \[\leadsto \log \color{blue}{1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-276}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.35 \cdot 10^{-240}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-154}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.74:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333}{x} - 0.5}{x}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.74:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333}{x} - 0.5}{x}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.74)
   (/ (- x (log x)) n)
   (if (<= x 5.5e+165)
     (/ (+ 1.0 (/ (- (/ 0.3333333333333333 x) 0.5) x)) (* x n))
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.74) {
		tmp = (x - log(x)) / n;
	} else if (x <= 5.5e+165) {
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.74d0) then
        tmp = (x - log(x)) / n
    else if (x <= 5.5d+165) then
        tmp = (1.0d0 + (((0.3333333333333333d0 / x) - 0.5d0) / x)) / (x * n)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.74) {
		tmp = (x - Math.log(x)) / n;
	} else if (x <= 5.5e+165) {
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.74:
		tmp = (x - math.log(x)) / n
	elif x <= 5.5e+165:
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.74)
		tmp = Float64(Float64(x - log(x)) / n);
	elseif (x <= 5.5e+165)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) / Float64(x * n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.74)
		tmp = (x - log(x)) / n;
	elseif (x <= 5.5e+165)
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.74], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 5.5e+165], N[(N[(1.0 + N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.73999999999999999

   1. Initial program 48.0%

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

   3. Taylor expanded in n around inf 51.4%

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

      1. log1p-define 51.4%

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

   5. Simplified 51.4%

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

   6. Taylor expanded in x around 0 50.5%

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

   if 0.73999999999999999 < x < 5.4999999999999998e165

   1. Initial program 52.1%

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

   3. Taylor expanded in n around inf 50.5%

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

      1. log1p-define 50.5%

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

   5. Simplified 50.5%

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

   6. Taylor expanded in x around -inf 70.7%

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

   7. Taylor expanded in n around -inf 70.7%

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

      1. mul-1-neg 70.7%

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

      2. unsub-neg 70.7%

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

      3. associate-*r/ 70.7%

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

      4. metadata-eval 70.7%

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

      5. *-commutative 70.7%

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

   9. Simplified 70.7%

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

   if 5.4999999999999998e165 < x

   1. Initial program 95.9%

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

      1. add-log-exp 95.9%

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

      2. pow-to-exp 95.9%

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

      3. un-div-inv 95.9%

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

      4. +-commutative 95.9%

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

      5. log1p-define 95.9%

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

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in x around inf 95.9%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.74:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333}{x} - 0.5}{x}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333}{x} - 0.5}{x}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.6)
   (/ (log x) (- n))
   (if (<= x 3.55e+165)
     (/ (+ 1.0 (/ (- (/ 0.3333333333333333 x) 0.5) x)) (* x n))
     0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.6) {
		tmp = log(x) / -n;
	} else if (x <= 3.55e+165) {
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 0.6d0) then
        tmp = log(x) / -n
    else if (x <= 3.55d+165) then
        tmp = (1.0d0 + (((0.3333333333333333d0 / x) - 0.5d0) / x)) / (x * n)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.6) {
		tmp = Math.log(x) / -n;
	} else if (x <= 3.55e+165) {
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 0.6:
		tmp = math.log(x) / -n
	elif x <= 3.55e+165:
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.6)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 3.55e+165)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) / Float64(x * n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 0.6)
		tmp = log(x) / -n;
	elseif (x <= 3.55e+165)
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.6], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 3.55e+165], N[(N[(1.0 + N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.599999999999999978

   1. Initial program 48.0%

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

   3. Taylor expanded in n around inf 51.4%

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

      1. log1p-define 51.4%

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

   5. Simplified 51.4%

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

   6. Taylor expanded in x around 0 50.2%

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

      1. neg-mul-1 50.2%

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

   8. Simplified 50.2%

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

   if 0.599999999999999978 < x < 3.54999999999999988e165

   1. Initial program 52.1%

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

   3. Taylor expanded in n around inf 50.5%

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

      1. log1p-define 50.5%

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

   5. Simplified 50.5%

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

   6. Taylor expanded in x around -inf 70.7%

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

   7. Taylor expanded in n around -inf 70.7%

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

      1. mul-1-neg 70.7%

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

      2. unsub-neg 70.7%

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

      3. associate-*r/ 70.7%

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

      4. metadata-eval 70.7%

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

      5. *-commutative 70.7%

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

   9. Simplified 70.7%

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

   if 3.54999999999999988e165 < x

   1. Initial program 95.9%

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

      1. add-log-exp 95.9%

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

      2. pow-to-exp 95.9%

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

      3. un-div-inv 95.9%

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

      4. +-commutative 95.9%

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

      5. log1p-define 95.9%

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

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in x around inf 95.9%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{+165}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333}{x} - 0.5}{x}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 49.6% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+166}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333}{x} - 0.5}{x}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.2e+166)
   (/ (+ 1.0 (/ (- (/ 0.3333333333333333 x) 0.5) x)) (* x n))
   0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.2e+166) {
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.2d+166) then
        tmp = (1.0d0 + (((0.3333333333333333d0 / x) - 0.5d0) / x)) / (x * n)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.2e+166) {
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.2e+166:
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.2e+166)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(0.3333333333333333 / x) - 0.5) / x)) / Float64(x * n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.2e+166)
		tmp = (1.0 + (((0.3333333333333333 / x) - 0.5) / x)) / (x * n);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.2e+166], N[(N[(1.0 + N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996e166

   1. Initial program 49.2%

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

   3. Taylor expanded in n around inf 51.1%

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

      1. log1p-define 51.1%

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

   5. Simplified 51.1%

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

   6. Taylor expanded in x around -inf 39.7%

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

   7. Taylor expanded in n around -inf 39.8%

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

      1. mul-1-neg 39.8%

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

      2. unsub-neg 39.8%

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

      3. associate-*r/ 39.8%

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

      4. metadata-eval 39.8%

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

      5. *-commutative 39.8%

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

   9. Simplified 39.8%

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

   if 1.19999999999999996e166 < x

   1. Initial program 95.9%

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

      1. add-log-exp 95.9%

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

      2. pow-to-exp 95.9%

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

      3. un-div-inv 95.9%

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

      4. +-commutative 95.9%

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

      5. log1p-define 95.9%

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

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in x around inf 95.9%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+166}:\\ \;\;\;\;\frac{1 + \frac{\frac{0.3333333333333333}{x} - 0.5}{x}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.4% accurate, 14.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in n around inf 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 41.2%

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

7. Taylor expanded in x around inf 32.7%

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

9. Simplified 41.2%

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

10. Final simplification 41.2%

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

Alternative 20: 46.4% accurate, 15.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in n around inf 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 41.2%

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

7. Taylor expanded in n around 0 41.2%

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

   1. sub-neg 41.2%

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

   2. associate-*r/ 41.2%

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

   3. sub-neg 41.2%

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

   4. metadata-eval 41.2%

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

   5. distribute-lft-in 41.2%

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

   6. neg-mul-1 41.2%

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

   7. associate-*r/ 41.2%

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

   8. metadata-eval 41.2%

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

   9. distribute-neg-frac 41.2%

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

   10. metadata-eval 41.2%

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

   11. metadata-eval 41.2%

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

   12. metadata-eval 41.2%

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

9. Simplified 41.2%

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

10. Final simplification 41.2%

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

Alternative 21: 45.8% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in n around inf 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 41.2%

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

7. Taylor expanded in n around -inf 41.0%

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

   1. mul-1-neg 41.0%

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

   2. unsub-neg 41.0%

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

   3. associate-*r/ 41.0%

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

   4. metadata-eval 41.0%

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

   5. *-commutative 41.0%

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

9. Simplified 41.0%

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

10. Final simplification 41.0%

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

Alternative 22: 40.6% 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 56.1%

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

3. Taylor expanded in n around inf 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 41.2%

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

7. Taylor expanded in x around inf 34.7%

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

   1. associate-/r* 34.9%

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

9. Simplified 34.9%

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

Alternative 23: 40.1% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in n around inf 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 34.7%

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

   1. *-commutative 34.7%

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

8. Simplified 34.7%

   \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
9. Add Preprocessing

Alternative 24: 4.5% accurate, 70.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in 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 34.7%

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

   1. *-commutative 34.7%

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

8. Simplified 34.7%

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

   1. associate-/r* 34.9%

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

   2. div-inv 34.9%

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

   3. add-exp-log 34.0%

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

   4. rec-exp 34.0%

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

   5. add-sqr-sqrt 11.1%

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

   6. sqrt-unprod 12.3%

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

   7. sqr-neg 12.3%

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

   8. sqrt-prod 1.3%

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

   9. add-sqr-sqrt 4.6%

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

   10. add-exp-log 4.6%

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

10. Applied egg-rr 4.6%

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

    1. associate-*r/ 4.6%

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

    2. *-rgt-identity 4.6%

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

12. Simplified 4.6%

    \[\leadsto \color{blue}{\frac{x}{n}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))